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1Lubrication model for the flow driven by high surface tension 1Emilia Rodica Borsa, Diana-Luiza Borsa
2Structures bicategorielles complementaires 5Dumitru Botnaru
3Some accelerated flows for an Oldroyd-B fluid 29Ilie Burdujan
4A fuzzy algorithm for reliability simulation of an electric station 49Simona Dzitac, Tiberiu Vesselenyi, Ioan Dzitac, Maria Parv
5A seven equation model for relativistic two fluid flows-I 59Sebastiano Giambo, Serena Giambo
6A general mountain-pass theorem for local Lipschitz functions 71Georgiana Goga
7Some results on simultaneous algebraic techniques in image reconstruction
from projections79
Lacramioara Grecu, Aurelian Nicola
8Branching equation in the root subspace for equations nonresolved with
respect to derivative and stability of bifurcating solutions97
Irina V. Konopleva, Boris V. Loginov, Yuri B. Rousak
9Solvability of Hammerstein equations with angle-bounded kernel 109Irina A. Leca
v
vi10Numerical k - busy periods algorithms for Polling systems with semi -
Markov switching119
Gheorghe Mishkoy, Diana Bejenari
11‘Fragile Bits’ vs. Multi-Enrollment - a case study of iris recognition on
Bath University Iris Database127
Nicolaie Popescu-Bodorin
12Bifurcation in a nonlinear business cycle model 145Carmen Rocsoreanu, Mihaela Sterpu
13Existence and uniqueness of fuzzy solution for linear Volterra fuzzy inte-
gral equations, proved by Adomian decomposition method153
Hamid Rouhparvar, Tofigh Allahviranloo, Saeid Abbasbandy
14On a certain differential inequality 163Roxana Sendrutiu
15Note on a paper on nonlinear inverse time heat equation in the unbounded
region169
Nguyen Huy Tuan, Dang Duc Trong, Pham Hoang Quan
16Stability of elastic elements of wing profile with time delay of bases
reactions181
Petr A. Velmisov, Andrey V. Ankilov
LUBRICATION MODEL FOR THE FLOWDRIVEN BY HIGH SURFACE TENSION
ROMAI J., 5, 2(2009), 1–4
Emilia Rodica Borsa, Diana-Luiza BorsaUniversity of Oradea, RomaniaStudent of Jacobs University, Bremen, [email protected], [email protected]
Abstract In this paper the flow of a thin fluid film, driven by a surface tension, down aninclined plane, is considered. By using the Navier-Stokes equations for thin filmflow, the continuity equation, the no-slip condition and the boundary condi-tions, we obtain the horizontal velocity, the vertical velocity and the governingequation of the film height. In general, the introduction of surface tension intostandard lubrication theory leads to a nonlinear parabolic equation.
Keywords: fluid mechanics, lubrication theory.
2000 MSC: 76M99.
1. INTRODUCTIONThe lubrication approximation of the Navier-Stokes equations (NSE) has
been used to describe a multitude of situations. Our attention has been fo-cussed on the situations where surface tension plays an important role, suchas: rain running down a window, the evolution of drying paint layers or thespreading of a fluid drop on a surface.
Research on lubrication equations with non-negligible surface tension ap-pears into two distinct situations. Firstly, there are the physical studies wherethe work is directly motivated by a specific problem. In this case, after deriv-ing the equation for film thickness, the mathematical treatment is generallylimited to asymptotic or numerical methods. Secondly, there are the math-ematical studies which are not directly motivated by a specific problem butdelve into the lubrication equation in greater detail.
In this paper we study the flow of a thin fluid layer lies on a plane whichis at an angle α to the horizontal. The flow is driven by a surface tension.In this case the flow is governed by the nonlinear degenerate parabolic equa-tion. Implicit in the derivation of this equation is the assumption that surfacetension and gravity effects are of the same order.
1
2 Emilia Rodica Borsa, Diana-Luiza Borsa
2. MAIN RESULTSConsider the flow of a thin-layer, of an incompressible Newtonian fluid with
constant density ρ and dynamic viscosity µ, down a plane inclined at an angleα to the horizontal (fig.1). The flow is driven simultaneously by gravity and asurface tension gradient Σ = ∂γ
∂x . The velocity is −→u = u(x, z, t)−→i +w(x, z, t)
−→k .
Fig. 1.
In the thin-film approximation the NSE reduce to [1], [4]:
0 = −1ρpx + νuzz + g sinα, (1)
0 = −1ρpz − g cosα, (2)
where p is the pressure in the fluid, g is the gravitational acceleration. Herez = h(x, t) is the unknown equation of the free surface and ν = µ
ρ is thecoefficient of kinematic viscosity. Here the subscripts denote differentiationwith respect to the corresponding variable. The motion of the fluid is governedby equations (1)-(2) and some initial and boundary conditions.
The non-slip conditions must be satisfied:
~u = 0, at z = 0. (3)
On the free surface z = h(x, t) the condition that the normal stress be equalto the atmospheric pressure p0 reduces to
p = p0, at z = h(x, t). (4)
The condition for the tangential stress (Marangoni effect) at the free surface[2] is
µuz = Σ, at z = h(x, t). (5)
Lubrication model for the flow driven by high surface tension 3
and the kinematic boundary condition [3]
w = ht + uhx, at z = h(x, t). (6)
The continuity equation is
ux + wz = 0. (7)
From equation (2) and condition (5), we obtain for fluid pressure
p = −ρ · g · cosα · z + C1.
For z = h we haveC1 = p0 + ρ · g · cosα · h,
and thusp = −ρ · g · cosα · (z − h) + p0. (8)
By integrating (1) we get for u
uz =1
ρ · ν · px · z − 1ν· g · sinα · z + C2.
The boundary condition (5) implies for z = h
C2 =1µ
Σ +1ν· (g · sinα− 1
ρ· px)h
anduz =
1ν· (g · sinα− 1
ρ· px)(h− z) +
1µ
Σ.
By integrating the last relation we obtain
u =1ν· (g · sinα− 1
ρpx)(hz − z2
2) +
1µ
Σz + C3
and by using the condition (3) the horizontal velocity is
u =ρg
2µ(− sinα + hx · cosα) z2 +
(Σµ
+ρg
µh sinα− ρg
µ· hx · h cosα
)z. (9)
This may be used in the continuity equation (7) to determine w
wz = −ux,
w(h) = −∫ z
0ux · dz
and by using for z = 0 the relation (3) we have for the vertical velocity
4 Emilia Rodica Borsa, Diana-Luiza Borsa
w = −ρg
6µ·hxx ·z3 ·cosα+
ρg
2µcosα
[h · hxx + (hx)2
]z2− ρg
2µ·hx ·z2 ·sinα. (10)
This expression together with the kinematic condition (6) leads to the gov-erning equation for film height h(x, t)
ρg
3µcosα
(h3 · hx
)x
= ht +Σµ· h · hx +
ρg
µ· sinα · h2 · hx. (11)
If the thin fluid layer lies on a plane which is at an angle α to the horizontalthen a surface tension driven flow is governed by the nonlinear degenerateparabolic equation (11).
Appropriate form of equation (11) have been used to model fluid flows ina number of physical situations such as coating, draining of foams and themovement of contact lenses.
3. CONCLUSIONSIn the lubrication approximation we have considered the flow of a thin layer
on an inclined plane. The flow is driven simultaneously by gravity and agradient of surface tension. This gradient implies a non-zero tangential stressboundary condition (Marangoni effect). We have estimated the response ofthe fluid to such a stress.
We have determined the horizontal velocity, the vertical velocity and thegoverning equation for the film height, which is a nonlinear degenerate parabolicequation. This equation can be linearized [5] and in some special cases we canobtain implicit solutions of the linearized system.
References[1] D. J. Acheson, Elementary Fluid Dynamics, Oxford University Press, United Kingdom,
1990.
[2] E. Chifu, C.I. Gheorghiu, I. Stan, Surface Mobility of Surfactant Solutions. Numericalanalysis for the Marangoni and the Gravity Flow in a Thin Liquid Layer of TriangularSection, Revue Roumaine de Chimie, 29(1984), 31-42.
[3] P. Grindrod, The Theory and Applications of Reaction-Diffusion Equations, Patternsand Waves, Clarendon Press, Oxford, 1996.
[4] H. Ochendon, J. R. Ochendon, Viscous Flow, Cambridge University Press, 1995.
[5] P. V. O’Neil, Advanced Engineering Mathematics, Wadsworth Publishing CompanyBelmont, California, 1991.
STRUCTURES BICATEGORIELLESCOMPLEMENTAIRES
ROMAI J., 5, 2(2009), 5–27
Dumitru BotnaruUniversite Technique de la Republique de Moldavie, Chisinau, Republique de [email protected]
Abstract We examine the complete lattice of the bicategory structure in the categoryof the vector locally convex Hausdorff spaces and the category of the locallyconvex groups. We construct the complementary elements for some elementsof this lattice.
Keywords: bicategory structure, reflective subcategory, classes of hereditary end cohered-
itary morphisms, spaces with weak topology, complete spaces, Schwartz spaces, nuclear
spaces, strict nuclear spaces.
2000 MSC: 18 B30; 46 M 15.
Resume On examine la latice complete des structures bicategorielles dans la categorie
des espaces vectoriels local-convexes Hausdorff et la categorie des groupes
local-convexes. On construit des elements complementaires pour une classe
propre des elements de cette latice.
Mots cles: structure bicategorielle, sous-categorie reflective, classe de morphismes hereditaire
et cohereditaire, espace a topologie faible, espace complet, espace Schwartz, espace nucleaire,
espace strictement nucleaire.
1. INTRODUCTION1.1 Dans une categorie local et colocal-petite a limites projectives ou in-
ductives, la latice des structures bicategorielle B est complete avec le primeelement (Ef , Mono) = (la classe de tous epis stricts, la classe de tous monos)et l’ultime element (Epi, Mf ) = (la classe de tous epis, la classe de tous monosstricts). On considere (P1, I1) ≤ (P2, I2) si (P1 ⊂ P2). Toute structure bi-categorielle (P, I) est determinee uniquement par chacune de ses classes: parla classe de projections P, de meme que par la classe d’injections I. Ainsi, dansune pareille categorie, pour deux elements (P1, I1) et (P2, I2) de la latice B,notons par P1∧P2 la classe de projections du minimum de ces deux elements,et par P1 ∨ P2 - la classe des projections du maximum.
5
6 Dumitru Botnaru
Il est clair que P1∧P2 = P1∩P2, et la structure bicategorielle correspon-dante est
(P1 ∩ P2, (P1 ∩ P2)x).
D’autre part, a la classe P1 ∨ P2 correspond la structure bicategorielle
((I1 ∩ I2)q, I1 ∩ I2).
Dans ces deux structures bicategorielles, une des classes represente l’inter-section des classes correspondantes: P1 ∩ P2 et I1 ∩ I2. Mais l’autre classe,en general, ne peut pas etre construite d’une maniere si simple. On introduitdans l’ouvrage la notion de classe hereditaire par rapport a une classe demorphismes de meme par rapport a une structure bicategorielle (Definitions2.1 et 2.14). Ces notions permettent de decrire aussi la classe (I1∩I2)q commecomposition des classes P1 et P2 (Theoreme 3.2).
Dans la categorie C2V des espaces locallement convexes (topologiquesvectoriels), de meme dans la categorie C2Ab des groupes local-convexes [15]sont connues les sous-categories reflectives formees par rapport a chaque sous-espace dont le foncteur reflecteur est exactement a gauche. Une telle sous-categorie R determine uniquement une sous-categorie coreflective K de tellemaniere que εR = µK ou
εR = f ∈ C2V | r(f) ∈ Iso,µK = f ∈ C2V | k(f) ∈ Iso.
Une telle classe de morphismes sert de classe de projections d’une struc-ture bicategorielle de droite (εR, (εR)⊥) et comme classe d’injections d’unestructure bicategorielle a gauche ((εR)>, εR). Ces classes sont nommees bi-completes (Definition 4.1) et elles permettent de construire les structures bi-categorielles complementaires (Theoreme 5.4).
1.2 Categories.Soient: C une categorie abstraite, C2V la categorie des espaces local-convexes
(topologiques vectoriels) Hausdorff [15], C2Ab la categorie des groupes local-convexes Hausdorff [16], D une des categories C2V ou C2Ab, Th la categoriedes espaces Tikhonov.
1.3 Sous-categories dans la categorie D.Soient: Π la sous-categorie des espaces complets a topologie faible [16],
[12], Γ0 la sous-categorie des espaces complets, S la sous-categorie des es-paces a topologie faible, N la sous-categorie des espaces nucleaires, sN lasous-categorie des espaces strictement nucleaires [11], Sc la sous-categorie desespaces Schwartz (voir [11]), M sous-categorie des espaces a topologie Mackey[16].
1.4 Morphismes orthogonaux. Le morphisme f est nomme orthogonalde haut au morphisme g, et g est nomme orthogonal de bas a f , et est note
Structures bicategorielles complementaires 7
par f ⊥ g si pour tout carre commutatif
gu = vf,
il existe un unique morphisme (diagonal) h de sorte que
v = gh
u = hf
Figure 1.1
Mentionnons que l’unicite du morphisme h avec les proprietes indiqueesa lieu si f est un epi ou g est un mono. Pour deux classes de morphismes K
et L de la categorie C, nous ecrirons K ⊥ L, si tout morphisme de la classe K
est orthogonal de haut a tout morphisme de la classe L. Notons
K⊥ = f ∈ C | K ⊥ f, Kx = K⊥ ∩MonoC,
K> = f ∈ C | f ⊥ K, Kq = K⊥ ∩ EpiC.
1.5 Latices.Soient: R la latice de toutes les sous-categories reflectives non-nulles dans
la categorie D, Ri la latice de toutes les sous-categories reflectives dans lacategorie D qui contient la sous-categorie Γ0, Rb la latice de toutes les sous-categories reflectives dans la categorie D qui contient la sous-categorie S, Rc
la latice de toutes les sous-categories c-reflectives, Bic la latice des classes demorphismes bicomplets dans la categorie D.
1.6 Structures bicategorielle dans la categorie D.Soient: (Epi,Mf ) = (la classe de tous les epis, la classe de tous les monos
stricts)=(operateurs continus a image dense, inclusions topologiques a imagefermee),
(Ef , Mono) = (la classe de tous les epis stricts, la classe de tous les monos)=(laclasse des applications factorielles, des applications injectives),
(Eu, Mp) = (la classe de tous les epis universels; la classe de tous lesmonos precis)=(la classe des applications surjectives, la classe des inclusionstopologiques),
8 Dumitru Botnaru
(Ep, Mu) = (la classe de tous les epis precis, la classe de tous le monosuniversels),
(E′p, M′u) = ((M′
u)q, la classe de tous les monos universels a image fermee).Decrivons plus en details les deux dernieres structures bicategorielles (voir
[7], [9], etant donne qu’elles jouent un role important dans l’ouvrage.1.7 THEOREME [8]. Soit f : (E, u) −→ (F, v) un mono de la categorie D.
Les affirmations suivantes sont equivalentes:1. f ∈ Mu.2. Toute fonctionel continu defini sur l’espace (E, u) s’extend par f .3. Les topologies u et v′ (celle indue de l’espace (F, v) sur l’espace vectorial
E) sont compatibles dans une et la meme dualite.1.8 THEOREME [8], [10]. Soit f : (E, u) −→ (F, v) ∈ D, τ(v) - la topologie
Mackey compatible avec la topologie v. Alors les affirmations suivantes sontequivalentes.
1. f ∈ E′p (respectivement f ∈ Ep).2. f est un epi (respectivement f ∈ Eu) et v =min u; τ(v).
2. CLASSES DE MORPHISMES EXTREMAUX
2.1 Definition [1]. Soient A et E deux classes de morphismes de lacategorie C. La classe A s’appelle E-hereditaire, si du fait que fg ∈ A etf ∈ E, il resulte que g ∈ A.
Notion duale: La classe A s’appelle E-cohereditaire, si du fait que fg ∈A et g ∈ E, il resulte que f ∈ A.
Examinons des situations quand la classe des injections d’une structurebicategorielle est hereditaire, et la classe de projections est cohereditaire parrapport a certaines classes de morphismes.
2.2 Definition. Une classe A s’appelle stable de droite si du fait quea′b = b′a est un carre cocartesien et a ∈ A, il resulte que a′ ∈ A.
Notion duale: la classe stable a gauche.2.3 Exemples. 1. Soit (P, I) une structure bicategorielle de droite (de
gauche), A ⊂ P, B ⊂ I. Alors la classe I est A-cohereditaire (la classe P estB-hereditaire).
2. Toute classe stable a droite est Epi-cohereditaire. Dans toute categorieC, la classe Mu est Epi-cohereditaire. Dans la categorie D, la classe Mp commeclasse stable a droite est aussi Epi-cohereditaire.
3. Dans la categorie C2V est bien connue l’affirmation suivante ([16], cap.VI, prop. 5):
Soit (E, t) un espace complet. Alors l’espace E reste complet dans toutetopologie compatible a la topologie t et plus forte qu’elle.
Cette affirmation peut etre generalisee de la maniere suivante:
Structures bicategorielles complementaires 9
Soient R une sous-categorie reflective dans la categorie C2V et Γ0 ⊂ R. Sif : X −→ Y ∈ Eu ∩Mu et Y ∈| R |, alors et X ∈| R |.
La derniere proposition est une simple consequence de l’affirmation suivante:la classe Eu ∩Mu est Epi-cohereditaire.
Mentionnons que cette classe est aussi Mono-hereditaire.2.4 LEMME. Soit (P, I) une structure bicategorielle de droite. Alors la
classe P est Epi-cohereditaire.Demonstration. Soit
p = fe (1)
avec p ∈ P et e ∈ Epi. Considerons la (P, I) - factorisation de morphisme f
f = i1p1 (2)
Comme p ⊥ i1, il existe un morphisme h ainsi que
p1e = hp (3)
i1h = 1 (4)
De l’hypothese et de l’egalite (3), il resulte que h ∈ Epi. Alors de l’egalite (1),il resulte que i1 ∈ Iso.
Figure 2.1
2.5 REMARQUE. La classe de projection P d’une structure bicategorielle dedroite (P, I) de la categorie C n’est pas toujours C-cohereditaire. Par exemple,dans la categorie C2V, la classe εR pour R 6= 0.
2.6 LEMME. Soit C une categorie a carres cocartesiens. Alors la classe Epiest Mu-hereditaire.
Demonstration. Soit uv ∈ Epi, u ∈ Mu et
fv = gv (5)
10 Dumitru Botnaru
Figure 2.2
Construisons les carres cocartesiens suivants:
u′1f = f ′u (6)
sur les morphismes u et f ;u′2g = g′u (7)
sur les morphismes u et g;u′′1u
′2 = u′′2u
′1 (8)
sur les morphismes u′1 et u′2.Alors les morphismes u′1, u
′2, u
′′1, u
′′2, de meme que u′′1u
′2(= u′′2u
′1) sont monos.
Nous avons
u′′1g′uv
(7)= u′′1u
′2gv
(5)= u′′1u
′2fv
(8)= u′′2u
′1fv
(6)= u′′2f
′uv
et comme uv est un epi, deduisons que
u′′1g′ = u′′2f
′ (9)
Ensuiteu′′1u
′2f
(8)= u′′2u
′1f
(6)= u′′2f
′u(9)= u′′1g
′u(7)= u′′1u
′2g
c’est-a-direu′′1u
′2f = u′′1u
′2g
et comme u′′1u′′2 est mono, deduisons que f = g.
2.7 Dans la categorie Th des espaces Tikhonov existe la structure bicategori-elle (Eu,Mp) =(applications surjectives, inclusions topologiques).
LEMME. Soit Mp ⊂ I ⊂ Mono. Alors la classe I n’est pas Epi-cohereditaire.Demonstration. Pour l’espace X = (0; 2π) existent plusieurs extensions
compactes: X peut etre realise comme un sous-espace dense du cercle unitaireaussi du segment [0; 2π]. Soit Y un espace qui possede plusieurs extensionscompactes, et b : Y −→ Z - une d’entre elles. Dans ce cas
b = fβY
Structures bicategorielles complementaires 11
pour un certain morphisme f , ou βY est la compactification Stone-Cech.
Figure 2.3
Dans cette egalite b, βY ∈ Mp ∩ Epi. Si f est un mono, alors il est un iso.2.8 LEMME. Dans la categorie Th la classe Epi n’est pas (Eu ∩ Mono)-
hereditaire.Demonstration. Soit X un espace Tikhonov et Y un sous-espace dense a
application canonique i : Y −→ X. Soit dY et dX les memes ensembles dotesde la topologie discrete, et dY : dY −→ Y et dX : dX −→ X les applicationsidentiques qui sont continues. Alors l’application d(i) : dY −→ dX, genereede l’application i, est aussi continue
Figure 2.4
Dans l’egalite idY = dXd(i), l’application idY est epi, l’application dX ∈(Eu ∩Mono) et d(i) n’est pas epi si Y 6= X.
2.9 LEMME. Dans la categorie C2V, la classe Mono n’est pas (Mu ∩ Epi) -cohereditaire.
Demonstration. Soit sur l’espace vectorial E deux topologies local-convexesu et v comparables u > v, et qui ne sont pas compatibles avec l’une et la memedualite:
(E, v)′ 6= (E, u)′
Soit, p1 : (E, u) −→ p(E, u) et p2 : (E, v) −→ p(E, v) Π-repliques des objetscorrespondants, ou Π est la sous-categorie des espaces local-convexes completsaussi avec la topologie faible.
12 Dumitru Botnaru
Pour l’application canonique f : (E, u) −→ p(E, v), nous avons la dia-gramme commutative
Figure 2.5
Il y ap(f)p1 = p2f (10)
ou p2f est mono, et p1 ∈ (Mu∩Epi). Supposons que p(f) est mono. Dans ce casp(f) est une inclusion topologique ([12], pr. 11, p.151). Mais de l’egalite (10), ilresulte que p(f) est epi. Donc p(f) est iso. Ce qui signifie que (E, u)′ = (E, v)′la contradiction obtenue montre que p(f) n’est pas mono.
2.10 LEMME. Dans la categorie C2V la classe Epi n’est pas (Eu ∩Mono)-hereditaire.
Demonstration. Soit (E, u) un espace local-convexe, et (F, u′) un sous-espace dense propre. Examinons sur les espaces vectoriels E et F les plusfines topologies local-convexes σ′ et σ. Nous avons la suivante diagrammecommutative et les applications canoniques.
Figure 2.6
De l’egaliteiσF = σEσ(i)
comme iσF est epi et σE ∈ (Eu ∩Mono), il ne resulte pas que σ(i) est un epi.Ainsi σ(i), et avec lui i, et un iso ([12], pr. 10, p.150-151).
2.11 Soit K une sous-categorie coreflective de la categorie C avec le foncteurcorrespondant k : C −→ K. Notons
µK = m ∈ MonoC | k(m) ∈ IsoC.Dual. Soit R est une sous-categorie reflective avec le foncteur r : C −→ R.
NotonsεR = e ∈ EpiC | r(e) ∈ IsoC.
Structures bicategorielles complementaires 13
Dans une categorie local-petite a limites projectives ((µK)>, µK) est unestructure bicategorielle de gauche.
Dual: (εR, (εR)⊥) est une structure bicategorielle de droite. Les mor-phismes de la classe (εR)⊥ sont nommes R-parfaits, et les morphismes dela classe εR sont nommes R-estensions [17].
LEMME [1]. La classe εR est Epi-cohereditaire, et la classe µK est Mono-hereditaire.
2.12 Soit C une categorie a carres cartesiens et cocartesiens, la classe Mu
stable a gauche, et R - une sous-categorie monoreflective.Pour le morphisme arbitraire f : X −→ Y ∈ C soit rX et rY R-replique des
objets correspondants. Alors nous avons l’egalite
r(f)rX = rY f (11)
Sur le morphisme r(f) et rY construisons le carre cartesien
r(f)v = rY u (12)
AlorsrX = vt (13)
f = ut (14)
pour certain morphisme t.Comme R ⊂ (εR)⊥ il resulte que r(f) ∈ (εR)⊥. Ainsi, u ∈ (εR)⊥. Ainsi
rY est mono universel. Conformement a l’hypothese concernant la classe Mu,deduisons que v ∈ Mu l’est aussi. Alors de l’egalite (13), conformement aulemme 2.6∗, t est un epi. On verifie facilement que v est R-replique de l’objetP . Alors t ∈ εR.
Figure 2.7
Ainsi, nous l’avons demontre.THEOREME. Soit C une categorie a carres cartesiens et cocartesiens, la
classe Mu stable a gauche et R une sous-categorie monoreflective.
14 Dumitru Botnaru
Alors:1. (εR, (εR)⊥) est une structure bicategorielle de droite.2. Pour tout morphisme f ∈ C l’egalite f = ut est sa (εR, (εR)⊥) - factori-
sation.3. f ∈ (εR⊥), alors et seulement alors quand le carre r(f)rX = rY f est
cartesien.2.13 Nous examinerons encore une construction qui permet d’obtenir des
exemples de classes hereditaires (voir [10]). Soit (P, I) une structure bi-categorielle dans la categorie C, et A - une classe de morphismes. Notons
I′(A) = m ∈ MonoC | ∃a ∈ A ainsi que ∃ ma et ma ∈ IP′(A) = (I′(A))q
THEOREME. Soient C une categorie local-petite a limites projectives, (P, I)- une structure bicategorielle.
1. Si (A>,A) est une structure bicategorielle de gauche, alors (P′(A), I′(A))est une structure bicategorielle dans la categorie C.
2. De plus, si A ⊂ EpiC, alorsa) I′(A) est - cohereditaire.b) I′(A) est la plus petite classe A-cohereditaire qui contient la classe I.2.14 Definition. Soient A une classe de morphismes, et (E,M) - une
structure bicategorielle de droite dans la categorie C. La classe A est nommee(E, M)-cohereditaire si du fait que a = me est (E,M)-factorisation du mor-phisme a ∈ A, il resulte que m ∈ A.
Notion duale. La classe A-hereditaire par rapport a la structure bi-categorielle de gauche (E, M).
2.15 LEMME. Soient (P, I) et (E, M) deux structures bicategorielles dedroite et I ⊂ M. La classe A est (E,M)-cohereditaire, alors et seulementalors quand la classe A ∩ P est (E, M)-cohereditaire.
3. COMPOSITION DES CLASSES DE MORPHISMES
3.1 Definition. Soient A et B deux classes de morphismes de la categorieC. La composition des classes A et B s’appelle la classe A B de tous lesmorphismes de la categorie C de forme ab a elements a ∈ A et b ∈ B pourlaquelle existe la composition correspondante ab.
3.2 THEOREME. Soit (P, I) et (E,M) deux structures bicategorielles dedroite dans la categorie C. Examinons les affirmations suivantes:
1. La classe I est E-cohereditaire.2. La classe I est (E, M)-cohereditaire.3. P E ⊂ E P.4. (E P, M ∩ I) est une structure bicategorielle de droite dans la categorie
C.
Structures bicategorielles complementaires 15
Alors 1 ⇒ 2 ⇔ 3 ⇔ 4.Demonstration. 1 ⇒ 2. Conformement aux definitions correspondantes.2 ⇒ 3. Soit p ∈ P, e ∈ E et soit qu’il existe la composition pe. Examinons
(P, I)-factorisation du morphisme pe
pe = i1p1 (15)
Soiti1 = m2e2 (16)
(E, M)-factorisation du morphisme i1. Il resulte des egalites ci-dessus que
pe = m2(e2p1) (17)
et comme e ⊥ m2, il existe un morphisme t ainsi que
e2p1 = te (18)
m2t = p (19)
De l’hypothese et de l’egalite (19) il resulte que m2 ∈ I. De l’egalite (18) ilresulte que t ∈ Epi. Du lemme 2.4 et de l’egalite (19), il resulte que m2 ∈ P.Ainsi m2 ∈ P ∩ I = Iso, et le morphisme pe peut etre ecrit
pe = (m2e2)p1 (20)
avec m2e2 ∈ E et p1 ∈ P.
Figure 3.1
3 ⇒ 4. Les deux classes E P et M ∩ I sont fermees par rapport a la com-position et (E P) ⊥ (M ∩ I). Ainsi, il reste a demontrer que tout morphismepossede (E P, M ∩ I)-factorisation. Soit f ∈ C, et
f = ip (21)
est sa (P, I)-factorisation,i = me (22)
16 Dumitru Botnaru
est (E, M) - factorisation du morphisme i, et
m = i1p1 (23)
est (P, I)-factorisation du morphisme m. Conformement a l’hypothese le mor-phisme p1e peut etre ecrit
p1e = e2p2 (24)
avec e2 ∈ E et p2 ∈ P. Nous avons
i(22)= me
(23)= i1p1e
(24)= i1e2p2
i = i1e2p2 (25)
d’ou il resulte que p2 ∈ I. Ainsi p2 ∈ I ∩ P = Iso. Alors e2p2 ∈ E, et de l’egalite(24), deduisons que p1e ∈ E. Du lemme 2.4 il resulte, il que p1 ∈ E. Commem ∈ M de l’egalite (23), il resulte que p1 ∈ M. Donc p1 ∈ E ∩M = Iso, et lemorphisme f peut etre factorise
f = (i1p1)(ep) (26)
avec i1p1 = m ∈ M ∩ I, et ep ∈ E P.
Figure 3.2
4 ⇒ 2. Soit i ∈ I, eti = me (27)
est (E, M)-factorisation du morphisme i. Soit
i = t(e1p1) (28)
(E P, M ∩ I)-factorisation du meme morphisme, ou t ∈ M ∩ I, et e1 ∈ E etp1 ∈ P. De l’egalite (28), comme i ∈ I, il resulte que p1 ∈ Iso. Donc, lesegalites (27) et (28) sont deux (E, M)-factorisations du morphisme i. Donc
e = re1p1 (29)
Structures bicategorielles complementaires 17
mr = t (30)
pour un certain isomorphisme r. De la derniere egalite, comme r ∈ Iso, ett ∈ I, il resulte que m ∈ I.
Figure 3.3
3.3 COROLLAIRE. 1. Si (E P,M ∩ I) est une structure bicategorielle dedroite, f = ip (P, I)-factorisation du morphisme arbitraire f ∈ C, eti = me (E, M)-factorisation du morphisme i, alors f = m(ep) est (E P,M ∩ I)-factorisation du morphisme f .
Figure 3.4
2. Si (P, I) est une structure bicategorielle, alors M ∩ I ⊂ MonoC.3.4 Exemples. 1. Soit R une sous-categorie reflective non-nulle dans la
categorie D. Puisque la classe Mu est Epi-cohereditaire (exemple 2.3), elle estaussi (εR)-cohereditaire. Ainsi les structures bicategorielles (P, I) = (Ep,Mu)et (E, M) = (εR, (εR)⊥) verifient la condition 1 du Theoreme 3.2. Donc
((εR) Ep, (εR)⊥ ∩Mu)
est une structure bicategorielle dans la categorie D.2. De meme,
((εR) Eu, (εR)⊥ ∩Mp)
est une structure bicategorielle dans la categorie D.
18 Dumitru Botnaru
3. Soit S ⊂ R. Ainsi la classe Mono est εR-cohereditaire (lemme 2.6), et
((εR) Ef , (εR)⊥ ∩Mono)
est une structure bicategorielle dans la categorie D.4. ((εS) Ep, (εR)⊥ ∩Mu) = (Eu,Mp).5. ((εΓ0) Ep, (εΓ0)⊥ ∩Mu) = (E′p, M′
u).6. ((εΠ) Ep, (εΠ)⊥ ∩Mu) = (Epi,Mf ).3.5 Examinons, comme exemple au theoreme precedent, la construction
suivante. Soit (P, I) une structure bicategorielle dans la categorie D ayant lesproprietes:
1. La classe I est Epi-cohereditaire.2. (I ∩ Epi, (I ∩ Epi)⊥) est une structure bicategorielle de droite dans la
categorie D.La structure bicategorielle de droite (I ∩ Epi, (I∩Epi)⊥) determine une sous-
categorie L de la categorie D qui est (I ∩ Epi)-reflective
L = (I ∩ Epi)⊥(Π).
Pour tout objet X de la categorie D soit πX sa Π-replique, et
πX = mX lX . (31)
(I ∩ Epi, (I ∩ Epi)⊥)-factorisation de ce morphisme. Alors lX est L-repliquede l’objet X. Notons par RI la latice de toutes les sous-categories I-reflectivesdans la categorie D. Il est evident que RI = R ∈ R | L ⊂ R.Notons par BI la latice de toutes les structures bicategorielles (E,M) pourlesquelles P ⊂ E et la classe E est (I ∩Epi)-hereditaire, ou, ce qui est la memechose, la classe E est I-hereditaire.
THEOREME. Les latices RI et BI sont antiisomorphes.Demonstration. Soit (E, M) un element de la latice BI. Pour tout objet X
de la categorie D considerons L-replique lX : X −→ lX et (E, M)-factorisationdu morphisme correspondant
lX = sXrX . (32)
La correspondance X 7−→ rX definit la sous-categorie R = M(L) connue unesous-categorie E-reflective. De plus, comme lX ∈ I, il resulte que rX ∈ I aussi.Donc R ∈ RI. Ainsi nous avons etabli la correspondance
ϕ : BI −→ RI,
pour laquelle nous construirons celle inverse
ψ : RI −→ BI.
Structures bicategorielles complementaires 19
Soit R ∈ RI. Comme εR ⊂ Epi, conformement a la premiere hypothese con-cernant la classe I, elle est (εR)-cohereditaire. Conformement au Theoreme3.2, le couple ((εR P, I ∩ (εR)⊥) est une structure bicategorielle dans lacategorie D. Pour montrer qu’elle appartient a la latice BI, il reste a demontrerque la classe (εR) P est (I ∩ Epi)-hereditaire. Soit
ep = if, (33)
ou ep ∈ ((εR) P), c’est-a-dire e ∈ εR, p ∈ P, et i ∈ I ∩ Epi. Comme p ⊥ i du(33), il resulte l’existence d’un morphisme t avec les proprietes
e = it, (34)
f = tp. (35)
La classe I ∩ Epi est stable a droite (hypothese 2). Donc I ∩ Epi ⊂ Mu. Ainsi,dans l’egalite (34) t ∈ Epi (lemme 2.6). Donc e ∈ εR et t ∈ Epi. Alorsde l’egalite (34), il resulte que t ∈ εR, et de l’egalite (35) - que f ∈ (εR) P. De telle maniere, nous avons montre que la classe (εR P) est (I ∩ Epi)-hereditaire. Donc ((εR) P, I ∩ (εR)⊥) est un element de la latice BI.
Figure 3.5
ϕψ = 1. Pour tout objet X de la categorie D, L-replique lX appartient a laclasse I. Ainsi ((εR P, I∩(εR)⊥)-factorisation du morphisme lX coıncide avec(εR, (εR)⊥)-factorisation (corollaire 3.3). Donc ϕψ est l’application identique.
ψϕ = 1. Soit maintenant (E, M) ∈ BI,R = M(L) et nous allons demontrerque E = (εR) P.
(εR) P ⊂ E. Comme P ⊂ E, il reste de demontrer que εR ⊂ E. Soit f :X −→ Y ∈ εR. Alors
gf = rX (36)
pour un certain morphisme g, ou rX est R-replique de l’objet X. Commef ∈ Epi, deduisons que g ∈ εR ⊂εL ⊂ I. Ainsi, rX ∈ E (lX = sXrX est
20 Dumitru Botnaru
(E, M)-factorisation du morphisme lX), g ∈ (I∩ Epi) et comme la classe E est(I ∩ Epi)-hereditaire, il resulte que f ∈ E.
Figure 3.6
E ⊂ (εR) P. Soit f : X −→ Y ∈ E, et
f = ip (37)
est (P, I)-factorisation du morphisme correspondant, ou i : Z −→ Y . Demontronsque i ∈ εR. Comme i ∈ (I ∩ Epi) ⊂ Mu ∩ Epi = εΠ, il resulte que
πZ = gi, (38)
pour un certain morphisme g. L’egalite (38) peut etre ecrite aussi de la manieresuivante (voir l’egalite(31)):
gi = mZ lZ , (39)
ou i ∈ (I ∩ Epi), et mZ ∈ (I ∩ Epi)⊥. Donc
lZ = ti, (40)
g = mZt, (41)
pour un certain morphisme t. L’egalite (40), conformement a l’egalite (32),peut etre ecrite
ti = sZrZ , (42)
ou i ∈ E, et sZ ∈ M. DoncrZ = hi, (43)
t = sZh, (44)
Structures bicategorielles complementaires 21
Figure 3.7
pour un certain morphisme h. L’egalite (43) montre que i ∈ εR. Le theoremeest demontre.
3.6 Examinons a present certaines structures bicategorielles qui verifient lesdeux hypotheses du p. 3.5.
1. (P, I) = (Ep, Mu). Alors RI = R, L = Π.2. (P, I) = (E′p,M′
u), ou M′u est la classe des monomorphismes universels
a l’image fermee (voir [8], [9]). Il est evident que M′u ∩ Epi = Mu ∩ Eu, et il
est clair que cette structure bicategorielle verifie les hypotheses p. 3.5. Nousavons RI = Rb, L = S.
3. (P, I) = (Eu, Mp). Alors RI = Ri, L = Γ0.
Figure 3.8
3.7 REMARQUE. En ce qui concerne la latice R et ses sous-latices men-tionnees dans le diagramme precedent (voir [5] et [9]).
22 Dumitru Botnaru
4. CLASSES DE MORPHISMES BICOMPLETES4.1 Definition. La classe de morphismes B de la categorie C sera nommee
bicomplete si (B, B⊥) est une structure bicategorielle de droite, et (B>, B) estune structure bicategorielle de gauche.
4.2 Remarque. Soit B une classe bicomplete de morphismes de la categorieC, conformement a la definition, B ⊂ Eu∩Mu. Ainsi, dans plusieurs categories(par exemple, dans les categories abeliennes) Iso est l’unique classe bicomplete.
4.3 Dans la categorie C2V de meme que dans la categorie C2Ab [15] existeune classe propre de sous-categories reflectives qui contient la sous-categorieS des espaces a topologie faible dont le foncteur reflecteur est exactement agauche (voir [2-4], [10], [11], [14]).
Une telle sous-categorie R determine de maniere unique une sous-categoriecoreflective K avec les proprietes suivantes
εR = µK.
Ainsi εR ⊂ Eu ∩ Mu et εR est la classe de projections de la structure bi-categorielle de droite (εR, (εR)⊥) et la classe d’injections de la structure bi-categorielle de gauche ((µK)>, µK) = ((εR)>, εR).
4.4 THEOREME ([3], voir [7] Theoreme 2.7). Soit R une sous-categoriereflective dans la categorie D avec le foncteur reflectif r : D −→ R. Lesaffirmations suivantes sont equivalentes:
1. R est une sous-categorie Eu-reflective et r(Mf ) ⊂ Mf .2. R est une sous-categorie Eu-reflective et r(Mp) ⊂ Mp.3. εR est une classe bicomplete.4. Il existe une sous-categorie coreflective K dans la categorie D avec le
foncteur coreflectif k : D −→ K, ainsi quea) rk ∼ r;b) kr ∼ k.5. Il existe une sous-categorie coreflective K dans la categorie D ainsi que
εR = µK
4.5 Definition [3]. Une sous-categorie reflective de la categorie D qui verifieles conditions equivalentes du theoreme precedent est nommee c-reflective,et le couple de sous-categories (K, R) est nomme couple conjugue de sous-categories.
4.6 Exemples. (M, S) est un couple conjugue de sous-categories. Les sous-categories sN, Sh sont c-reflectives. La sous-categorie N n’est pas c-reflective.La sous-categorie Eu-reflective generee d’un espace non-nul Mp-injectif est c-reflective (voir [11], [14], [4]).
4.7 THEOREME [4]. 1. La correspondance R 7−→ εR etablit un isomor-phisme de la latice complete Rc des sous-categories c-reflectives et de la laticeBic des classes bicompletes de la categorie D.
Structures bicategorielles complementaires 23
2. Les elements de ces latices Rc et Bic forment des classe propres (ils nesont pas d’ensembles).
4.8 THEOREME. Soit C une categorie a carres cartesiens et cocartesiens,dans laquelle (Ef , Mono) et (Epi,Mf ) sont des structures bicategorielles.
1. Pour toute classe bicomplete B, les couples (B Ef , Bx) et (Bq,Mf B)sont des structures bicategorielles dans la categorie C.
2. Soit B1 et B2 deux classes bicompletes distinctes dans la categorie C.Alors les structures bicategorielles (B1Ef , Bx
1) et (B2Ef ,Bx2) et les structures
bicategorielles (Bq1, Mf B1) et (Bq
2,Mf B2) sont distinctes.Demonstration. 1. Conformement au lemme 2.6∗, la classe Mono est Eu-
cohereditaire. Comme B ⊂ Eu, il resulte que la classe Mono est aussi B-cohereditaire. Conformement au theoreme 3.2, deduisons que (Ef B, Bq) estune structure bicategorielle. Pour le couple (Bx,B Mf ) la demonstration estduale.
2. Soit f ∈ B1 et f /∈ B2. Alors f ∈ B1 Ef . Supposons que f ∈ B2 Ef .Alors ce morphisme se presente
f = b2 · eavec b2 ∈ B2 et e ∈ Ef . Comme f ∈ Mono, il resulte que e ∈ Mono. Alorse ∈ Ef ∩Mono = Iso, et f ∈ B2.
4.9 Mentionnons encore d’autres paires de structures bicategorielles quiverifient la condition 2 du Theoreme 3.2.
LEMME ([10], lemme 3.2). Soit R une sous-categorie Eu-reflective de lacategorie D. Alors:
1. r(Mu) ⊂ Mu.2. r(Mono) ⊂ Mono.4.10 THEOREME ([10], theoreme 3.2). Soit (K, R) une paire conjuguee de
sous-categorie de la categorie D, et (P, I) - une structure bicategorielle. Alorsles affirmations suivantes sont equivalentes:
a) r(I) ⊂ I;b) k(P) ⊂ P.4.11 Exemples. Soit (K,R) une paire conjuguee de sous-categorie de la
categorie D.1. Conformement au theoreme 4.4, nous avons r(Mf ) ⊂ Mf et k(Ef ) ⊂ Ef .
Ainsi, conformement au theoreme 4.10, il resulte que r(Mono) = Mono etk(Epi) ⊂ Epi i.e. r est un monofoncteur, et k est un epifoncteur.
Le premiere affirmation resulte de le lemme 4.9 et la deuxieme est unepropriete topologique (voir [16]).
2. Conformement au theoreme 4.4 pour la paire (K, R), nous avons r(Mp) ⊂Mp. Donc r(Eu) ⊂ Eu, ce qui est evident.
3. Conformement a le lemme 4.9, nous avons r(Mu) ⊂ Mu. Donc k(Ep) ⊂Ep. Cela resulte directement de la description de la classe Ep (voir [8], [10]).
24 Dumitru Botnaru
4.12 THEOREME. Soit C une categorie a carres cortesiens et cocartesiens,dans laquelle la classe Mu est stable a gauche, R une sous-categorie monore-flective, et (P, I) - une structure bicategorielle a droite dans la categorie C
ainsi que r(I) ⊂ I. Alors1. La classe I et (εR, (εR)⊥) - cohereditaire.2. ((εR) P, I ∩ (εR)⊥) est une structure bicategorielle a droite dans la
categorie C.Demonstration. 1. Soit f ∈ I et examinons le diagramme du p. 2.12.
Conformement a l’hypothese r(f) ∈ I, et le carre r(f)v = rY u est cartesien.Donc u ∈ I. Comme f = ut est (εR, (εR)⊥) - factorisation du morphisme f(theoreme 2.12), il resulte que la classe I a la propriete respective.
2. Cette affirmation resulte des demonstrations precedentes et du theoreme3.2.
5. STRUCTURES BICATEGORIELLES COMPLEMENTAIRE
5.1 Dans une categorie C local et colocal-petite a limites projectives ouinductives, la classe B des structures bicategorielles est une latice completeavec un element minimal et un element maximal.
Soit (P1, I1), (P2I2) ∈ B. Considerons
(P1, I1) ≤ (P2I2) ⇔ P1 ⊂ P2.
Comme une structure bicategorielle de droite est uniquement determineepar sa classe de projections, la relation (P1, I1) ≤ (P2, I2) sera notee plussimplement P1 ≤ P2.
Ainsi Ef est l’element minimal, et Epi est l’element maximal de la latice B.Mentionnons les suivantes regles simples:
1. P1 ∧ P2 = P1 ∩ P2.2. P1 ∪ P2 ⊂ P1 ∨ P2.3. (P1 ∨ P2)x = I1 ∩ I2.4. (P1 ∧ P2)x = I1 ∨ I2.5. P1 ∨ P2 = (I1 ∩ I2)q.5.2 En appelant a la terminologie de la theorie des latices, introduisons la
notion (voir [13], cap. I §6).Definition. Les elements (P1, I1), (P2, I2) de la latice B s’appellent recipro-
quement complementaires si P1 ∧ P2 = Ef et P1 ∨ P2 = Epi.Mentionnons que, dans toute categorie, les structures bicategorielles
(Epi,Mf ) et (Ef ,Mono) sont reciproquement complementaires.Conformement aux relations 5.1.1 et 5.1.5, nous pouvons dire que les
elements (P1, I1) et (P2, I2) de la latice B sont reciproquement complementairessi P1 ∩ P2 = Ef et I1 ∩ I2 = Mf .
Structures bicategorielles complementaires 25
5.3 Soit B une classe bicomplete dans la categorie C, (P1, I1) - une struc-ture bicategorielle avec la classe d’injections I1 (B,B⊥) - cohereditaire;(P2, I2) - une structure bicategorielle avec la classe de projections P2(B>, B) -hereditaire. Alors, conformement au theoreme 3.2 (P′1, I
′1) = (B P1, B
⊥ ∩ I1)est une structure bicategorielle; et conformement au theoreme dual 3.2∗, lecouple (P′2, I
′2) = (P2 ∩B>, I2 B) est aussi une structure bicategorielle.
THEOREME. 1. La classe I′2 est P2 - cohereditaire.2. La classe P′1 est I1 - hereditaire.3. P′1 ∩ P′2 ⊂ P1 ∩ P2.4. P1 ∪ P2 ⊂ P′1 ∨ P′2.
Particulierement, si les structures bicategorielles (P1, I1) et (P2, I2) sontreciproquement complementaires, alors le sont aussi les structures bicategorielles(P′1, I
′1) et (P′2, I
′2).
Demonstration. 1. Soit i2b ∈ I′2 avec i2 ∈ I2 et b ∈ B.Soit
i2b = fp2 (45)
avec p2 ∈ P2. Comme p2 ⊥ i2 il y a un morphisme g de telle maniere que
b = gp2, (46)
f = i2g. (47)
Dans l’egalite (46), p2 est un epi, donc le carre
g · p2 = 1 · b (48)
est cocartesien. Ainsi g ∈ B, et dans l’egalite (47) i2 ∈ I2 et g ∈ B. Ainsinous avons demontre que f ∈ I′2.
Figure 5.1
2. Dual.3. Nous avons P′1 ∩ P′2 = (B P1) ∩ (P2 ∩B>).
26 Dumitru Botnaru
Soit que f ∈ P′1 ∩ P′2. Alors f = bp1 avec b ∈ B et p1 ∈ P1. De memebp1 ∈ P2 et bp1 ∈ B>. De la relation bp1 ∈ B>, il resulte que b ∈ B>. Doncb ∈ B ∩B> = Iso. Alors bp1 ∈ P2 et bp1 ∈ P1, i.e. bp1 ∈ P1 ∩ P2.
4. Comme P1 ⊂ P′1, il reste a montrer que P2 ⊂ P′1 ∨ P′2. Soit p2 ∈ P2, et
p2 = i1p1 (49)
(P1, I1) - factorisation du morphisme p2,
i1 = bb′ (50)
(B>, B) - factorisation du morphisme i1. De l’egalite (49), il resulte que i1 ∈P2, et de l’egalite (50), il resulte que b′ ∈ P2, comme la classe P2 est (B>, B)-hereditaire. Ainsi dans l’egalite p2 = bb′p1 nous avons p1 ∈ P1, b ∈ B, etb′ ∈ P2 ∩B>. Ainsi b, b′, p1 ∈ P′1 ∪ P′2 et bb′p1 ∈ P′1 ∨ P′2
Figure 5.2
5.4 THEOREME. Soit C une categorie a carres cartesiens et cocartesiens,et B - une classe bicomplete de morphismes. Soit que (Ef , Mono) et (Epi,Mf )sont des structures bicategorielles dans la categorie C. Alors
1. Les structures bicategorielles
(B Ef , Bx), (Bq, Mf B)
sont reciproquement complementaires.2. La classe Mf B est Epi-cohereditaire.3. La classe B Ef est Mono-hereditaire.Demonstration. 1. Les couples correspondants forment des structures bi-
categorielles conformement au theoreme 4.8. Ils sont complementaires con-formement au theoreme precedent.
2 et 3 resultent du theoreme 5.3.5.5 COROLLAIRE. Dans la categorie D, il existe une classe propre de
structures bicategorielles qui possede des complements.Demonstration. Conformement aux theoremes 5.4 et 4.7.
Structures bicategorielles complementaires 27
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[14] V. A. Geiler, V. R. Gisin, Dualite generalisee pour les espaces local-convexes, Analysefonctionnelle, Ulianovsk, 11 (1978), 41-50 (en ruse).
[15] P. Kenderov, Groupes Vectoriaux Topologiques, Mat. Sbornic, 81, (123), 3 (1970), 580-599 (en russe).
[16] A. P. Robertson, W. Robertson, Topological Vector Spaces, Cambridge University Press,1964.
[17] G. E. Strecker, On caracterization of perfect morphisms and epireflective hulls, LectureNotes in Math., 378 (1974), 468-500.
SOME ACCELERATED FLOWS FOR ANOLDROYD-B FLUID
ROMAI J., 5, 2(2009), 29–48
Ilie BurdujanDepartment of Mathematics, University of Agricultural and Veterinary Medicine“Ion Ionescu de la Brad”, Iasi, Romaniaburdujan [email protected]
Abstract This paper deals with an important problem in physics and engineering, namelywith Taylor-Couette flow formation in an Oldroyd-B fluid filling the annularregion between two infinitely long coaxial circular cylinders, due to a constantor time-dependent axial shear applied on the outer surface of the inner cylinder.The obtained solution is presented as the sum of the corresponding Newtoniansolution and the non-Newtonian contribution. Afterwards, it was specialized togive the solution for second grade fluids and Maxwell fluids, as well. Very simpleforms of some exact solutions, that either have already been known or areobtained for the first time, are presented as limit cases of our solution. Theseresults were established as limit cases of the solution of an initial-boundaryproblem in fractional derivatives which was obtained, in its turn, by using theLaplace and Hankel transforms.
Keywords: Taylor-Couette flow, Oldroyd-B fluid, Maxwell fluid, second grade fluid.
2000 MSC: 76A05.
1. INTRODUCTIONThe wide variety of non-Newtonian fluids arising in a large variety of in-
dustrial applications - such as chemical process industry (e.g. synthetic fibres,polymer solutions, petroleum production, etc.), food industries, constructionengineering and so on - motivates the great interest in their study. It iswell known that the analysis of the behavior of the fluid motion for the non-Newtonian fluids is more complicated and subtle in comparison with that ofNewtonian fluids. For a wide class of flows of Newtonian fluids it is possible togive a closed form for analytical solution, while for non-Newtonian fluids suchsolutions are seldom found. On the other hand, some of the mathematicalmodels do not fit well with experimental data. That is why, some mathemat-ical objects obtained by placing some fractional derivatives instead of sometime derivatives into the rheological constitutive equations, that describe therheological properties of some classes of materials, were tested. On this linewe can quote the papers of Bagley [1], Friedrich [6], Makris and Constantinou[15], Glokle and Nonnenmacher [8], Mainardi [13], Mainardi and Gorenflo [14],Rossikhin and Y. A., Shitikova [17], [18] and so on. They had obtained results
29
30 Ilie Burdujan
which are in a good agreement with experimental data. Unfortunately, aninitial-boundary problem for an equation with fractional derivatives (shortly,IBPEFD) cannot be considered as being a mathematical model for a real dy-namical system, because the fractional derivatives have no tensorial character.Nevertheless, some of its limit cases may be the mathematical models for realphenomena. Therefore, it becomes important to solve such an IBPEFD be-cause its solution gives the possibility to find the solutions for all its limitcases, among them being the solutions of problems modelling real dynamicalsystems. Two important situations may arise in the limiting processes. Aresult of such a limit can be the disappearance of all fractional derivatives.As example, in the problem under consideration in the present paper (i.e., theIBPEFD [(7), (10), (12)]), the Newtonian solution is obtained by making therelaxation time λ and the retardation time λr tend to zero. The second kindof results corresponds to the case when the orders of fractional derivatives,here α or/and β, tend to 1. This time the limiting process is considered inthe sense of Schwartz’s distribution theory with respect to some appropriatelyclasses of testing functions. Indeed, the solution for ordinary Maxwell fluidsis obtained from the before mentioned IBPEFD when λr → 0 and α → 1.
These remarks will be used in what follows in order to find the exact solu-tion for Taylor-Couette flow of an incompressible Oldroyd-B fluid in a circularpipe. More exactly, the main purpose of this paper is to provide exact so-lutions for the velocity field and the shear stress corresponding to the largeclass of unsteady flows of incompressible Oldroyd-B fluids between two in-finite coaxial circular cylinders, one of them being subject to a constant ortime-dependent rotational shear stress. To this end, into the governing equa-tions, corresponding to an Oldroyd-B fluid in the absence of body forces anda pressure gradient in the flow direction, some time derivatives are replacedby fractional derivatives. The obtained mathematical object was named byTong and Liu [20] the governing equations of an incompressible ”generalized”Oldroyd-B fluid. After making the similar replacement in the initial-boundaryconditions, an IBPEFD is obtained. The governing equations for an incom-pressible ”generalized” Maxwell fluid or a ”generalized” second grade fluid aresimilarly obtained. The attribute ”generalized” will be used here for designingthe hypothetical fluids that would be characterized by such IBPEFDs.
The solution of IBPEFD [(7), (10), (12)] is presented as a sum of the New-tonian solution and the corresponding non-Newtonian contribution. It can beeasily specialized to give the similar solutions for the second grade and Maxwellfluids. As it was already remarked, the Newtonian solutions can be also ob-tained as limit cases of general solutions. Furthermore, the non-Newtoniancontributions to the general solutions have been expressed in terms of thetime derivatives of a Newtonian solution.
Some accelerated flows for an Oldroyd-B fluid 31
2. MODEL AND BASIC EQUATIONSLet us consider an incompressible Oldroyd-B fluid at rest in the annular re-
gion between two infinitely long coaxial circular cylinders of radii R1, R2 (R1 <R2). The outer cylinder is always at rest, while at time t = 0+ the innercylinder is suddenly set in rotation around its axis by a constant or a time-dependent shear stress.
The Cauchy stress tensor T for an incompressible Oldroyd-B fluid, is givenby
T = −pI + S, S + λDSDt
= µ
(A + λr
DADt
), (1)
where −pI is the indeterminate spherical stress, S is the extra-stress tensor,A is the first Rivlin-Ericksen tensor, µ is the dynamic viscosity of the fluid, λand λr(< λ) are material constants, and
DSDt
=dSdt
+(V ·∇)S−LS−SLT ,DADt
=dAdt
+(V ·∇)A−LA−ALT . (2)
Into above equation (2), V denotes the velocity,∇ is the gradient operator, L isthe velocity gradient and the superscript T indicates the transpose operation.For the problem under consideration, we assume a velocity field V and anextra-stress tensor S of the form
V = V(r, t) = ω(r, t)eθ, S = S(r, t) (3)
where eθ is the unit vector along the θ-direction of the cylindrical coordinatesystem r, θ and z. For such flows the constraint of incompressibility is auto-matically satisfied. Furthermore, if the fluid is at rest up to the moment t = 0,and therefore
V(r, 0) = 0, S(r, 0) = 0, (4)
then the governing equations for an Oldroyd-B fluid, in the absence of bodyforces and a pressure gradient in the flow direction, are given, for all r ∈(R1, R2) and t > 0, by
λ∂2ω(r, t)
∂t2+
∂ω(r, t)∂t
= ν
(1 + λr
∂
∂t
)(∂2
∂r2+
1r
∂
∂r− 1
r2
)ω(r, t), (5)
(1 + λ
∂
∂t
)τ(r, t) = µ
(1 + λr
∂
∂t
)(∂
∂r− 1
r
)ω(r, t), (6)
where τ(r, t) = Srθ(r, t) is the nonzero shear stress, µ is the dynamic viscosityof the fluid, ρ is its constant density, ν(= µ/ρ) is the kinematic viscosity, andλ and λr are the relaxation and retardation times.
32 Ilie Burdujan
By replacing some inner time derivatives by the fractional differential oper-ators Dα
t and Dβt (β ≥ α), the governing equations (5) and (6) of an incom-
pressible Oldroyd-B fluid become (see [20]), for all r ∈ (R1, R2) and t > 0,
(1 + λDαt )
∂ω(r, t)∂t
= ν(1 + λrDβt )
(∂2
∂r2 +1r
∂
∂r− 1
r2
)ω(r, t), (7)
(1 + λDαt )τ(r, t) = µ(1 + λrD
βt )
(∂
∂r− 1
r
)ω(r, t), (8)
where the fractional derivatives are defined by [16]
Dpt [f(t)] =
1Γ(1− p)
d
dt
∫ t
0
f(τ)(t− τ)p dτ, 0 < p < 1, (9)
(here Γ(·) is Euler’s Gamma function).The momentum equation (5) must be solved subject to the initial and
boundary conditions
ω(r, 0) =∂ω(r, 0)
∂t= 0, τ(r, 0) = 0, (10)
respectively,(
1 + λ∂
∂t
)τ(R1, t) = µ
(1 + λr
∂
∂t
)(∂ω(R1, t)
∂r− 1
R1ω(R1, t)
)= fta, (11)
for r ∈ (R1, R2), t > 0, f ∈ R (a fixed real number), and a ≥ 0. As before,by replacing in (11) the inner derivatives with respect to t by the fractionaldifferential operators Dα
t and Dβt (β ≥ α), we get
(1 + λDαt )τ(R1, t) = µ(1 + λrD
βt )
(∂ω(R1, t)
∂r− 1
R1ω(R1, t)
)= fta, (12)
for t > 0, a ≥ 0. Consequently, an IBPEFD, consisting of equations [(7),(10), (12)], is associated with the model of the Taylor-Couette flow of anOldroyd-B fluid in an annulus due to a constant or time depending couple andcharacterized by simultaneously equations [(5), (10), (11)]. It will be solved byusing the integral transform techniques. More exactly, the Laplace and finiteHankel transform are used to transform the IBPEFD [(7), (10), (12)] into analgebraic system.
Moreover, the equations (7) and (8) contain as limit cases the governing equa-tions of the so called (see [20]) ”generalized” second grade and Maxwell models(i.e. the models obtained by replacing some inner time derivatives by somefractional differential operators in the governing equations of a second grade
Some accelerated flows for an Oldroyd-B fluid 33
or a Maxwell fluid), as well as the ordinary Oldroyd-B, Maxwell and secondgrade models.
In this paper, we are especially interested in the cases a = 0 and a = 1.
COMMENT. Of course, in order to enssure the dimensional consistency ofequations (7) and (8), the material constants λ and λr must have necessarilythe dimensions of tα and tβ, respectively. In some papers, e.g. [11], the authors(correctly) used λα and λβ
r instead of λ and λr. However, for simplicity, weshall keep the notations λ and λr, having in mind their correct significations.
3. EXACT SOLUTIONS FOR THE VELOCITYFIELD
In what follows, we shall use the modified Hankel transform, with respectto r, defined by means of the Bessel functions of index 1
B(r, rn) = J1(rrn)Y2(R1rn)− J2(R1rn)Y1(rrn),
where J1(·), J2(·), Y1(·) and Y2(·) are Bessel functions (of index 1 and 2),(rn)n∈N∗ is the increasing sequence of the positive roots of the transcedentalequation J1(R2x)Y2(R1x) − J2(R1x)Y1(R2x) = 0 (i.e. B(R2, rn) = 0 for alln ∈ N∗). We shall denote by ωH(rn, t) the image of ω(r, t) by the modifiedHankel transform defined by
ωH(rn, t) =
R2∫
R1
r ω(r, t) B(r, rn) dr . (13)
Standard computations, similar to those used for proving the identity (13.4.31)in [5], give the identity
R2∫R1
r B(r, rn)[
∂2
∂r2 + 1r
∂∂r− 1
r2
]ω(r, t) dr =
= −R1B(R1, rn)[∂ω(R1, t)
∂r− 1
R1ω(R1, t)
]− r2
nωH(rn, t).(14)
Multiplying Eq. (7) by rB(r, rn), then integrating it with respect to r fromR1 to R2 and taking into account the identity (14) it follows
(1 + λDαt )
∂ωH(rn, t)∂t
= −f
ρtaR1B(R1, rn)− νr2
n(1 + λrDβt )ωH(rn, t), t > 0,
(15)where ωH(rn, t), due to Eqs. (10), have to satisfy the initial conditions
ωH(rn, 0) = 0, n = 1, 2, 3.... . (16)
34 Ilie Burdujan
Now, applying the Laplace transform to Eq. (15), using the Laplace transformformula for fractional derivatives [16] and taking into account the initial con-ditions (16), as well as Eq. (A1) from Appendix, we find the Laplace imageωH(rn, q) of ωH(rn, t) as being
ωH(rn, q) =2f
πρrn
Γ(1 + a)q1+a
1q + λq1+α + νr2
n(1 + λrqβ)
. (17)
We shall apply the inverse Laplace transform to (17) in order to obtainωH(rn, t). However, for a suitable presentation of the final results, we firstlyrewrite Eq. (17) in the form
ωH(rn, q) =2f
πρrn
1q + νr2
n
Γ(1 + a)q1+a
[1− λq1+α + λr · νr2
n · qβ
q + λq1+α + νr2n(1 + λrq
β)
]. (18)
In its turn, the last term in Eq. (18), can be written in an equivalent formas the series
1
q1+aλq1+α + λr · νr2
n · qβ
q(1 + λqα) + νr2n(1 + λrq
β)=
λq1+α + λr · νr2n · qβ
q1+a
∞∑k=0
(−νr2n)k(1 + λrq
β)k
(λq)k+1 (qα + 1/λ)k+1 =
=∞∑
k=0
k∑m=0
(−νr2
n
λ
)kk!λm
r
m!(k −m)!qαm−a
(qα + 1/λ)k+1−
−λr
∞∑k=0
k∑m=0
(−νr2
n
λ
)k+1k!λm
r
m!(k −m)!qβm−a
(qα + 1/λ)k+1 ,
(19)
where αm = mβ + α− k − 1 and βm = mβ + β − k − 2.Introducing (19) into (18), inverting the result by means of the inverse
Laplace transform and using Eq. (A2), we find that
ωH(rn, t) =2f
πρrn
∫ t
0
sae−νr2n(t−s) ds− 2f
πρrnΓ(1 + a)
∞∑
k=0
(−νr2
n
λ
)k ∫ t
0
Fk(s)e−νr2n(t−s) ds,
(20)where
Fk(t) =
k∑m=0
k!λmr
m!(k −m)!
[Gα,αm−a,k+1
(− 1
λ, t
)+ νr2
nλr
λGα,βm−a,k+1
(− 1
λ, t
)](21)
and (see [12])
Ga,b,c(d, t) =∞∑
j=0
Γ(j + c)ta(j+c)−b−1
Γ(j + 1)Γ(c)Γ[a(j + c)− b]dj = L−1
qb
(qa − d)c
, (22)
Re (ac− b) > 0,
∣∣∣∣dqa
∣∣∣∣ < 1.
Some accelerated flows for an Oldroyd-B fluid 35
Recall that the inverse of the modified Hankel transform (13) is defined by
f(r) =π2
2
∞∑
n=1
r2nJ2
1 (R2rn)B(r, rn)J2
2 (R1rn)− J21 (R2rn)
fH(rn) =π2
2
∞∑
n=1
r2nCFnfH(rn), (23)
where fH(rn) denotes the image of f(r) by Hankel transform (13) and
CFn =J2
1 (R2rn)B(r, rn)J2
2 (R1rn)− J21 (R2rn)
= DFnB(r, rn). (24)
By applying the inverse Hankel transform (23) to Eq. (20) we get, for thevelocity field ω(r, t), the expression
ω(r, t) = ωN,a(r, t)− πf
ρΓ(1+a)
∞∑
n=1
∞∑
k=0
(−νr2
n
λ
)k
rnCFn
∫ t
0Fk(s)e−νr2
n(t−s)ds,
(25)where
ωN,a(r, t) =πf
ρ
∞∑
n=1
rnCFn
∫ t
0sae−νr2
n(t−s)ds, (26)
represents the velocity field corresponding to a Newtonian fluid performingthe same motion. In particular, by using (A3) one gets
ωN (r, t) = ωN,0(r, t) = ϕ0(r)−πf
µ
∞∑
n=1
1rn
CFne−νr2nt, (27)
where
ϕ0(r) = − f
2µ
(R1
R2
)2 (R2
2
r− r
).
Indeed, making λ = λr = 0 into Eq. (17) and following the same way asbefore, we obtain (26). In particular, for a = 0 we attain to ωN,0(r, t) givenby (27). On the another hand, Eq. (27) gives
∂ωN (r, t)∂t
=πf
ρ
∞∑
n=1
rnCFne−νr2nt
and consequently (26) becomes
ωN,a(r, t) = ta ∗ ∂tωN (r, t). (28)
Further, for a = 1, one obtains
ωN,1(r, t) =∫ t
0ωN (r, s) ds = 1 ∗ ωN (r, t)
36 Ilie Burdujan
and the corresponding Newtonian solution takes the following form
ωN,1(r, t) = ϕ0(r)t + ϕ1(r) +πf
µν
∞∑
n=1
1r3n
CFne−νr2nt, (29)
where
ϕ1(r) =A
r+ Br + Cr3 + E r ln r,
with
A = − fR41
8R22µν
(2R22 −R2
1), B = fR21
8R22µν
[4R4
2 ln R2 − (R22 −R2
1)2],
C = f8µν
(R1
R2
)2
, E = −fR21
2µν.
It is important to notice that the Newtonian solution ωN (r, t) given by Eq.(27) is coincident with that obtained in [2] (p. 829) by a different technique.
Finally, having in mind the expression (27) of the Newtonian solutionωN (r, t), it is easy to show that the general solution ω(r, t) can be writtenin a suitable form in terms of its time derivatives, namely
ω(r, t) = ta ∗ ∂tωN (r, t)−−Γ(1 + a)
∞∑k=0
k∑m=0
k!λmr
m!(k −m)!λk ∂k+1t ωN (r, t) ∗Gα,αm−a,k+1
(− 1
λ, t
)+
+λr
λΓ(1 + a)
∞∑k=0
k∑m=0
k!λmr
m!(k −m)!λk ∂k+2t ωN (r, t) ∗Gα,βm−a,k+1
(− 1
λ, t
).
(30)
4. CALCULATIONS OF THE SHEAR STRESSApplying the Laplace transform to Eq. (8) and using the initial conditions
(10), we find that
τ(r, q) = µ1 + λrq
β
1 + λqα
[∂ω(r, q)
∂r− 1
rω(r, q)
], (31)
where τ(r, q) denotes the Laplace transform of τ(r, t). The last factor, con-taining the velocity field, may be obtained using Eq. (30) and (25) as well.
Some accelerated flows for an Oldroyd-B fluid 37
Indeed, from (30) we have
∂ω(r, t)∂r
− 1rω(r, t) = ta ∗ ∂tΩN (r, t)−
−Γ(1 + a)∞∑
k=0
k∑m=0
k!λmr
m!(k −m)!λk ∂k+1t ΩN (r, t) ∗Gα,αm−a,k+1
(− 1
λ, t
)+
+λr
λΓ(1 + a)
∞∑k=0
k∑m=0
k!λmr
m!(k −m)!λk ∂k+2t ΩN (r, t) ∗Gα,βm−a,k+1
(− 1
λ, t
),
(32)where
ΩN (r, t) =∂ωN (r, t)
∂r− 1
rωN (r, t). (33)
Applying the Laplace transform to Eq. (32) and having in mind Eq. (A4),it results that
∂ω(r, q)∂r
− 1rω(r, q) = Γ(1 + a)
q1+a L∂tΩN (r, t)−
−Γ(1 + a)∞∑
k=0
k∑m=0
k!λmr
m!(k −m)!λkqαm−a
(qα + 1/λ)k+1 L∂k+1t ΩN (r, t)+
+λr
λΓ(1 + a)
∞∑k=0
k∑m=0
k!λmr
m!(k −m)!λkqβm−a
(qα + 1/λ)k+1 L∂k+2t ΩN (r, t).
(34)
Now, introducing (33) into (31), using the simple decomposition
1 + λrqβ
1 + λqα = 1 +λr
λ
qβ
qα + 1/λ− qα
qα + 1/λ
and applying the inverse Laplace transform we get, taking into account of(A5), the next form for the shear stress
τ(r, t) = τN,a(r, t) + µΓ(1 + a)A(− 1
λ, t
)∗ ∂tΩN (r, t)−
−µΓ(1 + a)∞∑
k=0
[Bk
(− 1
λ, t
)∗ ∂k+1
t ΩN (r, t)− λr
λCk
(− 1
λ, t
)∗ ∂k+2
t ΩN (r, t)]
(35)where
τN,a(r, t) = µta ∗ ∂tΩN (r, t) =
= −πfν∞∑
k=0
r2n[J2(rrn)Y2(R1rn)− J2(R1rn)Y2(rrn)]
J22 (R1rn)− J2
1 (R2rn)
t∫0
sae−νr2n(t−s) ds
(36)
38 Ilie Burdujan
represents the shear stress corresponding to a Newtonian fluid and
A
(− 1
λ, t
)=
λr
λRα,β−a−1
(− 1
λ, t
)−Rα,α−a−1
(− 1
λ, t
),
Rα,β(a, t) =∞∑
k=0
akt(k+1)α−β−1
Γ((k + 1)α− β), Re (α− β) > 0, |atα| < 1,
Bk
(− 1
λ, t
)=
1
λk
k∑m=0
k!λmr
m!(k −m)!
[Gα,αm−a,k+1
(− 1
λ, t
)+
λr
λGα,αm+β−a,k+2
(− 1
λ, t
)−
−Gα,αm+α−a,k+2
(− 1
λ, t
)],
Ck
(− 1
λ, t
)=
1
λk
k∑m=0
k!λmr
m!(k −m)!
[Gα,βm−a,k+1
(− 1
λ, t
)+
λr
λGα,βm+β−a,k+2
(− 1
λ, t
)−
−Gα,βm+α−a,k+2
(− 1
λ, t
)].
The equivalent form of the shear stress, resulting from (25) and (31) is
τ(r, t) = τN,a(r, t) + µΓ(1 + a)A(− 1
λ, t
)∗ ∂tΩN (r, t)+
+πfν Γ(1 + a)∞∑
n=1
∞∑k=0
rnC ′Fn(−νr2
n)k
[Bk
(− 1
λ, t
)+ νr2
nλr
λC
(− 1
λ, t
)]∗ e−νr2
nt,
(37)where
C ′Fn =
J2(rrn)Y2(R1rn)− J2(R1rn)Y2(rrn)J2
2 (R1rn)− J21 (R2rn)
J21 (R2rn),
Making a = 0 and 1 into (35) and (37), the shear stresses corresponding tof and ft into (10) are obtained. For instance, the Newtonian solutions are
τN (r, t) = τN,0(r, t) = τ0(r) + πf
∞∑
n=1
C ′Fn e−νr2
nt, (38)
τN,1(r, t) = tτ0(r) + τ1(r)− πfν
∞∑n=1
C ′Fne−νr2
nt, (39)
where
τ0(r) =fR2
1
r2 , τ1(r) =fR2
1
[(R2
2 − r2)2 − (R22 −R2
1)2]
8νR22r
2 .
Some accelerated flows for an Oldroyd-B fluid 39
5. LIMIT CASES1. By taking the limit of Eqs. (25), (30), (35) and (37) as λr → 0, we get
the similar solutions corresponding to the so-called generalized Maxwell fluids,namely
ω(r, t) =
= ωN,a(r, t)− πfρ
Γ(1 + a)∞∑
n=1
rnCFn
∞∑k=0
(−νr2
n
λ
)k t∫0
Gα,γk−a,k+1
(− 1
λ, t
)e−νr2
n(t−s) ds =
= ωN,a(r, t)− Γ(1 + a)∞∑
k=0
1
λk Gα,γk−a,k+1
(− 1
λ, t
)∗ ∂k+1
t ωN (r, t),
(40)
τ(r, t) = τN,a(r, t)− µΓ(1 + a)t∫0
∂sΩN (r, s)Rα,α−a−1
(− 1
λ, t− s
)ds+
+πfνΓ(1 + a)∞∑
n=1
∞∑k=0
rnC′Fn
( − νr2n
λ
)k [Gα,γk−a,k+1
(− 1
λ, t
)−
−Gα,γk+α−a,k+2
(− 1
λ, t
)]∗ e−νr2
nt =
= τN,a(r, t)− µΓ(1 + a)∂tΩN (r, t) ∗Rα,α−a−1
(− 1
λ, t
)−
−µΓ(1 + a)∞∑
k=0
1
λk ∂k+1t ΩN (r, t) ∗
[Gα,γk−a,k+1
(− 1
λ, t
)−Gα,γk+α−a,k+2
(− 1
λ, t
)],
(41)
where γk = α − k − 1. Furthermore, by making λ → 0 in (40) and (41) andtaking into account of Eqs. (A6), the Newtonian solutions
ωN,a(r, t) = ta ∗ ∂tωN (r, t), τN,a(r, t) = µta ∗ ∂tΩN (r, t) (42)
are recovered.
2. By taking now α → 1 into (40) and (41), the solutions for ordinary Maxwellfluids are obtained, namely
ω(r, t) = ωN,a(r, t)−−πf
ρΓ(1 + a)
∞∑n=1
∞∑k=0
(−νr2
n
λ
)k
rnCFn
t∫0
G1,−k−a,k+1
(− 1
λ, t
)e−νr2
n(t−s) ds =
= ωN,a(r, t)− Γ(1 + a)∞∑
k=0
1
λk ∂k+1t ωN (r, t) ∗G1,−k−a,k+1
(− 1
λ, t
),
(43)
40 Ilie Burdujan
τ(r, t) = τN,a(r, t)− µΓ(1 + a)t∫0
∂sΩN (r, s)R1,−a
(− 1
λ, t− s
)ds+
+πfνΓ(1 + a)∞∑
n=1
∞∑k=0
rnC′Fn
( − νr2n
λ
)k
×
×t∫0
e−νr2nt
[G1,−k−a,k+1
(− 1
λ, t− s
)−G1,−k−a+1,k+2
(− 1
λ, t− s
)]ds =
= τN,a(r, t)− µΓ(1 + a)∂tΩN (r, t) ∗R1,−a
(− 1
λ, t
)−
−µΓ(1 + a)∞∑
k=0
1
λk ∂k+1t ΩN (r, t) ∗
[G1,−k−a,k+1
(− 1
λ, t
)−G1,−k−a+1,k+2
(− 1
λ, t
)].
(44)Direct computations implying suitable grouping of terms shows that Eq. (43),corresponding to the special cases a = 0 and 1, can be respectively written inthe forms (see also Eq. (A7),
ω0(r, t) = ωN,0(r, t)− 2fλ
ρ
∞∑n=1
(−νr2
n
λ
)k
rnCFne−νr2nt ∗ L
−1
(q
λq2 + q + νr2n
), (45)
ω1(r, t) = ωN,1(r, t)− 2fλ
ρ
∞∑n=1
(−νr2
n
λ
)k
rnCF ne−νr2nte−νr2
nt ∗L−1
(1
λq2 + q + νr2n
), (46)
where ω0(r, t) and ω1(r, t) denote the velocity field respectively correspondingto a = 0 and a = 1.
By using formulae (A8), one obtains
ω0(r, t) = ωN (r, t) +πfλ
µΓ(1 + a)
∞∑n=1
1
rnCFn
[e−νr2
nt +qn2e
qn1t − qn1eqn2t
qn1 − qn2
], (47)
ω1(r, t) = ωN,1(r, t)− πfλ
µνΓ(1 + a)
∞∑n=1
1
r3n
CFn
[e−νr2
nt − q2n2e
qn1t − q2n1e
qn2t
qn1 − qn2
], (48)
where qn1, qn2 are the solutions of equation λq2 + q + νr2n = 0. Taking into
account Eq. (27), the solutions (47) and (48) can be written in the followingsimpler forms:
ω0(r, t) = − f
2µ
(R1
R2
)2 (r − R2
2
r
)+
πf
µ
∞∑n=1
1
rnCFn
qn2eqn1t − qn1e
qn2t
qn1 − qn2, (49)
ω1(r, t) =ft2µ
(R1
R2
)2 (r − R2
2
r
)− πf
µν
∞∑n=1
1
r3n
CFn + λπfµν
∞∑n=1
1
r5n
CFnq2
n2eqn1t − q2
n1eqn2t
qn1 − qn2.
(50)
Some accelerated flows for an Oldroyd-B fluid 41
A similar procedure, applied to Eq. (44), yields
τ0(r, t) = τN (r, t)− µR1,0
(− 1
λ, t
)∗ ∂ΩN (r, t)+
+πfνλ∞∑
n=1
rnC′FnL−1
q
λ
(q +
1
λ
)(q + νr2
n)(λq2 + q + νr2n)
,
(51)
τ1(r, t) = τN,1(r, t)− µR1,−1
(− 1
λ, t
)∗ ∂ΩN (r, t)+
+πfνλ∞∑
n=1
rnC′FnL−1
1
λ
(q +
1
λ
)(q + νr2
n)(λq2 + q + νr2n)
.
(52)
After a straightforward computation we get
τ0(r, t) =fR2
1
r2
1− e
− tλ
+
πf
λ
∞∑n=1
C′Fneqn1t − eqn2t
qn1 − qn2, (53)
τ1(r, t) =fR2
1
r2
t−
1− e
− tλ
+
πf
ν
∞∑n=1
1
r2n
C′Fn
[1− qn2e
qn1t − qn1eqn2t
qn1 − qn2
]. (54)
3. In the special case when λ → 0 into (25), (30), (32) and (34), the solutions(see also Eq. (A6))
ω(r, t) = ωN,a(r, t)+
+πfρ
λrΓ(1 + a)∞∑
n=1
∞∑k=0
k∑m=0
CFnk!λm
r (−νr2n)k+1
m!(k −m)!Γ(a− βm)
t∫0
sa−βm−1e−νr2n(t−s) ds =
= ωN,a(r, t) + λrΓ(1 + a)∞∑
k=0
k∑m=0
k!λmr
m!(k −m)!
t∫0
∂k+2s ωN (r, s) (t− s)a−βm−1
Γ(a− βm) ds
(55)
42 Ilie Burdujan
and
τ(r, t) = τN,a(r, t) + µΓ(1 + a)λr
t∫0
(t− s)a−β
Γ(a− β + 1)∂sΩN (r, t) ds−
−πfνλrΓ(1 + a)∞∑
n=1
rnC′Fn
∞∑k=0
k∑m=0
k!λmr (−νr2
n)k+1
m!(k −m)!
t∫0
[sa−βm−1
Γ(a− βm)+ λr
sa−βm−β−1
Γ(a− βm − β)
]e−νr2
2(t−s) ds =
= τN,a(r, t) + µλrΓ(1 + a)t∫0
(t− s)a−β
Γ(a− β + 1)∂sΩN (r, s) ds+
+µΓ(1 + a)λr
∞∑k=0
k∑m=0
k!λkr
m!(k −m)!
t∫0
[(t− s)a−βm−1
Γ(a− βm)+ λr
(t− s)a−βm−β−1
Γ(a− βm − β)
]∂k+2
s ΩN (r, s) ds
(56)
corresponding to a ”generalized” second grade fluid are obtained.Of course, by taking λr → 0 into Eqs. (53) and (54), we again obtain the
Newtonian solutions given by Eq. (42). Moreover, in the special case whenβ → 1, Eqs. (55) and (56) reduce to the solutions for an ordinary secondgrade fluid, namely
ω(r, t) = ωN,a(r, t)+
+πfλr
ρΓ(1 + a)λr
∞∑n=1
∞∑k=0
k∑m=0
CFnk!λm
r (−νr2n)k+1
m!(k −m)!Γ(k + a−m + 1)
t∫0
sk+a−me−νr2n(t−s) ds =
= ωN,a(r, t) + λrΓ(1 + a)∞∑
k=0
k∑m=0
k!λmr
m!(k −m)!
t∫0
∂k+2s ωN (r, s)
(t− s)k+a−m
Γ(k + a−m + 1)ds
(57)and
τ(r, t) = τN,a(r, t) + µaλr
t∫0
(t− s)a−1∂sΩN (r, t) ds−
−πfνλrΓ(1 + a)∞∑
n=1
rn C′Fn
∞∑k=0
k∑m=0
k!λmr
m!(k −m)!(−νr2
2)k+1×
× ∫ t
0
[sk+a−m
Γ(k + a−m + 1)+ λr
sk+a−m−1
Γ(k + a−m)
]e−νr2
n(t−s) ds =
= τN,a(r, t) + µλrata−1 ∗ ∂tΩN (r, t)+
+µΓ(1 + a)λr
∞∑k=0
k∑m=0
k!λkr
m!(k −m)!
[tk+a−m
Γ(k + a−m + 1)+ λr
tk+a−m−1
Γ(k + a−m)
]∗ ∂k+2
t ΩN (r, t)
(58)for a 6= 0, while for a = 0 we have
τ(r, t) = τN (r, t) + µλrΩN (r, t)−−πfνλr
∞∑n=1
rn C′Fn
∞∑k=0
k∑m=0
k!λmr
m!(k −m)!(−νr2
2)k+1
∫ t
0
[sk−m
(k −m)!+ λr
sk−m−1
(k −m− 1)!
]e−νr2
n(t−s) ds =
(59)
Some accelerated flows for an Oldroyd-B fluid 43
= τN (r, t) + µλrΩN (r, t))+
+µλr
∞∑k=0
k∑m=0
k!λkr
m!(k −m)!
tk−m−1
(k −m− 1)![t + λr(k −m)]
∗ ∂k+2
t ΩN (r, t).
For a = 0 and 1 these solutions take the simple forms.Case a = 0.
ω(r, t) = ωN (r, t)− πfµ
∞∑n=1
1rn
CFn
e
− νr2nt
1 + νr2nλr − e−νr2
nt
=
= − f2µ
R1
R2
(R2
2
r− r
)− πf
µ
∞∑n=1
1rn
CFne− νr2
nt
1 + νr2nλr ,
τ(r, t) =fR2
1
r2 + πf∞∑
n=1
C′Fn1
1 + νr2nλr
e− νr2
nt
1 + νr2nλr =
= τN (r, t) + πf∞∑
n=1
C′Fn1
1 + νr2nλr
e
− νr2nt
1 + νr2nλr − e−νr2
nt
.
(60)
Case a = 1.
ω(r, t) = ωN,1(r, t)+
+fλr
2µ
(R1
R2
)2 (R2
2
r− r
)+
πfµν
∞∑n=1
CFn
r3n
e−νr2nt +
πfµν
∞∑n=1
1 + νr2nλr
r3n
CFne− νr2
nt
1 + νr2nλr =
= − f2µ
(R1
R2
)2 (R2
2
r− r
)(t− λr)− πf
µν
∞∑n=1
1
r3n
CFn +πfµν
∞∑n=1
1 + νr2nλr
r3n
CFne− νr2
nt
1 + νr2nλr ,
τ(r, t) =fR2
1
r2 t +πfν
∞∑n=1
1
r2n
C′Fn − πfν
∞∑n=1
1
r2n
C′Fne− νr2
nt
1 + νr2nλr =
= τN,1(r, t) +fR2
1[(R22 − r2)2 − (R2
2 −R1)2]
8νR22r
2 − πfν
∞∑n=1
1
r2n
C′Fn
e− νr2
nt
1 + νr2nλr − e− νr2
nt
.
(61)
4. In the special case when α → 1 and β → 1 into (25), (30), (35) and(37), the solutions (see also Eq. (A6)) for an Oldroyd-B fluid are obtained,
44 Ilie Burdujan
namely:
ω(r, t) = ωN,a(r, t)− Γ(1 + a)∞∑
k=0
k∑m=0
k!λmr
m!(k −m)!λk ∂k+1t ωN (r, t) ∗G1,m−k−a,k+1
(− 1
λ, t
)+
+λr
λΓ(1 + a)
∞∑k=0
k∑m=0
k!λmr
m!(k −m)!λk ∂k+2t ωN (r, t) ∗G1,m−k−a−1,k+1 =
= ωN,a(r, t)− πfρ
Γ(1 + a)∞∑
n=1
rn CFn
∞∑k=0
k∑m=0
k!λmr
m!(k −m)!
(−νr2
n
λ
)k
×
×[G1,m−k−a,k+1
(− 1
λ, t
)+ νr2
nλr
λG1,m−k−a−1,k+1
(− 1
λ, t
)],
(62)
τ(r, t) = τN,a(r, t) + µΓ(1 + a)λr − λ
λR1,−a
(− 1
λ, t
)∗ ∂tΩN (r, t)− µΓ(1 + a)
∞∑k=0
k∑m=0
k!λmr
m!(k −m)!λk×
×[G1,m−k−a,k+1 (−1/λ, t) +
λr
λG1,m−k−a+1,k+2 (−1/λ, t)−G1,m−k−a+1,k+2
(− 1
λ, t
)]∗ ∂k+1
t ΩN (r, t)+
+µΓ(1 + a)λr
λ
∞∑k=0
k∑m=0
k!λmr
m!(k −m)!λk×
×[G1,m−k−a−1,k+1 (−1/λ, t) +
λr
λG1,m−k−a,k+2 (−1/λ, t)−G1,m−k−a,k+2 (−1/λ, t)
]∗ ∂k+2
t ΩN (r, t) =
= τN,a(r, t) + µΓ(1 + a)λr − λ
λR1,−a
(− 1
λ, t
)∗ ∂tΩN (r, t)+
+πfνΓ(1 + a)∞∑
n=1
rnC′Fn
∞∑k=0
k∑m=0
k!λmr
m!(k −m)!
(−νr2
n
λ
)k
×
×[
G1,m−k−a,k+1
(− 1
λ, t
)+
λr − λλ
G1,m−k−a+1,k+2
(− 1
λ, t
)]+
+ νr2n
λr
λ
[G1,m−k−a−1,k+1
(− 1
λ, t
)+
λr − λλ
G1,m−k−a+1,k+2
(− 1
λ, t
)]∗ e−νr2
nt
(63)
Using (A8) we respectively get, for a = 0 and a = 1, the following solutions.
Case a = 0
ω(r, t) = ωN (r, t)− πfµ
∞∑n=1
1rn
CFn
(e−νr2
nt − qn2eqn1t − qn1e
qn2t
qn1 − qn2
)=
= f2µ
(R1
R2
)2 (R2
2r− r
)+ πf
µ
∞∑n=1
1rn
CFnqn2e
qn1t − qn1eqn2t
qn1 − qn2,
(64)
Some accelerated flows for an Oldroyd-B fluid 45
τ(r, t) = τN (r, t)+
+fR2
∞∑n=1
J2(rrn)rnJ1(R2rn)
[(λrνr2
n − 1)e−νr2nt + (1 + λqn1)eqn1t − (1 + λqn2)eqn2t
λ(qn1 − qn2)
]=
= 2fR2
∞∑n=1
J2(rrn)rnJ1(R2rn)
e
− tλ − 1 + 1
λeqn1t − eqn2t
qn1 − qn2
.
(65)Case a = 1
ω(r, t) = ωN,1(r, t)−−fλr
2µ
(R1
R2
)2 (R2
2r− r
)− πf
µν
∞∑n=1
1r3n
CFn
(e−νr2
nt − λq2n2e
qn1t − q2n1e
qn2t
qn1 − qn2
)=
= f2µ
(R1
R2
)2 (R2
2r− r
)(t− λr)− πf
µν
∞∑n=1
1r3n
CFn
[1− λ
q2n2e
qn1t − q2n1e
qn2t
qn1 − qn2
],
(66)
τ(r, t) = τN,1(r, t) + λfR2
1
r2
1− e
− tλ
− πf
ν
∞∑n=1
1r2n
C ′Fn
(1− e−νr2
nt)−
−πfν
∞∑n=1
1r2n
C ′Fn + πf
ν
∞∑n=1
1r2n
C ′Fn
(1− qn2e
qn1t − qn1eqn2t
qn1 − qn2
)=
= fR21
r2
t− λ + λe
− tλ
+ πf
ν
∞∑n=1
1r2n
C ′Fn
(1− qn2e
qn1t − qn1eqn2t
qn1 − qn2
).
(67)
6. CONCLUSIONSThe main purpose of this work is to provide exact solution for the unsteady
flow of an incompressible Oldroyd-B fluid filling the annular region betweentwo infinitely long co-axial cylinders subject either to a constant or to a time-dependent shear stress. Such solutions, obtained by using both the Hankeland Laplace transforms, are presented as a sum between the Newtonian so-lutions and the corresponding non-Newtonian contributions. Furthermore,the non-Newtonian contributions of the general solutions are also presentedin equivalent forms, under series form in terms of the time derivative of thesimplest Newtonian solution ωN . For λr → 0 and λ → 0 these contributions
46 Ilie Burdujan
tend to zero, such that the general solutions become Newtonian solutions cor-responding to the given initial-boundary conditions.
It is remarkable that the general solutions can be easily specialized to giveboth the similar solutions for ”generalized” second grade and Maxwell fluidsand the solutions for all ordinary fluids (Oldroyd-B, Maxwell and second grade)performing the same motions. Direct computations shows that the solutionswhich have been obtained certainly satisfy both the governing equations andall imposed initial and boundary conditions. Furthermore, the solutions cor-responding to ordinary Maxwell and second grade fluids can be also obtainedas limit cases of those for ordinary Oldroyd-B fluids. As regard the Newto-nian solutions, given under simple forms (27), (29), (38) and (39), they canbe obtained as limit cases of the previous solutions.
From our general solutions, corresponding to non-Newtonian fluids, it clearlyresults that the non-Newtonian contributions of these solutions exponentiallydecrease in time, the motion of the non-Newtonian fluids being well approx-imated, for large values of t, by the motion of the corresponding Newtonianfluid.
Appendix A
LDpt f(t) = qpLf(t) − q1−pf(0+); 0 < p < 1, (A1)
Ga,b,c(d, t) = L−1
qb
(qa − d)c
; Re(ac−b) > 0, Re(q) > 0,
∣∣∣∣d
qa
∣∣∣∣ < 1, (A2)
∫ R2
R1
(r2 −R22) B(rrn) dr =
(R2
R1
)2 4πr3
n
, (A3)
L(f ∗ g)(t) = Lf(t) · Lg(t), (A4)
Ra,b(c, t) = L−1
qb
qa − c
, Re(a−b) > 0, Re(q) > 0, Et(ν, a) = L
(s−ν
s− a
).
(A5)
limλ→0
1λk
Ga,b,k(−1/λ, t) =t−b−1
Γ(−b), lim
λ→0
1λ
Ra,b(−1/λ, t) =t−b−1
Γ(−b), (A6)
limλ→0
1λ
Fa(−1/λ, t) = δ(t).
Some accelerated flows for an Oldroyd-B fluid 47
∞∑
k=0
(−νr2
n
λ
)k
G1,−k−a,k+1
(− 1
λ, t
)= λL−1
(1
qa−1
1λq2 + q + νr2
n
)(A7)
e−νr2nt ∗ L−1
(q
λq2 + q + νr2n
)= 1
νr2n
(e−νr2
nt + qn2eqn1t − qn1e
qn2t
qn1 − qn2
),
e−νr2nt ∗ L−1
(1
λq2 + q + νr2n
)= 1
(νr2n)2
[e−νr2
nt + λq2n2e
qn1t − q2n1e
qn2t
qn1 − qn2
],
eat ∗ ebt = eat − ebt
a− b.
(A8)
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48 Ilie Burdujan
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[16] Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.
[17] Rossikhin, Y. A., Shitikova, M. V., A new method for solving dynamic problems offractional derivative viscoelasticity, Int. J. Engng Sci. 39(2000), 149–176.
[18] Rossikhin, Y. A., Shitikova, M. V., Analysis of dynamic behavior of viscoelastic rodswhose rheological models contain fractional derivatives of two different orders, ZAMP81, 6(2001), 363–376.
[19] Srivastava, P. N., Non-steady helical flow of a viscoelastic liquid, Arch. Mech. Stos.,18(1966), 145-150.
[20] Tong, D., Liu, Y., Exact solutions for the unsteady rotational flow of non-Newtonianfluid in an annular pipe, Int. J. Eng. Sci., 43(2005), 281-289.
[21] Tong, D., Ruihe, Y., Heshan, W., Exact solutions for the flow of non-Newtonian fluidwith fractional derivative in an annular pipe, Science in China, Ser. G Physics, Me-chanics & Astronomy, 48(2005), 485-495.
[22] Wood, W. P., Transient viscoelastic helical flows in pipes of circular and annular cross-section, J. Non-Newtonian Fluid Mech., 100(2001), 115-126.
A FUZZY ALGORITHM FOR RELIABILITYSIMULATION OF AN ELECTRIC STATION
ROMAI J., 5, 2(2009), 49–58
Simona Dzitac, Tiberiu Vesselenyi, Ioan Dzitac, Maria ParvEnergy Engineering Faculty, University of Oradea, Romania,Management and Technological Eng. Faculty, University of Oradea, Romania,Exact Sciences Faculty, ”Aurel Vlaicu” University of Arad, and Cercetare Dezvoltare Agora/AgoraR & D, Oradea, Romania,University of Agricultural Sciences and Veterinary Medicine, Cluj-Napoca, [email protected]
Abstract In this paper we present an applied study of reliability simulation for an electricstation, based on a soft computing simulation method, namely using a fuzzyalgorithm in MATLAB environment. This study revealed that the values ofreliability obtained through this method is accurate compared to the valuesobtained by Monte Carlo method or by direct computation.
Keywords: failure tree, reliability, fuzzy simulation, electric station (ES).
2000 MSC: 03E75, 93C42.
1. INTRODUCTIONThe soft computing paradigm is based on fuzzy logic and is tolerant to
imprecision, uncertainty, partial truth, and approximation. Fuzzy logic repre-sents an extremely useful tool in modeling the behavior of electrical equipment.Fuzzy set theory is using multi state systems and multi criteria decisions, form-ing a mathematical instrument which is flexible and easily adaptable to reality.This theory is useful for modeling electromagnetic systems and also for energyequipment reliability evaluation [1, 10]. In reliability studies a bivalent op-erational evolution mode is generally accepted: normal operation state andfailure state. In reality transitions between states are not swift, which impliesa nuanced expression of system’s performance (very good, good,..., medium,poor). This paper presents the development of reliability simulation softwarefor electric stations. The software is based on failure trees method and is usingfuzzy logic in the MATLAB environment. The MATLAB programming en-vironment has predefined functions for development of fuzzy computing steps(fuzzyfication, inference, defuzzyfication) [6, 8]. These functions are linked to2 external C++ modules, the ”inference system” and the ”fuzzy engine”. Typ-ical structures of fuzzy inference systems can be represented by a model whichreveals a correspondence between: crisp input value - membership functions
49
50 Simona Dzitac, Tiberiu Vesselenyi, Ioan Dzitac, Maria Parv
- inference rules output characteristics - output membership functions - crispoutput values. Similar studies and program variants are presented in [3] and[4]. In Section 2 we describe the development of a computer simulation pro-gram for the study of complex electric system’s reliability, fuzzy algorithms,definition of asymmetric Gauss input and output membership functions, rulesets and result display methods. The third Section is focused on a case studyfor the electric station in Voivozi, (Bihor county) using the developed simu-lation program under MATLAB environment. Section 4 presents the conclu-sions, showing the importance and efficiency of fuzzy modeling in reliabilityanalysis by comparing fuzzy and Monte Carlo methods also shown in equiva-lent reliability diagrams and highlighting the contribution of the authors.
2. DEVELOPMENT OF THE SIMULATIONSOFTWARE USING FUZZY LOGIC
A frequently used analysis method in a system’s reliability study is basedon failure probability evaluation. In this method the crisp values of failureprobabilities for electrical components are generally used in order to computethe system’s reliability, based on equivalent reliability diagrams [2,3,4,6,8].
2.1 Definition of the input membership functions
No. Linguistic variable Acronym Value
1. Not acceptable NA 0
2. Almost acceptable AA 0.167
3. Close to acceptable CA 0.333
4. Acceptable A 0.5
5. Good G 0.667
6. Almost very good AVG 0.833
7. Very good VG 1
Table 1.
The developed software is using Gaussian membership functions. For this kindof function the mean and the standard deviation (σ) must be specified. So forevery component of the system seven degrees were defined (see Table 1), ona linear interval of failure and repair intensity values, and then the functionvalues were established (λ and µ).
A fuzzy algorithm for reliability simulation of an electric station 51
Fig. 1. Schematic function blocks for fuzzy analysis set up.
Fig. 2. Flowchart of the fuzzy program.
The fuzzy method is presented schematically in Figure 1.The fuzzy analysis program generates input membership functions (on basis
of specified failure intensity λ and repair intensity µ) and then generates theoutput membership functions and the rule set. The flowchart of the algorithmis presented in Figure 2.
All program functions are launched from the ”d.fuzzy.m” module. TheGraphical User Interface (GUI) window is presented in Figure 3.The ”d.date.m” module is launched on action of ”Input data” button. Thesimulation data input includes λ and µ specification, data saving and datareload. The input system equation is introduced from the ”d.param” moduleand the input λ and µ are instantiated from a separate window which allowsas many parameters as many components were specified. The program alsoallows data saving (the data are saved in a .mat type file) in files with op-
52 Simona Dzitac, Tiberiu Vesselenyi, Ioan Dzitac, Maria Parv
Fig. 3. Main program GUI (Translation: Fuzzy simulation...; Data; Inputs; Fuzzy simu-lation; Decision surfaces; Exit).
tional names. The saved data can be reloaded in a separate interface fromtheir files which contain all system parameters and also the system equation.Once established the input system parameter values for specific runs can beinstantiated from the ”d.param.intr.m” module.
Mean values for membership functions are computed on basis of the relation
Fi =λi
λi + µi
, (1)
where i =1...7, is the number of the membership function according to theearlier defined grade. The standard deviation (σ) is computed with an asym-metric Gauss function based on the relations:
σi,1 =|Fi−1 − Fi|
3, (2)
σi,2 =|Fi − Fi+1|
3. (3)
After the introduction or reload of data from saved files, we can reenter themain module which gives us two options: fuzzy simulation or decision surfacesdisplay. After computing the input membership functions parameters for eachsystem component the program generates these functions. The decision sur-face display facilitates the evaluation of the fuzzy outputs.2.2 Definition of the output membership function
In order to compute the output membership functions we start with thereduced system schematics from which the failure tree is generated. From
A fuzzy algorithm for reliability simulation of an electric station 53
the flowchart we can derive the system’s characteristic equation and then theprogram generates the output membership function.
2.3 Definition of the rule set
The rule set of the fuzzy inference system defines the way in which theinputs and the outputs are linked. The rules are described in form of logicalrelations having as variables linguistic degrees of the inputs and as operatorsthe ”and” and ”or” logical operators.
An example of fuzzy rule is:If elem1 is VW and elem2 is W and elem3 is A then the system is W.
After the establishment of the rule set, the program can generate inferencesurfaces in the input-output space which are in fact the values of the outputsfor the whole range of given inputs. Due to the limitations of 3D represen-tation, these surfaces can be represented only as 2 inputs simultaneously, theremaining inputs being considered static for that case. The 2 inputs which wewant to represent can be selected in the program interface.
2.4 Simulation results
After generating the membership functions and the rule sets the programalso generates the so called ”fuzzy inference system” information structure. Ifthis structure is used for a single run, then the crisp values of the inputs arespecified and the ”evalfis” function is used for the computation of the crispoutput values. The program displays this value in a separate window.
3. CASE STUDY VOIVOZI ELECTRIC STATIONCase study is performed for the normal form of Electric Stations (ES)
Voivozi, Bihor County scheme. The evaluation of reliability is realized con-sidering the Padurea Neagra user, positioned on BC1- 20kV collector bar andthe study criteria is considered in the absence of the consumer. Analyzingthe operative mono cable 110 kV scheme of the Bihor County energy system,it can be concluded that Suplac is considered output and Oradea Vest andMarghita are considered inputs.Using statistical data representing median values of reliability indicators forthe equipments in ES and also using the equivalent reliability diagram we havereduced the ES Voivozi scheme to an Equivalent Reliability Diagram (ERD)presented in Figure 7. This diagram was used to formulate the system of equa-
54 Simona Dzitac, Tiberiu Vesselenyi, Ioan Dzitac, Maria Parv
Fig. 4. λ and µ parameter editing window for ES Voivozi.
tion for the fuzzy simulation. The reduction of normal scheme has been madeby transposing it in a scheme in which the elements are connected in seriesor parallel considering the dimensioning and the connection of elements. Allfeeds for Padurea Neagra consumer, on all path, from the source have beenconsidered.
In Figures 4-5 input data of analyzed electric station are presented. In Fig-ure 6 the obtained membership function diagram are presented for ES Voivozi.Similarly we have representations of last nine elements of ERD. For computingoutput membership functions we start from the system scheme from which thefailure tree is generated - Figures 7, 8.The characteristic equation of the system is deduced from the schemes pre-sented in Figures 7, 8 and is given by relation
FV OI = 1−(1−F1F2)(1−F3)(1−F4F5)(1−F6F7)(1−F8)(1−F9)(1−F10) (4)
Relation (4) is used by the program to generate the output membershipfunction presented in Figure 9. In Figure 10 is presented, for example, a deci-sion surface. The program displays the obtained output values in a separatewindow presented in Figure 11.
A fuzzy algorithm for reliability simulation of an electric station 55
Fig. 5. Simulation data editing window for ES Voivozi.
Fig. 6. Membership functions for element 1.
Fig. 7. The reduced scheme for ES Voivozi.
56 Simona Dzitac, Tiberiu Vesselenyi, Ioan Dzitac, Maria Parv
Fig. 8. The failure tree for ES Voivozi.
Fig. 9. Output membership functions generated for the analyzed system.
Fig. 10. Output values for inputs 1 and 2.
A fuzzy algorithm for reliability simulation of an electric station 57
Fig. 11. Reliability output window for the reliability of ES Voivozi.
4. CONCLUSIONSThe use of fuzzy sets theory in the study of the reliability of the electric
energy systems and equipments is justified by the possibilities offered by thequantification and the modeling of the qualitative enounces - incomplete andaltered information, subjective appreciations - in flexible forms, more close tothe way of thinking that the engineers operates with. The program developedunder MATLAB environment for the fuzzy simulation of reliability of electri-cal equipments permits the step by step definition of the fuzzy model and it isrealized in a versatile manner, object oriented and modular. The program canmake diverse simulations, in small times, for a given scheme, in the analyzedfuzzy intervals making possible the visualization of values range in which thenon-reliability and the reliability of the system can evolve. In Table 2 wecan see that the realized evaluations, obtained with the ES reliability fuzzysimulation program, are accurate, in comparison with the values obtained byMonte Carlo method and the direct ERD computation [2, 4].
ES/R FUZZY MONTE CARLO (10.000 sim.) ERD
VOIVOZI 0,99942759 0,99951 0,99947
Table 2.
The development of ES fuzzy reliability simulation program by using theMATLAB programming environment, based on the failure tree method, ap-plication of this program for ES Voivozi, Bihor County, and the comparativeevaluation with the Monte Carlo simulation method results and with the ERDanalytical method results, are the contributions of the authors in this article.
58 Simona Dzitac, Tiberiu Vesselenyi, Ioan Dzitac, Maria Parv
References[1] R. Billinton, R. Allan, Reliability Evaluation of Power Systems, Plenum Press, New
York and London, 1984.
[2] S. Dzitac, Contributions to modeling and simulation of reliability and disposability per-formance simulation for electric energy distribution systems, PhD Thesis, Universityof Oradea, 2008.
[3] S. Dzitac, C. Hora, Fuzzy simulation in reliability analysis, Ann. of the Faculty of Eng.Hunedoara - Journal of Engineering, VII, 3(2009), 318-323.
[4] S. Dzitac, I. Felea, Application of fuzzy modeling in reliability analysis of the electricstations, International World Energy System Conference, Iasi, June 30 -July 2, 2008.
[5] S. Dzitac, T. Vesselenyi, I. Dzitac, E. Valeanu, Electrical Power Station Reliability Mod-eling Procedure using The Monte Carlo Method, IFAC MCPL 2007, 4th IFAC Confer-ence on Mangement and Control of Production and Logistic, Sibiu 27-30 September ,Romania, September 27-30, 2007, Preprints, ISBN 978-973-739-481-1, vol. III, 625-700.
[6] I. Felea, N. Coroiu,Reliability and maintenance of electrical equipment, Ed. Tehnica,Bucuresti, 2001.
[7] E. Sofron , N. Bizon, S. Ionica, R. Raducu, Fuzzy control systems. Computer aidedmodeling and design, All, Bucuresti, 1998.
[8] H. Tanaka, L. Fan, F. Lai, K. Toguchi, Fault-Tree Analysis by fuzzy probability, IEEETransactions on Reliability, 32(5), 453-457, 1983.
[9] T. Vesselenyi, S. Dzitac, I. Dzitac, M.-J. Manolescu, Fuzzy and Neural Controllers fora Pneumatic Actuator, Int. J. of Computers, Communications & Control (IJCCC), 2,4(2007), 375-387, 2007.
[10] L.A. Zadeh, D. Tufis, F.G. Filip, I. Dzitac (eds.), From Natural Language to Soft Com-puting: New Paradigms in Artificial Intelligence, Editura Academiei Roamane, 2008.
A SEVEN EQUATION MODEL FORRELATIVISTIC TWO FLUID FLOWS-I
ROMAI J., 5, 2(2009), 59–70
Sebastiano Giambo, Serena GiamboDepartment of Mathematics, University of Messina, [email protected], [email protected]
Abstract An interface-capturing method is used to describe relativistic two-fluid flows.The conservation equations for the particle number of each fluid and for thetotal momentum-energy tensor of the mixture are the starting point of thisapproach. A model for relativistic two-fluid flow without friction and heatconduction and differential equations, plus additional algebraic relations, con-sistent with this model, are derived. The weak discontinuities propagatingin this relativistic two-fluid system are examined and the expressions for thespeeds of propagation are obtained.
Keywords: general relativity, relativistic fluid dynamics, two-fluid mixtures, nonlinear
waves.
2000 MSC: 83C99, 80A10, 80A17, 76T99, 74J30.
1. INTRODUCTIONThere are many topics in General Relativity where matter is represented as
a mixture of two fluids. In fact, some astrophysical and cosmological situationsneed to be described by an energy tensor consisting of the sum of two or moreperfect fluids. For most of the history of the universe, the dominant mattercontent is a mixture of matter and radiation [1]-[12]; other examples are a nullfluid with string fluid [13], or a radiation fluid in addiction to a string fluid[14]-[17]. It was also shown [18]-[22] that an anisotropic relativistic fluid canbe consistently described by two-perfect-fluid components and inflationarymodels have been deduced as mixtures of two relativistic fluids [23], [24].Moreover, the acoustic modes [25] and the wave fronts [26], [27] have beenstudied in some of the cases quoted above.
The purpose of this paper is to build up a relativistic formulation of somerecent results on the classical dynamics of a mixture of two perfect fluids basedon the papers of H. Guillard and A. Murrone [28]-[30]. The model presentedhere is a two-phase flow model, in which the entire flow domain is filled witha mixture of the two fluids. However, in this underlying two-phase model, thefluids are not mixed on the molecular level: the mixture consists of very smallelements of the two pure fluids, arranged in an irregular pattern. So the fluidis a mixture in the macroscopic sense.
59
60 Sebastiano Giambo, Serena Giambo
Moreover, both fluids are assumed to be present everywhere in the flowdomain and the interfaces between the two fluids are considered as gradualtransitions from fluid 1 to fluid 2. In this way, the concept of interface betweenthe two fluids disappears from the model [28]-[32].
Each fluid still has its own particle number density, rk, its specific internalenergy, εk, and its energy density ρk, [33], [34]:
ρk = rk (1 + εk) , k = 1, 2. (1)
In what follows, the units are such that the velocity of light is unitary: c = 1.Conversely, a single pressure, p, and a single four-velocity, uα, are assumedfor the two fluids. Here, uα is the fluid unit four-vector defined to be future-pointing
gαβuαuβ = 1, (2)
where gαβ are the covariant components of Lorentz metric tensor with signa-ture +,−,−,−.
In this paper we derive the complete system of governing differential equa-tions and we determine the propagation speed of weak discontinuity wavefrontsin this relativistic two-fluid model.
The paper is organized as follows. Section 2 starts with a description ofthe relativistic mixture and the derivation of the flow equations. Section 3analyzes the source term appearing in the flow equations. In Section 4, theevolution equation for the pressure is derived in order to perform the closureof the system of differential equations obtained in Section 2. In Section 5, thepropagation of weak discontinuities admitted by the model under considerationare examined and the expressions for their speeds of propagation are obtained.Section 6 concerns a special case, that may be physically relevant, in whichthe expression of the velocity is the relativistic version of the Wallis formula[39].
2. RELATIVISTIC FLOW MODELThe standard equations for simple relativistic fluid flow hold for the two-
fluid model. The total energy-momentum conservation is
∇αTαβ = 0 , (3)
where the stress energy tensor is given by
Tαβ = rfuαuβ − pgαβ, (4)
being r the total particle number density, f the relativistic total specific en-thalpy
f = 1 + h = 1 + ε +p
r, (5)
A seven equation model for relativistic two fluid flows-I 61
with h = ε+ pr the “classical” specific enthalpy, of the mixture, ε and p denoting
the total energy density and pressure, respectively.Moreover, the balance law for the total particle number is
∇α (ruα) = 0 . (6)
The projection of equation (3) along uα is
uβ∇αTαβ ≡ uα∂αρ + (ρ + p)∇αuα = 0, (7)
being ρ = r (1 + ε) the total energy density, whereas the spatial projection ofequation (3) is
γλβ∇αTαβ ≡ rfuα∇αuλ − γαλ∂αp = 0, (8)
where γαβ = gαβ − uαuβ is the projection tensor onto the three-space orthog-onal to uα (the rest space of an observer moving with four-velocity uα).
However, we have to determine suitable expressions for the bulk quantitiesr, ε, ρ and f .
First, the volume fraction X and the mass fraction Y ,
Y =Xr1
r(9)
of fluid 1 are chosen as field variables. The variables X and Y allow to defineany bulk quantity; the particle number density r, the specific internal energyε, the energy density ρ and the relativistic specific enthalpy f are defined as:
r = X1r1 + X2r2 ,
ε = Y1ε1 + Y2ε2 ,
f = Y1f1 + Y2f2 ,
ρ = X1ρ1 + X2ρ2 ,
rf = X1r1f1 + X2r2f2 ,
(10)
with
X1 = X , X2 = 1−X ,
Y1 = Y , Y2 = 1− Y .(11)
Thus, for regular solutions, the mathematical study of the model can be per-formed using the following set of seven independent field variables uα, r, p, X,Y . The governing system (6)-(8) is a set of five equations for seven variables.Thus, two more equations are to be determined in order to close the system.
62 Sebastiano Giambo, Serena Giambo
Since all the bulk equations have already been used, the only option is totake into account quantities characterizing one of the two fluids.
The first equation to be considered is, of course, the balance law for theparticle number density for fluid 1. Using the partial density X1r1, the corre-sponding equation is
∇α (Xr1uα) = 0. (12)
From the conservation equation (12) (written using the relation Xr1 = Y r)
∇α (Y ruα) = 0 ,
taking into account equation (6), we obtain the following evolution law for thevariable Y
uα∂αY = 0 . (13)
Observe that equation (12), together with equation (6), implies the followingbalance law for the particle number density of fluid 2
∇α (X2r2uα) = 0 , (14)
where, as already said, X2 = 1−X.At this point, it is clear that the only option in order to get one more
equation, and then the closure of the governing system, is to determine thebalance equation for the energy-momentum tensor of fluid 1
Tαβ1 = (ρ1 + p) uαuβ − pgαβ . (15)
Since exchanges of energy and momentum between the fluid components areallowed, there will be no local energy-momentum conservation for each fluidcomponent separately. Then, the equation for the energy-momentum tensorsof each of the two fluids, Tαβ
k , k = 1, 2, has the following form
∇α
(XkT
αβk
)= F β
k , k = 1, 2 , (16)
where F βk represents the loss and source term in the separate balance. Now,
since the total energy-momentum tensor is conserved, according to (3), andtaking into account the expression (15) of Tαβ
1 and the relation
Tαβ = X1Tαβ1 + X2T
αβ2 (17)
it is easily shown thatF β
1 = −F β2 = F β .
A seven equation model for relativistic two fluid flows-I 63
3. DERIVATION OF THE SOURCE TERMThis section is devoted to handling the source term F β in equations (16).The projection along uα and the spatial projection of equation (16) for fluid
1 are, respectively,
Xuα∂αρ1 + ρ1uα∂αX + X (ρ1 + p)∇αuα = uαFα (18)
andX
(ρ1 + p) uα∇αuβ − γαβ∂αp
− pγαβ∂αX = γβ
αFα . (19)
Equation (18), taking into account equations (1) and (12), yields the followingequation
Xr1uα
(∂αε1 + p ∂α
1r1
)= p uα∂αX + uαFα . (20)
We assume the following axiom: the entropy Sk of each fluid component isa function of the energy εk and the specific volume 1/rk
Sk = Sk (εk, rk) , k = 1, 2 . (21)
By thermodynamic arguments, the derivatives of entropy can be related tosome observable variables. Thus, we can write
(∂Sk
∂εk
)
rk
=1Tk
,
(∂Sk
∂rk
)
εk
= − p
r2kTk
,
(22)
where Tk is the temperature of fluid component k. From equation (??), itfollows that
TkdSk = dεk + pd1rk
(23)
and then
Tkuα∂αSk = uα
(∂αεk + p ∂α
1rk
). (24)
We now also suppose that the entropy Sk is conserved along the flow lines
uα∂αSk = 0 , (k = 1, 2) .
Thus, from equation (24) we can deduce that
uα
(∂αεk + p ∂α
1rk
)= 0 (25)
64 Sebastiano Giambo, Serena Giambo
and equation (20) allows to write the following relation involving Fα
uαFα = −p uα∂αX . (26)
Next, using equations (19) and (8), we obtain
X (r1f1 − rf) γαβ∂αp− prfγαβ∂αX = rfγβαFα . (27)
Now, introducing the relativistic enthalpy concentration
χ =f1
fY ,
we haver1f1 =
χ
Xrf , (28)
thus, from (27) we deduce
γαβFα = (χ−X) γαβ∂αp− pγαβ∂αX . (29)
Therefore, using equations (26) and (29), the source term Fβ can now becomputed as
Fβ = (χ−X) γαβ∂αp− p∂βX , (30)
which represents the relativistic formulation of the classical expression of thesource terms obtained by Wackers and Koren [31], [32].
4. PRESSURE EQUATIONThe derivation of a pressure equation is rather involved, as it requires the
two energy equations (7) and (18). From equation (7), because ρ = r(1 + ε),we deduce that
r2uα∂αε + ρuα∂αr + r2f∇αuα = 0 . (31)
The total specific internal energy can be expressed in terms of variables r, p,X and Y by an equation of state. For this analysis, we use equations of stateof the most general form, writing it as
ε1 = ε1 (r1, p) , ε2 = ε2 (r2, p) , (32)
withr1 =
Y
Xr , r2 =
1− Y
1−Xr . (33)
Substituting (32) in equation (10)2, the bulk specific internal energy ε can bewritten as
ε (r, p, X, Y ) = Y ε1 (r1, p) + (1− Y ) ε2 (r2, p) . (34)
A seven equation model for relativistic two fluid flows-I 65
Now, thanks to this last expression (34) of ε, and using equations (13) and(6), the following form of the bulk energy equation (31) is deduced
r∂ε
∂puα∂αp + r
∂ε
∂Xuα∂αX +
(p− r2 ∂ε
∂r
)∇αuα = 0 . (35)
Conservation of energy for fluid 1 (equation (20)), with equation (26) and (12),becomes
r1uα∂αε1 +
p
Xuα∂αX + p∇αuα = 0 , (36)
and, using (33)1 and taking into account the equation of state (32)1, equation(36) writes as
r1∂ε1
∂puα∂αp +
(p
X+ r1
∂ε1∂X
)uα∂αX +
(p− rr1
∂ε1∂r
)∇αuα = 0 . (37)
From equations (35) and (37), we are able to deduce the following evolutionequations for the pressure p and the volume fraction α, respectively
uα∂αp + ω∇αuα = 0 ,
uα∂αX + ξ∇αuα = 0 ,(38)
where ω and ξ are defined by
ω =
(p− r2 ∂ε
∂r
) (r1
∂ε1∂X + p
X
)− r ∂ε
∂X
(p− r1r
∂ε1∂r
)
r ∂ε∂p
(r1
∂ε1∂X + p
X
)− rr1
∂ε∂X
∂ε1∂p
,
ξ =r ∂ε
∂p
(p− rr1
∂ε1∂r
)− r1
∂ε1∂p
(p− r2 ∂ε
∂r
)
r ∂ε∂p
(r1
∂ε1∂X + p
X
)− rr1
∂ε∂X
∂ε1∂p
.
(39)
To end this section, we note that the complete system of governing differ-ential equations may be written in term of variables (uα , r, p, X, Y ) as
uα∂αr = −r∇αuα ,
rfuα∇αuβ = γαβ∂αp ,
uα∂αp = −ω∇αuα ,
uα∂αX = −ξ∇αuα ,
uα∂αY = 0 .
(40)
66 Sebastiano Giambo, Serena Giambo
5. DISCONTINUITIESIn a domain Ω of space-time V4, let Σ be a regular hypersurface, not gener-
ated by the flow lines, being ϕ (xα) = 0 its local equation. We set Lα = ∂αϕ.As it will be clear below, the hypersurface Σ is a space-like one, i.e. LαLα < 0.In the following, Nα will denote the normalized vector
Nα =Lα√−LβLβ
, NαNα = −1 .
We consider a particular class of solutions of system (40) namely, weak dis-continuity waves Σ, on which the field variables uα, r, p, X, Y are continuous,but, conversely, jump discontinuities may occur in their normal derivatives.In this case, if Q denotes any of these fields, then there exists [33],[38] thedistribution δQ, with support Σ, such that
δ [∇αQ] = NαδQ ,
where δ is the measure of Dirac defined by ϕ with Σ as support, square bracketsdenote the discontinuity, δ being an operator of infinitesimal discontinuity; δbehaves like a derivative insofar as algebraic manipulations are concerned.
Then, from the system (40), we obtain the following linear homogeneoussystem in the distributions Nαδuα, δr, δp, δX and δY
Lδr + rNαδuα = 0 ,
rfLδuα − γαβNβδp = 0 ,
Lδp + ωNαδuα = 0 ,
LδX + ξNαδuα = 0 ,
LδY = 0 ,
(41)
where L = uαNα. Moreover, from the unitary character of uα we have
uαδuα = 0 . (42)
Now, we want to investigate the normal speeds of propagation of the variouswaves with respect to an observer moving with the mixture velocity uα. Thenormal speed λΣ of propagation of the wave front Σ, described by a timelikeword line having tangent vector field uα, that is with respect to the timedirection uα, is given by [33]-[38]
λ2Σ =
L2
1 + L2. (43)
A seven equation model for relativistic two fluid flows-I 67
The local causality condition, i.e. the requirement that the characteristichypersurface Σ has to be timelike or null (or equivalently that the normalNα be spacelike or null, that is gαβNαNβ ≤ 0), is equivalent to the condition0 ≤ λ2
Σ ≤ 1.From the above equations (41), we first obtain the solution L = 0, which
represents a wave moving with the mixture. For the corresponding disconti-nuities we find
Nαδuα = 0 , δp = 0 . (44)Since the coefficients characterizing the discontinuities exhibit five degrees offreedom, then system (41) admits five independent eigenvectors correspondingto L = 0 in the space of the field variables.
From now on we suppose L 6= 0. Equation (41)2, multiplied by Nβ, give us
rfLNαδuα − l2δp = 0 , (45)
where l2 = 1 + L2.As a consequence, (41)3 and (45) represent a linear homogeneous system
in the two scalar distributions Nαδuα and δp, which may have non trivialsolutions only if the determinant of the coefficients vanishes. Therefore, wefind the equation
H ≡ rfL2 − ωl2 = 0 , (46)which corresponds to the hydrodynamical waves propagating in such a two-fluid system. Their speeds of propagation are given by
λ2Σ =
ω
rf(47)
and the condition 0 < ωrf ≤ 1 ensures their spatial orientation.
The associated discontinuities can be written in terms of ψ = nαδuα asfollows
δuα = −ψnα ,
δr = −rl
Lψ ,
δp = −ωl
Lψ ,
δX = −ξl
Lψ ,
δY = 0 ,
(48)
where nα is the unitary space-like four-vector defined by
nα =1l
(Nα − Luα) . (49)
68 Sebastiano Giambo, Serena Giambo
Observe that if the above condition characterizing the space-like orientations ofthe surface is verified, then the governing equations represent a (not strictly)hyperbolic system. In fact, all velocities (eigenvalues) are real, and thereis a complete set of eigenvectors in the space of field variables, i.e. sevenindependent eigenvectors (5 from L = 0 and 2 from H = 0), for the sevenindependent field variables uα, r, p, X and Y .
6. APPLICATIONNow, we want to examine the application of the preceding method in order
to determine weakly discontinuous solutions in the case of a mixture of twofluids of cosmological interest. To this end, we assume that each fluid (k = 1, 2)satisfy the equation of state of perfect gases:
εk =1
γk − 1Xk
Yk
p
r, k = 1, 2, (50)
where γk is the ratio of the specific heat capacities at constant pressure andvolume of the k-th fluid. Using (50), (10)2 writes as
ε =(
X
γ1 − 1+
1−X
γ2 − 1
)p
r. (51)
Then, (39) can be written, respectively, in the following form
ω =γ1γ2
Xγ2 + (1−X) γ1
p ,
ξ = X (1−X)γ1 − γ2
Xγ2 + (1−X) γ1
.
(52)
Replacing this last expression (46)1 of ω into the equation (47), we get
λ2Σ =
1rf
γ1γ2
X1γ2 + X2γ1
p . (53)
Recalling that the normal speeds of propagation of hydrodynamical waves, λk,of each fluid k is given by
λ2k = γk
p
rkfk, k = 1, 2, (54)
equation (53) can be rewritten under the form
1rfλ2
Σ
=X1
r1f1λ21
+X2
r2f2λ22
. (55)
Equation (55) represents the relativistic generalization of the formula due toWallis [39], allowing to express the speed of acoustic modes for a two-fluidsystem as combination of the individual speeds of acoustic modes in eachspecies.
A seven equation model for relativistic two fluid flows-I 69
References
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[4] K. C. Jacobs, Friedmann cosmological model with both radiation and matter, Nature,215 (1967), 1156-1157.
[5] A. A. Coley, B. O. J. Tupper, Viscous fluid collapse, Phys. Rev., D 29 (1984), 2701-2704.
[6] A. A. Coley, D. J. McManus, New slant on tilted cosmology, Phys. Rev., D 54 (1996),6095-6100.
[7] K. Dunn, Two-fluid cosmological models in Godel-type spacetimes, Gen. Rel. Grav., 21(1989), 137-147 .
[8] J. J. Ferrando, J. A. Morales, M. Portilla, Two-perfect fluid interpretation of an energytensor, Gen. Rel. Grav., 22 (1990), 1021-1032.
[9] V. Husain, Exact solutions for null fluid collapse, Phys. Rev., D 53 (1996), 1759-1762.
[10] J. P. S.Lemos, Collapsing shells of radiation in anti-de Sitter spacetimes and the hoopand cosmic censorship conjectures, Phys. Rev., D 59 (1999), 044020 [gr-qc/9812078].
[11] A. Wang, Y. Wu, Generalized Vaidya solutions, Gen. Rel. Grav., 31 (1999), 107-114.
[12] T. Harko, K. S. Cheng, Collapsing strange quark matter in Vaidya geometry, Phys.Lett., A 266 (2000), 249-253.
[13] E. N. Glass, J. P. Krisch, Radiation and string atmosphere for relativistic stars, Phys.Rev., D 57 (1998), 5945-5947.
[14] E. N. Glass, J. P. Krisch, Two-fluid atmosphere for relativistic stars, Class. QuantumGrav., 16 (1999) 1175-1184.
[15] E. N. Glass, J. P. Krisch, Scale symmetries of spherical string fluids, J. Math. Phys.,40 (1999), 4056-4063.
[16] S. Kar, Stringy black holes and energy conditions, Phys. Rev., D 55 (1997), 4872-4879.
[17] F. Larsen, A string model of black hole microstates, Phys. Rev., D 56 (1997), 1005-1008.
[18] P. S. Letelier, Anisotropic fluids with two-perfect fluid components, Phys. Rev., D 22(1980), 807-813.
[19] P. S. Letelier, Solitary waves of matter in general relativity, Phys. Rev., D 26 (1982),2623-2631 .
[20] S. S. Bayin, Anisotropic fluid spheres in general relativity, Phys. Rev., D 26 (1982),1262-1274 .
[21] T. Harko, M. K. Mak, Anisotropic relativistic stellar models, arXiv:gr-qc/0302104 v126 Feb 2003.
[22] P. S. Letelier, P. S. C. Alencar, Anisotropic fluids with multifluid components, Phys.Rev., D 34 (1986), 343-351.
70 Sebastiano Giambo, Serena Giambo
[23] W. Zimdahl, Reacting fluids in the expanding universe: a new mechanism for entropyproduction, M on. Not. R. Astron. Soc., 288 (1997), 665-673.
[24] W. Zimdahl, D. Pavon, R. Maartens, Reheating and causal thermodynamics, Phys.Rev., D 55 (1997), 4681-4688.
[25] J. P. Krisch, L. L. Smalley, Two fluid acoustic modes and inhomogeneous cosmologies,Class. Quantum Grav., 10 (1993), 2615-2623.
[26] M. Cissoko, Wavefronts in a relativistic cosmic two-component fluid, Gen. Rel. Grav.,30 (1998), 521-534
[27] M. Cissoko, Wave fronts in a mixture of two relativistic perfect fluids flowing with twodistinct four-velocities, Phys. Rev., D 63 (2001), 083516.
[28] A. Murrone, Modeles bi-fluides a six et sept equations pour les ecoulement diphasiquesa faible nombre de Mach, PhD Thesis, Universite de Provence, Aix-Marseille I, 2004.
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A GENERAL MOUNTAIN-PASS THEOREMFOR LOCAL LIPSCHITZ FUNCTIONS
ROMAI J., 5, 2(2009), 71–77
Georgiana Goga” N. Rotaru” College, Constanta, Romaniageorgia [email protected]
Abstract A variant for locally Lipschitz functions of a general mountain pass principle,due to Ghoussoub and Preiss, which carries some information on the locationof critical points, is proved by using the Borwein-Preiss variational principle asthe main variational tool.
Keywords: critical points, locally Lipschitz functionals, Borwein-Preiss variational princi-
ple, Clarke subdifferential.
2000 MSC: 54C60, 58E30, 49J35, 49J52.
1. INTRODUCTIONIn this paper we present a new proof of a general mountain-pass theorem
for locally Lipschitz functions established in [5] , giving information about thelocation of the critical points for these kind of functions. Unlike the methodof localization used in [5], which replaces the Ghoussoub-Preiss techniques [6]by Ekeland’s variational principle, our proof is based on the Borwein-Preissvariational principle [1] and a lemma of Choulli, Deville and Rhandi [3] .
Let (E, ‖.‖) be a Banach space, S be a compact metric space, S0 be a closedsubset of S, C (S, E) be the Banach space of all E- valued bounded continuousmappings on S with the norm ‖γ‖ := sup
x∈S‖γ (x)‖ . Let γ0 ∈ C (S, E) be a fixed
element and put
Γ = γ ∈ C (S,E) | γ (s) = γ0 (s) , ∀s ∈ S0 ,
c = infγ∈Γ
sups∈S
f (γ (s)) ,
where f is a real-valued function defined on E.We give now our main result.
Theorem 1.1. . Let f : E → R be a locally Lipschitz function and F be aclosed nonempty subset of E. Assume that
a) γ (S) ∩ F ∩ Fc 6= ∅, ∀ γ ∈ Γ, where Fc = x ∈ E | f (x) ≥ c ,
71
72 Georgiana Goga
b) dist (γ0 (S0) , F ) > 0, where dist (., F ) is the distance function to F in E.
Then for every ε > 0 there exists xε ∈ E such that
i) dist (xε, F ) <3ε2 ,
ii) c ≤ f (xε) < c+5ε2
4 ,
iii) dist (0, ∂f (xε)) ≤ 2ε, where ∂f (x)is the Clarke subdifferential of f at x.
Before the start of the proof of theorem 1, we recall some known results usedin the sequel (the proofs can be found in standard convex analysis literature).
2. PRELIMINARIES
Let X be a Banach space and Φ : X → R a locally Lipschitz functional. Foreach x, v ∈ X, the generalized directional derivative of Φ at x in the directionv is
Φ (x; v) = limy→x,
supt0
Φ(y + tv) − Φ(y)t
.
It follows, by the definition of locally Lipschitz functionals, that Φ (x; v) isfinite and
|Φ (x; v)| ≤ kx ‖v‖ ,
where kx is the Lipschitz constant of Φ on a neighborhood of x.Moreover, v → Φ (x; v) is positively homogenous and subadditive and
(x; v) → Φ (x; v) is upper semi-continuous.The generalized gradient (Clarke subdifferential) of Φ at x is the multifunc-
tion ∂Φ : X → P (X∗) = 2X∗defined by
∂Φ(x) = x∗ ∈ X∗ | Φ (x; v) ≥ 〈x∗, v〉 , for all v ∈ X ,
where X∗ is the dual space of X. It enjoys the following properties:1) For each x ∈ X, ∂Φ(x) is a non-empty, convex weak * compact subset of
X∗.
2) For each x, v ∈ X, we have Φ (x; v) = max 〈x∗, v〉 | x∗ ∈ ∂Φ(x) .
3) ∂ (Φ + Ψ) (x) ⊂ ∂Φ + ∂Ψ, where Φ and Ψ are locally Lipschitz at x.
Theorem 2.1. (Lebourg mean value theorem). If x and y are two distinctpoints in X, then there exists z = (1− t) x + ty, 0 < t < 1, such that
Φ(y)− Φ(x) ∈ 〈∂Φ (z) , y − x〉 .The notion of critical point of a locally Lipschitz functional is the following.
A general mountain-pass theorem for local Lipschitz functions 73
Definition 2.1. Let Φ : X → R be locally Lipschitz. A point x ∈ X is acritical point of Φ if 0 ∈ ∂Φ(x) . A real number c is called a critical valueof Φ if Φ−1 (c) contains a critical point x.
We recall now some results on multivalued mappings. Let M and N be twotopological spaces.
Definition 2.2. A multivalued mapping T : M → 2N is a map whichassigns to each point m ∈ M a subset T (m) of N.
Definition 2.3. Let T : M → 2N be a multivalued mapping.
T is upper semi-continuous (u.s.c.) if T−1 (A) is closed for all closedsubsets A of N, where the preimage T−1 (A) is defined by
T−1 (A) = m ∈ M | T (m) ∩A 6= ∅ .
T is lower semi-continuous (l.s.c.) if T−1 (A) is open for all opensubsets A of N.
T is continuous if is both lower and upper semi-continuous.
Lemma 2.1. (Choulli, Deville, Rhandi). Let E be a Banach space, S bea compact metric space F : S → 2E∗ be a weak * - upper semicontinuousmultivalued mapping with weak *- compact and convex values and ε > 0.Denote:
µ = inf ‖x∗‖∗ | x∗ ∈ F (s) , s ∈ S .
Then there exists a continuous function h : S → E satisfying
1) ‖h (s)‖ ≤ 1, ∀s ∈ S,
2) 〈x∗, h (s)〉 ≥ µ− ε, ∀s ∈ S, ∀x∗ ∈ F (s) .
Now recall the Borwein-Preiss variational principle, which is the variationaltool of this article.
Theorem 2.2. (Borwein-Preiss variational principle). Let E be a Banachspace with the norm ‖·‖ ,and f : E → R ∪ +∞ be a l.s.c. function boundedfrom below, let λ > 0 and let p ≥ 1. Assume that ε > 0 and z ∈ E satisfy
f (z) < infE
f + ε.
Then there exist y and a sequence (xi) in E with x1 = z and a functionϕp : E → R of the form
ϕp (x) :=∞∑i=1
µi ‖x− xi‖p ,
74 Georgiana Goga
where µi > 0 for all i = 1, 2, . . . and∞∑i=1
µi = 1 such that
a) ‖xi − y‖ ≤ λ, i = 1, 2, . . . ,
b) f (y) +(
ελp
)ϕp (y) ≤ f (z) , and
c) f (x) +(
ελp
)ϕp (x) > f (y) +
(ε
λp
)ϕp (y) , for all x ∈ E \ y .
3. PROOF OF THE MAIN RESULT
Proof of Theorem 1.1. Let dist (γ0 (S0) , F ) > ε(b)> 0. Take γ′ε ∈ Γ such
thatsups∈S
f(γ′ε (s)
)< c + ε2 (1)
and putS1 =
s ∈ S; dist
(γ′ε (s) , F
) ≥ ε
,
Γr =y ∈ C (S, E) ;
∥∥γ (s)− γ′ε (s)∥∥ ≤ r, ∀s ∈ S0
,
cr = infγ∈Γr
sups∈S′
f (γ (s)) ,
where S′ = S \ S1. By (a), we obtain that there exists xε ∈ γ (S) ∩ F suchthat f (xε) ≥ c. Since xε ∈ γ (S) , there exists s ∈ S such that xε = γ (s) ,therefore
f (xε) = f (γ (s)) ≥ c. (2)
Now we prove the following.Claim. : c + ε2 > cr ≥ c, ∀r ∈ (0, ε) .Proof. We have
cr ≤ sups∈S′
f(γ′ε (s)
) ≤ sups∈S
f(γ′ε (s)
) (1)< c + ε2,
socr < c + ε2.
We assume now that cr < c. Then, for ε′ = 12 (c− cr) > 0, there exists γ1 ∈ Γr
such that
sups∈S′
f (γ1 (s)) < cr + ε′ = cr +12
(c− cr) =12
(c + cr) < c,
thereforef (γ1 (s)) ≤ sup
s∈S′f (γ1 (s)) < c,
A general mountain-pass theorem for local Lipschitz functions 75
which is in contradiction with (2) . Then, we have cr ≥ c and the Claim isproved.2
Let r = ε8 . By the Claim, since S′ ⊂ S, we have
sups∈S′
f(γ′ε (s)
) ≤ sups∈S
f(γ′ε (s)
)< c + ε2 ≤ cr + ε2. (3)
By compactness of S′, it is easy to prove that the function Φ : Γr → R, Φ (x) =sups∈S′
f (γ (s)) is locally Lipschitz in C (S, E) and, by (3) , we have
Φ(γ′ε
) ≤ infΓr
Φ + ε2.
Now, applying the Borwein-Preiss variational principle for the function Φon Γr with p = 2 and λ = ε
2 > 0, we obtain: there exists γε and a sequence(γi) in Γr with γ1 = γ′ε, and a function ϕ2 : Γr → R, defined by
ϕ2 (γ) :=∞∑i=1
µi ‖γ − γi‖2 ,
where µi > 0,∞∑i=1
µi = 1, such that
‖γi − γε‖ ≤ λ, i = 1, 2, . . . , (4)
Φ (γε) +(
ε2
λ2
)ϕ2 (γε) ≤ Φ
(γ′ε
), (5)
Φ (γ) +(
ε2
λ2
)ϕ2 (γ) > Φ(γε) +
(ε2
λ2
)ϕ2 (γε) , ∀γ ∈ Γr \ γε . (6)
Now, (4) shows that γi → γε in Γr and, by (6) , γε is a local minimum ofthe function
Φ (γ) := Φ (γ) +(
ε2
λ2
)ϕ2 (γ)
on C (S, E) . By the necessary conditions for a local minimum and the formulafor the Clarke directional derivative of the sum of two function, we obtain
0 ≤ Φ (γε; v) ≤ Φ (γε; v) + 8 limγ→γε
∞∑i=1
µi (γ − γi) · v, (7)
∀v ∈ C (S, E) . We use now Lemma 6, with F (·) = ∂f (γε (·)) and we obtainthat there exists a continuous function h : S′ → E such that
f (γε (s) ;h (s)) ≤ −d + ε, (8)
76 Georgiana Goga
‖h (s)‖ ≤ 1, ∀s ∈ S′,
where d = infs∈S′
dist (0, ∂f (γε)) . Since γi → γε, for ti 0 we have
δi :=Φ (γi + tih) − Φ(γi)
ti→ Φ (γε;h) (9)
and ∞∑i=1
µi (γε − γi) → 0. (10)
Then, (7) and (10) imply
0 ≤ Φ (γε;h) ≤ Φ (γε;h) . (11)
Let si be a maximum point of the function f1 (γi + tih, ·) over S′, i.e.
f1 (γi + tih, si) = maxs∈S′
f1 (γi + tih, s) .
Then by the theorem 2, we have
δi ≤ f1 (γi + tih, si) − f1 (γi , si)ti
= 〈γ∗i , h〉 ≤ max 〈γ∗i , h〉 = f1 (γi, si;h) ,
where γ∗i ∈ ∂γf1 (γi, si) , γi ∈ [γi, γi + sih] . By (9) and (11) we obtain
0 ≤ Φ (γε; h) ≤ Φ (γε; h) ≤ f1 (γi, si; h) .
It is easy to see that f (γ (t) ; v (t)) = f1 (γ, t; v) , ∀γ, v ∈ C (S, E) , t ∈ S. Bycompactness of S′, we assume that si → sε ∈ S′ and since γi → γε, by uppersemicontinuity of f (·, ·) , we obtain
Φ (γε;h) ≤ f (γε (sε) ; h (sε))(8)
≤ −d + ε. (12)
By Lemma 6, we have
〈γ∗i (sε) , h (sε)〉 ≥ µ− ε,
which means thatf (γε (sε) ; h (sε)) ≥ µ− ε. (13)
Then, (12) and (13) give us
µ− ε ≤ −d + ε ⇐⇒ µ < 2ε,
and we have dist (0, ∂f (γε)) < 2ε, which is (iii) .
A general mountain-pass theorem for local Lipschitz functions 77
Since sε ∈ S′ = S \ S1, we have sε /∈ intS1, therefore
dist (xε, F ) ≤ dist(γ′ε (sε) , F
)+
∥∥γ′ε (sε)− γε (sε)∥∥
≤ ε + λ =3ε
2,
which is (i) . Now by (5) , we have
f(γ′ε
) ≤ supsε∈S′
f (γε (sε)) + 4∞∑i=1
µi ‖γε (sε)− γi (sε)‖2
≤ supsε∈S′
f (γε (sε)) + 4∞∑i=1
µi
(∥∥γε (sε)− γ′ε (sε)∥∥ +
∥∥γ′ε (sε)− γi (sε)∥∥)2
≤ supsε∈S′
f (γε (sε)) + 16r2
(1)
≤ c + ε2 + 16 · ε2
64
= c +5ε2
4,
and (ii) is proved. 2
References[1] J. Borwein, Q. Zhu, Techniques of Variational Analysis, Springer, New York 2005
[2] H. Brezis, N. Nirenberg, Remarks of finding critical points, Comm. Pure Appl. Math.,Vol. XLIV (1991), 939-963.
[3] M. Choulli, R. Deville, A. Rhandi, A general mountain pass principle for nondifferen-tiable functions and applications, Rev. Math. Apl., 13 (1992),45-58.
[4] F. H. Clarke, Non-smooth analysis and optimization, Wiley - Interscience 1983.
[5] P. Georgiev, A short proof of a general mountain-pass theorem for locally Lipschity func-tions, Preprint, IC/95/369, International Centre for Theoretical Physics, Miramare-Trieste, 5 pp. (1995)
[6] N. Ghousoub, D. Preiss, A general mountain pass principle for locating and classifyingcritical points, Ann. Inst. H. Poincare, 6 (1989), 321-330.
[7] N. Ghousoub, Multiplicity and Morse Indices for min-max critical points, J. ReineAngew. Math., 417 (1991), 27-76
[8] N.K. Ribarska, T.Y. Tsachev, M.I. Krastanov, On the general mountain pass principleof Ghoussoub-Preiss, Mathematica Balkanica, 5 (1991).
SOME RESULTS ON SIMULTANEOUSALGEBRAIC TECHNIQUES IN IMAGERECONSTRUCTION FROM PROJECTIONS
ROMAI J., 5, 2(2009), 79–96
Lacramioara Grecu, Aurelian NicolaFaculty of Mathematics and Computer Science, “Ovidius” University of Constanta, [email protected], [email protected]
Abstract In this paper we make a comparative analysis of two projection based itera-tive algorithms for systems of linear equations arising from image reconstruc-tion in computerized tomography: Kaczmarz’s successive projection iteration(1937) and Simultaneous Algebraic Reconstruction Technique (SART; 1984).We start a theoretical analysis of the SART algorithm, which gives us the possi-bility to consider its extended and constrained versions. Systematic numericalexperiments and comparisons are made on two phantoms widely used in imagereconstruction literature, with the classical, extended and constrained versionsof both Kaczmarz and SART methods.
Keywords: image reconstruction.
2000 MSC: 94A08, 68U10.
1. INTRODUCTIONComputerized tomography (CT) is a medical method that uses digital ge-
ometry processing to generate a three-dimensional image of the internals ofan object from a large series of two-dimensional X-ray images taken arounda single rotation axis. For the reconstruction of a two-dimensional image,the section coresponding to this image is scaned with X-rays sent forth fromsources that move on an arc of a circle around that section (Fig. 1). Theregion ABCD coresponding to the two-dimensional image is divided in imageelements (pixels). Each X-ray corresponds to a line in the scanning matrix,it’s elements representing the length of the segments determined by the X-rayon each pixel (see Figure 2).
79
80 Lacramioara Grecu, Aurelian Nicola
Figure 1. Fan beam scanning Figure 2. Construction of A and b
So, if we use a number of M X-rays and a discretization of the image withN pixels, the scanning matrix A will be M × N . The matrix A is sparse,rank-deficient, with a nonzero null space and is ill-conditioned. Measuring theintensities of the source emissions (Si) and reception (in the detector Dji) ofthe i-th X-ray, we obtain the component bi of the right hand side b of thefuture discreet model for the reconstruction problem. This technique thatreduces the reconstruction to solving a least squares problem
‖Ax− b‖ = min! (1)
is called Algebraic Reconstruction. For solving the discret model (1) therehave been developed a class of iterative methods based on projections, calledAlgebraic Reconstruction Techniques (ART).In this paper we shall make a comparative study between two methods: theKaczmarz method (which uses orthogonal succesive projections on the hiper-planes of the equations of problem (1)) and the Simultaneous Algebraic Recon-struction Technique - SART (which uses simultaneous non-orthogonal projec-tions on the hiperplanes mentioned before). This paper is organized as follows:in Section 2 we describe the Kaczmarz and SART algorithm, and the conver-gence results. In Section 3 we present some special theoretical results for theSART algorithm. A development of these results allows us to approach inSection 4 (at least from a formal point of view) the possibility of extendingthe SART method to an inconsistent problem of type (1) and/or combiningit with constraining techniques. In Section 5 we present some reconstructionexperiments of Kaczmarz algorithm and SART algorithm with and withoutconstraints, with two phantoms frequently used in other articles.We shall use the following notations: LSS(A; b) =
x ∈ RN , ‖Ax− b‖ = min!
,
S (A; b) =x ∈ RN , Ax = b
, xLS the minimum norm solution of (1), Ai row
i from matrix A, Aj column j from matrix A, N(A) =x ∈ RN , Ax = 0
,
R(A) =Ax, x ∈ RN
, AT the transpose of A. PS will denote the orthogonal
(Euclidean) projection onto the non-empty closed convex subset S ⊂ RN .
Some results on simultaneous algebraic techniques in image reconstruction ... 81
2. CONVERGENCE RESULTS FOR KACZMARZAND SART ALGORITHM
2.1. KACZMARZ ALGORITHMThe Kaczmarz method, based on orthogonal succesive projections on hiper-
planes of problem (1) was proposed by it’s author in [5].Algorithm Kaczmarz (K). Initialization: x0 ∈ RN
Iterative step:xk+1 = (f1 · · · fM )(b; xk), (2)
wherefi(b; x) = x− 〈x,Ai〉 − bi
‖Ai‖2 Ai, i = 1, . . . ,M. (3)
It’s convergence was analyzed in [5] and [11].
Theorem 2.1. For any x0 ∈ RN , the sequence (xk)k≥0 generated by thealgorithm (K) converges to x(x0) ∈ RN . In the consistent case for (1),x(x0) ∈ S(A; b) (with x(0) = xLS).
2.2. SART ALGORITHMSART was introduced, by Andersen and Kak in 1984 (see [1]), to reduce the
salt and pepper noise commonly associated with ART-type reconstructions.This method consists of simultaneous application of the error correction termsas computed by ART for all rays in a given projection. In [1] the authorsargue that a simultaneous application of the correction term for the rays in aparticular view is preferable to the usual sequential one. Componentwise theSART algorithm can be written as follows
xk+1j = xk
j +λk
M∑i=1
|Aij |
M∑
i=1
Aij
N∑j=1
|Aij |
(bi −Aix
k)
, j = 1, . . . , N. (4)
If we define the matrices
V = diag (V11, . . . , VNN ) and W = diag (W11, . . . , WMM ) (5)
with
Vjj =M∑
i=1
|Aij | , j = 1, . . . N, (6)
1Wii
=N∑
j=1
|Aij | , i = 1, . . . , M, (7)
82 Lacramioara Grecu, Aurelian Nicola
the algorithm (4) can be written in the following matricial formAlgorithm SART. Initialization: x0 ∈ RN
Iterative step:xk+1 = xk + λkV
−1AT W(b−Axk
). (8)
In [4] Jiang and Wang proved the following convergence result.
Theorem 2.2. For 0 ≤ λk ≤ 2 and∞∑
n=0min (λk, 2− λk) = +∞, the sequence
(xk)k≥0 generated by the algorithm (SART) converges to
xV,W (x0) = xV,WLS + P V
N(A)
(x0
),
where P VN(A) is the projection onto N(A) w.r.t. the energy scalar product 〈·, ·〉V
and xV,WLS is the solution with minimal V -norm of the problem
‖Ax− b‖W = min! (9)
In the consistent case for (1), xV,W (x0) ∈ S(A; b) (but, in general xV,W (0) 6=xLS).
Remark 1. For the Kaczmarz algorithm we have limk→∞
xk = xLS for x0 = 0
in the consistent case. For SART algorithm we have limk→∞
xk = xV,WLS (0) = xV
LS
in the consistent case and x0 = 0, where xVLS is the minimal V -norm solution
of (1).
Example 1. For A =
−1 0 1 10 1 2 1−1 0 −1 −1
and b =
0−1−2
we obtain
xLS =(1,−5
3 ,−13 , 4
3
), and xV
LS =(1, 17
8 , 18 , 7
8
)(for our problem we have
V = diag(2, 1, 6, 3)).
3. SOME TEORETICAL RESULTS FOR SARTIn [8] the authors consider a general iterative method of the form
xk+1 = Txk + Rb, x0 ∈ RN , (10)
where T : N×N and R : N×M. The following basic asumptions are introduced
I − T = RA, (11)
if x ∈ N (A) then Tx = x ∈ N (A) , (12)
if x ∈ R(AT
)then Tx ∈ R
(AT
), (13)
Some results on simultaneous algebraic techniques in image reconstruction ... 83
∀ y ∈ Rm, Ry ∈ R(AT
), (14)
if T = TPR(AT ) then∥∥∥T
∥∥∥ < 1. (15)
In this paper it is shown that if the method (10) satisfies these propertiesit is possible to extended it to an inconsistent problem of type (1) as well ascombining it with special constraining techniques (see [6]). Now we shall provesome of these properties for the SART algorithm described in Section 2. But,because in the SART algorithm (8), appear the pozitive definite matrices Vand W , from (6) and (7), the matrix A will be considered (as a linear operator)as follows
A :(RM , 〈, 〉V
) → (RM , 〈, 〉W
). (16)
We will denote with Aτ :(RM , 〈, 〉W
) → (RN , 〈, 〉V
)the adjoint of A.
Lemma 1. For A from (16) we have
Aτ = V −1AT W, (17)
RN = N(A)⊕⊥VR(Aτ ), (18)
where ⊕⊥Vrepresents the orthogonal direct sum with respect to 〈·, ·〉V .
Proof. According to (16), Aτ is characterized by
〈Ax, y〉W = 〈x,Aτy〉V , ∀ x ∈ RN , y ∈ RM . (19)
From (19) we get
〈Ax, y〉W = 〈WAx, y〉 =⟨x,AT Wy
⟩ ⇔ 〈x,Aτy〉V = 〈x, V Aτy〉 , (20)
∀ x ∈ Rn, y ∈ RM ,
andAτ = V −1AT W. (21)
Because N(A) ⊂ RN , R(Aτ ) ∈ RN , for (18) we will prove only “⊃”. Letx ∈ RN and x′′ = P V
R (Aτ ) (x). We define x′ = x− x′′. We want to prove thatx′ ∈ N(A) i.e. Ax′ = 0. If x′′ = P V
R (Aτ ) (x) we know that x−x′′⊥V R(Aτ ), i.e.⟨x− x′′, y
⟩V
= 0, ∀ y ∈ R(Aτ ) ⇔ ⟨x− x′′, V y
⟩= 0, ∀ y ∈ R(Aτ ) ⇔
⟨x− x′′, V V −1AT Wz = 0, ∀ z ∈ RM
⟩ ⇔ ⟨A
(x− x′′
),Wz
⟩= 0, ∀ z ∈ RM ⇔
⟨Ax′,Wz
⟩= 0, ∀ z ∈ RM ⇔ ⟨
Ax′, t⟩ ∀ t ∈ RM ⇔ Ax′⊥RM ⇔ Ax′ = 0
HenceRN = N(A) + R(Aτ ). (22)
84 Lacramioara Grecu, Aurelian Nicola
Now we prove that N(A)⊥V R(Aτ ). Consider z ∈ R(Aτ ) and y ∈ N(A), i.e.z = Aτx and Ay = 0. We obtain
〈z, y〉V = 〈Aτx, y〉V = 〈V Aτx, y〉 =⟨V V −1AT Wx, y
⟩=
⟨AT Wx, y
⟩=
〈Wx, Ay〉 = 〈Wx, 0〉 = 0.
HenceN(A)⊥V R(Aτ ). (23)
From (22) and (23) we obtain (18). 2
Forλk = λ, ∀ k ≥ 0 (24)
in (8) we define the matrices T : N ×N and R : N ×M by
T = I − λV −1AT WA, (25)
andR = λV −1AT W. (26)
Lemma 2. The matrices T and R from (25) and (26) satisfy
(i) I − T = RA,
(ii) if x ∈ N (A) then Tx = x ∈ N (A) ,
(iii) if x ∈ R (Aτ ) then Tx ∈ R (Aτ ) ,
(iv) ∀ y ∈ RM , Ry ∈ R (Aτ ) .
Proof. (i) From (8) we have
xk+1 = xk + λV −1AT W(b−Axk
)= xk
(I − λV −1AT WA
)+ λV −1AT Wb,
(27)thus according to (10)
R = λV −1AT W, (28)
T = I − λV −1AT WA = I −RA, (29)
i.e. the equality (i).(ii) For x ∈ N(A), we obtain
Tx = (I −RA) x = x−RAx = x. (30)
(iii) For x ∈ R (Aτ ) we get from (25)
Tx =(I − λV −1AT WA
)x = (I − λAτA) x = x− λAτAx ∈ R (Aτ ) . (31)
Some results on simultaneous algebraic techniques in image reconstruction ... 85
(iv) From (26), we get for any y ∈ RM
Ry = λV −1AT WAy = λAτy ∈ R (Aτ ) , (32)
and the proof is complete. 2
Remark 2. From Lemma 2 we obtain that the matrices T and R from (25)and (26) satisfy the properties (11)-(14). Unfortunately we don’t have yet aproof for property (15).
4. EXTENDED AND CONSTRAINED SART-LIKEALGORITHMS
Although (see Remark 2) we don’t have yet a proof for a property analo-gous with (15) related to matrices T and R from (25) and (26), the fact thatthe properties (11)-(14) have been proved (see Lemma 2), as well as the ex-periments that we performed (see also Section 5) allowed us to consider thefollowing extended and constrained versions of SART algorithm.
4.1. CONSTRAINED SARTA box-constraint is a function C : RN → RN , which is a metric projection
operator onto the box [a, b] = [a1, b1]× . . . [aN , bN ] ⊂ RN , defined by
(Cx)i =
xi, if xi ∈ [ai, bi]ai, if xi < ai
bi, if xi > bi
. (33)
It satisfies‖Cx− Cy‖ ≤ ‖x− y‖ , (34)
if ‖Cx− Cy‖ = ‖x− y‖ then Cx− Cy = x− y, (35)
if y ∈ Im (C) then Cy = y. (36)
Algorithm Constrained SART (CSART).Initialization: x0 ∈ Im(C)Iterative step:
xk+1 = C[xk + λkV
−1AT W(b−Axk
)]. (37)
4.2. EXTENDED SARTFor Extended SART we shall use the model of Extended Kaczmarz algo-
rithm from [9].
86 Lacramioara Grecu, Aurelian Nicola
Algorithm Extended SART.Initialization: x0 ∈ RN , y0 = bIterative step:
yk+1 = yk − µkV−1AWAT yk, (38)
bk+1 = b− yk+1, (39)
xk+1 = xk + λkV−1AT W
(bk+1 −Axk
)(40)
where
Vii =N∑
j=1
|Aij | , i = 1, . . . M,1
Wjj
=M∑
i=1
|Aij | , j = 1, . . . , N. (41)
4.3. EXTENDED SART WITH CONSTRAINTSFor the Extended SART with constraints we follow the model from the
Constrained Kaczmarz Extended algorithm in [10].Algorithm Constrained Extended SART.Initialization: x0 ∈ Im(C), y0 = bIterative step:
yk+1 = yk − µkV−1AWAT yk, (42)
bk+1 = b− yk+1, (43)
xk+1 = C[xk + λkV
−1AT W(bk+1 −Axk
)]. (44)
5. NUMERICAL EXPERIMENTSWe have used in our experiments the Head and Mitochondrian phantoms
described in the paper [3]. For each phantom we have the exact picture(N = 3969) xex ∈ R3969, i.e. with a 63 × 63 pixel resolution, and a scan-ning matrix A : 1376× 3969 for Head phantom, respectively, A : 1378× 3969for Mitochondrian phantom and a right hand side b ∈ R1376, respectively,b ∈ R1378. We also used the following wellknown error measures used in imagereconstruction (see e.g. [3]): standard deviation, distance, relative error andnormal equation residual, defined below.
xex = (xex1 , . . . , xex
N )T - the mitochondrian phantom
xk = (xk1, . . . , x
kN )T - the current approximation
xex = 1N
N∑i=1
xexi ; xk = 1
N
N∑i=1
xki - mean values of the exact image and
current approximation, respectively
Some results on simultaneous algebraic techniques in image reconstruction ... 87
Standard deviation = 1√N
√N∑
i=1(xk
i − xk)2
Distance =
√√√√√N∑
i=1(xex
i −xki )2
N∑i=1
(xexi −xex)2
Relative error =
N∑i=1
|xexi −xk
i |N∑
i=1xex
i
Residual = 1√N||Axk − b|| (consistent) and 1√
N‖ AT (Axk − b) ‖ (incon-
sistent).
In our experiments, beside SART, Constrained SART, Extended SART, andConstrained Extended SART algorithms we used the Kaczmarz original method(see [5], [11] with), Constrained Kaczmarz (CK, from [6]), Extended Kacz-marz (from [9]) and Contrained Extended Kaczmarz (from [10]). In Fig. 1we present reconstructions for consistent case using Kaczmarz and SART forHead phantom. Similarly in Fig. 3 we show reconstructions for Mitochondrianphantom. Errors described above for these cases are shown in Figures 2 and4. In Fig. 5 we present reconstructions for consistent case using ConstrainedKaczmarz and Constrained SART for Head phantom. Similarly in Fig. 7we show reconstructions for Mitochondrian phantom using constrains. Errorsdescribed above for these cases are shown in Figures 6 and 8. In Fig. 9 wepresent reconstructions for inconsistent case using Extended Kaczmarz andExtended SART for Head phantom. Similarly in Fig. 11 we show inconsistentreconstructions for Mitochondrian phantom. Errors described above for thesecases are shown in Figures 10 and 12. In Fig. 13 we present reconstructionsfor inconsistent case using Constrained Extended Kaczmarz and ConstrainedExtended SART for Head phantom. Similarly in Fig. 15 we show inconsistentreconstructions for Mitochondrian phantom using constrains. Errors describedabove for these cases are shown in Figures 14 and 16.
88 Lacramioara Grecu, Aurelian Nicola
Fig. 1. Consistent case reconstructions: exact solution, Kaczmarz, SART; no constraints
Fig. 2. Errors for Head phantom, consistent case, no constraints
Fig. 3. Consistent case reconstructions: exact solution, Kaczmarz, SART; no constraints
Some results on simultaneous algebraic techniques in image reconstruction ... 89
Fig. 4. Errors for Mit phantom, consistent case, no constraints
Fig. 5. Consistent case reconstructions: exact solution, Kaczmarz, SART; with constraints
90 Lacramioara Grecu, Aurelian Nicola
Fig. 6. Errors for Head phantom, consistent case, with constraints
Fig. 7. Consistent case reconstructions: exact solution, Kaczmarz, SART; with constraints
Some results on simultaneous algebraic techniques in image reconstruction ... 91
Fig. 8. Errors for Mit phantom, consistent case, with constraints
Fig. 9. Inconsistent case reconstructions: exact solution, Extended Kaczmarz, ExtendedSART; no constraints
92 Lacramioara Grecu, Aurelian Nicola
Fig. 10. Errors for Head phantom, inconsistent case, no constraints
Fig. 11. Inconsistent case reconstructions: exact solution, Extended Kaczmarz, ExtendedSART; no constraints
Some results on simultaneous algebraic techniques in image reconstruction ... 93
Fig. 12. Errors for Mit phantom, inconsistent case, no constraints
Fig. 13. Inconsistent case reconstructions: exact solution, Extended Kaczmarz, ExtendedSART; with constraints
94 Lacramioara Grecu, Aurelian Nicola
Fig. 14. Errors for Head phantom, inconsistent case, with constraints
Fig. 15. Inconsistent case reconstructions: exact solution, Extended Kaczmarz, ExtendedSART; with constraints
Some results on simultaneous algebraic techniques in image reconstruction ... 95
Fig. 16. Errors for Mit phantom, inconsistent case, with constraints
96 Lacramioara Grecu, Aurelian Nicola
References[1] Andersen A.H., Kak A.C., Simultaneous Algebraic Reconstruction Techniques (SART):
a superior implementation of the ART algorithm, Ultrasonic imaging, 6(1984), 81-94.
[2] Censor Y., Stavros A. Z., Parallel optimization: theory, algorithms and applications,”Numer. Math. and Sci. Comp.” Series, Oxford Univ. Press, New York, 1997.
[3] Herman, G. T., Image reconstruction from projections. The fundamentals of comput-erized tomography, Academic Press, New York, 1980.
[4] Jiang M., Wang G., Convergence studies on iterative algorithms for image reconstruc-tion, IEEE Trans. Medical Imaging, 22(2003), 569-579.
[5] Kaczmarz S., Angenaherte Auflosung von Systemen linearer Gleichungen, Bull. Acad.Polonaise Sci. et Lettres A (1937), 355–357.
[6] Koltracht I. and Lancaster P., Constraining strategies for linear iterative processes,IMA Journal of Numerical Analysis, 10(1990), 555–567.
[7] Natterer F., The Mathematics of Computerized Tomography, John Wiley and Sons,New York, 1986.
[8] Nicola A., Petra S., Popa C., Schnorr C., On a general extending and constrainingprocedure for linear iterative methods, Preprint 9761(2009), IWR Heidelberg, Germany(http://www.ub.uni-heidelberg.de/archiv/9761).
[9] Popa C., Extensions of block-projections methods with relaxation parameters to incon-sistent and rank-defficient least-squares problems;B I T, 38(1)(1998), 151–176.
[10] Popa C., Constrained Kaczmarz extended algorithm for image reconstruction, LinearAlgebra and its Applications, 429(2008), 2247–2267.
[11] Tanabe K., Projection Method for Solving a Singular System of Linear Equations andits Applications, Numer. Math., 17(1971), 203–214.
BRANCHING EQUATION IN THE ROOTSUBSPACE FOR EQUATIONS NONRESOLVEDWITH RESPECT TO DERIVATIVE ANDSTABILITY OF BIFURCATING SOLUTIONS
ROMAI J., 5, 2(2009), 97–107
Irina V. Konopleva, Boris V. Loginov, Yuri B. RousakUlyanovsk State Technical University, Russia;Ulyanovsk State Technical University, Russia;Canberra University, [email protected], [email protected], [email protected]
Abstract For stationary and dynamic bifurcation problems some theorems on the inheri-tance of group symmetry of nonlinear equations are proved by A. M. Lyapunovand E. Schmidt branching equations in the root-subspaces (BEqRs), movingalong the orbit of bifurcation point x0. Also theorems about BEqRs of po-tential type reduction at the action of continuous group symmetry are proved.The case of isolated branching point is considered separately.Acknowledgement The obtained results are supported by grant RFBR–Acad. Sci. of Romania No. 07− 01− 91680−a and by the program ”Develop-ment of the Scientific Potential of Higher School” of the Ministry of Educationof Russian Federation (project No. 2.1.1/6194).
Keywords: Symmetry, potentiality, branching equations in the root subspace, stationary
and dynamic bifurcation problems, stability.
2000 MSC: 35B32, 35B35.
1. INTRODUCTIONIn real Banach spaces E1 and E2 the nonlinear equation, i.e. bifurcation
problem with small (numerical or functional) parameter ε ∈ Λ
F (x, ε) = 0, F (x0, ε) = 0,F ′
x(x0, ε) = −Bx0 + Bx0(ε), B0 = Bx0 = −F ′x(x0, 0) (1)
is considered, where Bx0 is a densely defined in E1 Fredholm operator, DB ⊂DB(ε), B(ε) = F ′
x(x0, ε) + Bx0 , N(Bx0) = spanϕin1 , ϕi = ϕi(x0) is the zero-
subspace (kernel) of the operator B0, N∗(Bx0) = spanψin1 , ψi = ψi(x0) is the
subspace of defect functionals, and γin1 , γi = γi(x0) ∈ E∗
1 , zin1 , zi = zi(x0)
– the corresponding biorthogonal systems 〈ϕi, γj〉 = δij , 〈ψi, zj〉 = δij .
97
98 Irina V. Konopleva, Boris V. Loginov, Yuri B. Rousak
Bifurcating solutions of the equation (1) are considered as stationary solu-tions of the autonomous differential equation in Banach spaces E1 and E2
Adydt = Bx0y −Bx0(ε)y −R(x0, y, ε), y = x− x0
R(x0, 0, ε) ≡ 0, ‖R(x0, x− x0, ε)‖ = 0(‖x− x0‖) (2)
with closed densely defined Fredholm operators A and B = Bx0 .
Definition 1.1. The solution y0(t) = x(t) − x0, t ≥ 0 of the equation (1) isLyapunov stable if for any ε > 0 there exists δ > 0, such that for any solutiony(t), ‖y(0) − y0(0)‖ < δ for t > 0 the inequality ‖y(t) − y0(t)‖ < ε is fulfilledand asymptotically stable if ‖y(t)− y0(t)‖ → 0 at t →∞.
The technique of branching equations in the root-subspaces (BEqR) wasintroduced in our articles [1]–[5] with the aim of stability investigation ofbifurcating solutions (see also the review articles [6, 7]). Everywhere belowthe terminology and notations of [6]-[8] are used. We will not stipulate inevery case used here subordination conditions of densely defined Fredholmoperators [6, 7], allowing to reduce the discussion to bounded operators: fora pair of densely defined Fredholm operators A : DA → E2, B : DB → E2, ifDB ⊂ DA, then A is subordinated to B, i.e. ‖Ax‖ ≤ ‖Bx‖+ ‖x‖ on DB; or ifDA ⊂ DB then B is subordinated to A, i.e. ‖Bx‖ ≤ ‖Ax‖+ ‖x‖ on DA. It issupposed also that A and B haven’t common zeros with the aim to avoid thecomplicated technique of nonfinished generalized Jordan chains (GJCh).
Definition 1.2. ([8] − [10]) Elements ϕ(s)k , s = 1, pk, k = 1, n form complete
canonical GJSet (B(ε)−GS), if
Bϕ(s)k (x0) =
s−1∑j=1
Bjϕ(s−j)k (x0), B(ε) = B1ε + B2ε
2 + . . . ,
〈ϕ(s)k (x0), γl(x0)〉 = 0, s = 2, pk,
Dp = det
[pk∑
j=1〈Bjϕ
(pk+1−j)k (x0), ψ
(1)l (x0)〉
]6= 0,
k, l = 1, n, ϕk(x0) = ϕ(1)k (x0), ψl(x0) = ψ
(1)l (x0).
(3)
This GJS is bicanonical, if the GJS of conjugate operator-function B∗ −B∗(ε) corresponding to elements ψln
1 (x0) is also canonical and three-canonical,if in addition
〈ϕ(j)i (x0), γ
(l)k (x0)〉 = δikδjl, γ
(l)k (x0) =
pk+1−l∑s=1
B∗sψ
(pk+2−l−s)k (x0),
〈z(j)i (x0), ψ
(l)k (x0)〉 = δikδjl, z
(j)i (x0) =
pi+1−j∑s=1
Bsϕ(pi+2−j−s)i (x0),
Φ = Φ(x0) = (ϕ(1)1 (x0), . . . , ϕ
(p1)1 (x0), . . . , ϕ
(1)n (x0), . . . , ϕ
(pn)n (x0)),
(4)
Branching equation in the root-subspace ... 99
vectors γ = γ(x0),Ψ = Ψ(x0) and Z = Z(x0) are defined analogously.
Lemma 1.1. ([1, 2], [8]− [10]) The elements of the basis of the kernel N(A) =spanφim
1 (N(B) = spanϕin1 ) can be chosen so that for the N(A∗) =
spanψm1 (N(B∗) = spanψn
1 ) corresponding elements of B- and B∗-Jordansets (A- and A∗-JS) of the operator-functions A− εB and A∗ − εB∗ (B − µAand B∗ − µA∗) would be three-canonical, i.e.
⟨φ
(j)i , ϑ
(l)k
⟩= δikδjl,
⟨ζ(j)i , ψ
(l)
k
⟩= δikδjl, j(l) = 1, . . . , qi(qk),
ϑ(l)k = B∗ψ
(qk+1−l)
k , ζ(j)i = Bφ
(qi+1−j)i , i, k = 1, . . . , m;
(5)
⟨ϕ
(j)i , γ
(l)k
⟩= δikδjl,
⟨z(j)i , ψ
(l)k
⟩= δikδjl, j(l) = 1, . . . , pi(pk),
γ(l)k = A∗ψ(pk+1−l)
k , z(j)i = Aϕ
(pi+1−j)i , i, k = 1, . . . , n.
(6)
The conditions Aφ(s)i = Bφ
(s−1)i , 〈φ(s)
i , ϑ(1)j 〉 = 0, s = 2, . . . , qi, i, j = 1, . . . , m;
(Bϕ(s)i = Aϕ
(s−1)i , 〈ϕ(s)
i , γ(1)j 〉 = 0, s = 2, . . . , pi, i, j = 1, . . . , n) determine B-
JS (A-JS) uniquely. Its elements are linearly independent and form the basis
of the root-subspaces K(A;B) (K(B;A)); kA = dimK(A;B) =m∑
i=1qi (kB =
dimK(B; A) =n∑
i=1pi) is called the root-number of the Fredholm point ε =
0 (µ = 0) of σB(A) (σA(B)). The relations (5),(6) allow to introduce theprojectors [1, 2]
p =m∑
i=1
qi∑j=1
⟨·, ϑ(j)
i
⟩φ
(j)i = 〈·, ϑ〉φ : E1 → EkA
1 = K(A,B),
q =m∑
i=1
qi∑j=1
⟨·, ψ(j)
i
⟩ζ(j)i =
⟨·, ψ
⟩ζ : E2 → E2,kA
= spanζ(j)i ,
(7)
P =n∑
i=1
pi∑j=1
⟨·, γ(j)
i
⟩ϕ
(j)i = 〈·, γ〉ϕ : E1 → EkB
1 = K(B; A),
Q =n∑
i=1
pi∑j=1
⟨·, ψ(j)
i
⟩z(j)i = 〈·, ψ〉 z : E2 → E2,kB
= spanz(j)i
(8)
(where φ = (φ(1)1 , · · · , φ
(q1)1 , · · · , φ
(1)m , · · · , φ
(qm)m ), and the vectors ϑ, ψ, ζ, ϕ, γ,
ψ, z are defined analogously) generating the following direct sums expansions
E1 = EkA1 +E∞−kA
1 , E2 = E2,kA+E∞−kA
,
E1 = EkB1 +E∞−kB
1 , E2 = E2,kB+E2,∞−kB
.(9)
The intertwining relations are realized
Ap = qA on DA, Bp = qB on DB,BP = QB on DB, AP = QA on DA,
(10)
100 Irina V. Konopleva, Boris V. Loginov, Yuri B. Rousak
Aφ = AAζ, Bφ = ABζ, B∗ψ = ABϑ,Bϕ = ABz, Aϕ = AAz, A∗ψ = AAγ)
(11)
with cell-diagonal matrices AA = (A1, . . . , Am), AB = (B1, . . . , Bm)(AB = (B1, . . . , Bn), AA = (A1, . . . , An)) where the qi × qi-cells (pi × pi-cells) have the forms
Ai =
0 0 0 . . . 0 00 0 0 . . . 0 1...
......
. . ....
...0 0 1 . . . 0 00 1 0 . . . 0 0
, Bi =
0 0 0 . . . 0 10 0 0 . . . 1 0...
......
. . ....
...0 1 0 . . . 0 01 0 0 . . . 0 0
( Ai (Bi) have the same form as Ai(Bi)). The following relations for theoperators A and B are valid
N(A) ⊂ EkA1 , AEkA
1 ⊂ E2,kA, A(E∞−kA
1 ∩DA) ⊂ E2,∞−kA,
N(B) ⊂ E∞−kA1 , BEkA
1 ⊂ E2,kA, B(E∞−kA
1 ∩DB) ⊂ E2,∞−kA.
(12)
where the mappings B : EkA1 → E2,kA
,uA= A : E∞−kA
1 ∩DA → E2,∞−kAare
one-to-one. Analogously,
N(B) ⊂ EkB1 , BEkB
1 ⊂ E2,kB, B(E∞−kB
1 ∩DB) ⊂ E2,∞−kB,
N(A) ⊂ E∞−kB1 , AEkB
1 ⊂ E2,kB, A(E∞−kB
1 ∩DA) ⊂ E2,∞−kB.
(13)
and the mappings A : EkB1 → E2,kB
,tB= B : E∞−kB
1 ∩DB → E2,∞−kBare one-
to-one. Thus the operators A and B (B and A) act as invariant pairs of sub-spaces EkA
1 , E2,kAand E∞−kA
1 , E2,∞−kA(EkB
1 , E2,kBand E∞−kB
1 , E2,∞−kB).
The first object of this article is to investigate the stability of stationarybifurcating solutions of (2) under intertwining condition of the nonlinear oper-ator F in the equation by representations of some group G or individual opera-tors. On this base the stability investigation of periodical bifurcating solutionsat Poincare-Andronov-Hopf bifurcation for differential equation non-resolvedwith respect to derivative can be realized
F (p, x, ε) = 0, p = dxdt , F (0, x0, ε) ≡ 0,
F ′p(0, x0, 0) = Ax0 = A0, F ′
x(0, x0, 0) = −Bx0 = −B0,F ′
p(0, x0, ε) = A0 + Ax0(ε) = A(ε), F ′x(0, x0, ε) = −B0 + Bx0(ε),
DB0 = E1, DB0 ⊂ D(A(ε))
(14)
under the same intertwining conditions. Stationary solutions to (2) are alsostationary solutions to (14). The indicated problems similarly to [1]-[3] willbe solved at the usage of BEqR techniques [1]-[5] presented in detail in [11].
Branching equation in the root-subspace ... 101
2. STABILITY STUDY OF BIFURCATINGSOLUTIONS UNDER GROUP SYMMETRYCONDITIONS OF NONLINEAR OPERATORS
It is assumed that for continuously differentiable x and sufficiently smoothb (project no. 07-01-9168-a), (y; ε) in some neighborhood of bifurcation point(x0; 0); operator F allows the group G, i.e. there exist its representations Lg
in E1 and Kg in E2 intertwining the operator F
KgF (x, ε) = F (Lgx, ε) (KgF (p, x, ε) = F (Lgp, Lgx, ε)). (15)
Differentiation of this equalities by x and p at the (x0; 0) gives the relations
KgF′x(x0, ε) = F ′
x(Lgx0, ε)Lg, (KgAx0 = ALgx0Lg, KgBx0 = BLgx0Lg)
showing that operators Bx0 and Bx0(ε) (Ax0 and Ax0(ε)) possess the symmetrywith respect only stationary subgroup of the point x0.
When x0 is the stationary point of representation Lg, the operators A0, Bx0 ,Bx0(ε) and R(x0, y, ε) are intertwined by the pair Lg, Kg: KgA0 = A0Lg,KgBx0 = Bx0Lg, KgBx0(ε) = Bx0(ε)Lg and KgR(x0, y, ε) = R(x0,Kgy, ε)and the following partially generalized results of the article [2] are valid.
Theorem 2.1. Assume that, in the general case of symmetry absence, theoperator F is continuously differentiable with respect to x up to the orderq, where q is the maximal length of B0-Jordan chains of the basic elementsφim
1 ∈ N(A0). If the Fredholm operator A0 has a complete B0-Jordan set andthe A0-spectrum σA0(B0) lies in the left halfplane (even one point of σA0(B0)gets into the right halfplane), then the trivial solution of the equation (1) isasymptotically stable (unstable). In these conditions the principle of linearizedstability is fulfilled: the bifurcating stationary solution y(ε) from (x0; 0) of theequation (1) is asymptotically stable if the A0-spectrum σA0(Bx0 − Bx0(ε) −Ry(x0, y(ε), ε)) of the Frechet derivative Bx0 −Bx0(ε)−Ry(x0, y(ε), ε) on thesolution y(ε) lies in the left halfplane, and it is unstable if there exists even onepoint µ(ε) ∈ σA0(Bx0 − Bx0(ε) − Ry(x0, y(ε), ε)) in the right halfplane. Thelast question is solved by the Newton diagram method applied to the BEqR ofthe eigenvalue problem
(Bx0 −Bx0(ε)−Ry(x0, y(ε), ε)− µA0)ϕ = 0. (16)
Here the relevant BEqR≡linear resolving system (LRS) [6] is constructed,the determinant of which determines the branching equation for eigenvaluebifurcation of the problem (16). If the bifurcating from x0 solution y(ε) isrepresented by the series on a fractional degree ε1/k this degree is denoted byε, i.e. the problem about eigenvalue branching of Fredholm operators [8] isconsidered
102 Irina V. Konopleva, Boris V. Loginov, Yuri B. Rousak
(B0 −Bx0(εk)−Ry(x0, y(ε), εk)− µA0)Φ =
(B0 −R(ε)− µA0)Φ = (B0 −∞∑
k=1
εkRk − µA0)Φ = 0. (17)
According to Lemma 1.1, we rewrite the equation (17) in the form of a
system by using the Schmidt regularizator_B0= B0 +
n∑i=1〈·, γ(1)
j (x0)〉z(1)j (x0),
Γx0 = Γ0 =_B−1
0
_B0 Φ−
∞∑
k=1
εkRkΦ = µA0Φ +n∑
s=1
ξs1z(1)s , ξs1 = 〈Φ, γ(σ)
s 〉 (18)
the solution of which is sought in the form Φ = w+n∑
r=1
pr∑ρ=1
ξrρϕ(ρ)r = w+ξ ·ϕ =
= w + v(x0, ξ).
Then_B0 w +
n∑r=1
pr∑ρ=2
ξrρB0ϕ(ρ)r = µA0w + µ
n∑r=1
pr∑ρ=1
ξrρA0ϕ(ρ)r + R(ε)(w +
n∑r=1
pr∑ρ=1
ξrρϕ(ρ)r ) and using the relations B0ϕ
(ρ)r = z
(ρr+2−ρ)r , Γ0z
(pr)r = ϕ(2)
r , Γ0z(pr−1)r =
ϕ(3)r , . . . , Γ0z
(2)r = ϕ(pr)
r , Γ0z(1)r = ϕ(1)
r , ϕ(pr+1)r = ϕ(1)
r , z(pr+1)r = z
(1)r one has
w = (I − Γ0(µA0 + R(ε)))−1−n∑
r=1
pr∑ρ=2
ξrρϕ(ρ)r +
+µ
n∑r=1
(ξr1ϕ(2)r + ξr2ϕ
(3)r + . . . + ξrpr−1ϕ
(pr)r + ξrpr
ϕ(1)r ) +
n∑r=1
pr∑ρ=1
ξrρΓ0R(ε)ϕ(ρ)r =
= −n∑
r=1
pr∑ρ=2
ξrρϕ(ρ)r + µ
n∑r=1
(ξr1ϕ(2)r + ξr2ϕ
(3)r + . . . + ξrpr−1ϕ
(pr)r + ξrpr
ϕ(1)r )+
+
n∑r=1
pr∑ρ=1
ξrρΓ0R(ε)ϕ(ρ)r + Γ0(µA0 + R(ε)))(I − Γ0(µA0 + R(ε)))−1. . ..
Substitution of the last expression in the second set of equalities (18) allowsthe system to determine ξrρ, r = 1, . . . , n, ρ = 1, . . . , pn,
〈w, γ(σ)s 〉 = 0, s = 1, . . . , n, σ = 1, . . . , ps (19)
Branching equation in the root-subspace ... 103
or, with Γ∗0γ(σ)s = Γ∗0A
∗0ψ
(ps+1−σ)s = ψ
(ps+2−σ)s , ψ
(ps+1)s = ψ
(1)s , in coordinate
form
µξsps+ 〈
n∑r=1
pr∑ρ=1
ξrρR(ε)ϕ(ρ)r + (µA0 + R(ε))(I − Γ0(µA0 + R(ε)))−1. . ., ψ(1)
s 〉 = 0,
−ξs2 + µξs1 + 〈n∑
r=1
pr∑ρ=1
ξrρR(ε)ϕ(ρ)r + (µA0 + R(ε))(I − Γ0(µA0 + R(ε)))−1. . ., ψ(ps)
s 〉 = 0,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
−ξsps+ µξsps−1 + 〈
n∑r=1
pr∑ρ=1
ξrρR(ε)ϕ(ρ)r + (µA0 + R(ε))(I − Γ0(µA0 + R(ε)))−1. . ., ψ(2)
s 〉 = 0.
(20)
Technical computations by using the expressions for (A∗0Γ∗0)
κγ(σ)s through γ
(τ)s
allow us to calculate the determinant of the linear resolving system (19) ((20))for ε = 0 : ∆ = (−µ)p1+...+pn(−1)n. Hence by the method of Newton’s diagramthe µ-BEqR
∆(µ, ε) ≡ det[〈w, γ(σ)
s 〉]
= 0, s = 1, . . . , n; σ = 1, . . . , ps (21)
has k(B, A) =n∑
s=1ps roots µj = µj(ε), the signs of real parts of which deter-
mine the asymptotic stability of the solution x0(ε).Remark 2.1. The LRS (19) or (20) is E. Schmidt BEqR. According
to [1, 2, 6, 11] the relevant A.M. Lyapunov’s LRS can be constructed, in [6]their equivalence for nonlinearity case is proved.
Assume now that the bifurcation point x0 has nontrivial stationary sub-group. When G is Lie group Gl = Gl(a), a = (a1, . . . , al) it is supposed to bel-dimensional differentiable manifold, satisfying the following conditions [15,16],[11]-[14]
c1) The mapping a 7→ Lg(a)x0 acting from a neighborhood of the uniqueelement in Banach space E1 belongs to the class C1, therefore Xx0 ∈ E1 forall infinitesimal operators Xx = lim
t→0t−1
[Lg(a(t))x− x
]in the tangent to Lg(a)
manifold T lg(a).
c2) The stationary subgroup of the element x0 determines the representationL(Gs) of the local Lie group Gs ⊂ Gl, s < l, with s-dimensional subalgebraT s
g(a) of infinitesimal operators. Thus the elements of the form in N(Bx0) formsome m = (l−s)-dimensional subspace, i.e. the bases in N(Bx0) and in algebraT s
g(a) can be ordered so that ϕk = ϕk(x0) = Xkx0, 1 ≤ k ≤ m, and Xkx0 = 0for k ≥ m + 1.
c3) For all X ∈ T lg(a) the mapping X : E1 → H is bounded in L(E1,H)-
topology. The dense embeddings E1 ⊂ E2 ⊂ H in a Hilbert space H areassumed with estimates ‖u‖H ≤ α2‖u‖E2 ≤ α1‖u‖E1 .
To shorten the article, in the further presentation we will use and cite theworks [12]-[14],[15, 16] without going into details.
104 Irina V. Konopleva, Boris V. Loginov, Yuri B. Rousak
According to Theorem 1 (see also [1, 2, 11]) the stability of bifurcating fromx = x0 solution y(ε) is determined by the signs of the eigenvalues contained inσA0(Bx0 − Fy(x0, y(ε), εk)), i.e. by the problem about eigenvalues branchingof Fredholm operators
Bx0y − Fy(x0, y(ε), εk)y − µA0y = 0. (22)
Lemma 2.1. LRS for the linear bifurcation problem (21) inherits the groupsymmetry of the equation (1)
Proof Transformation of the equation (22) with the aid of E. Schmidt reg-ularizer [8] gives
_Bx0 Φ− Fy(x0, y(ε), εk)Φ− µA0Φ =
n∑
j=1
ξj1z(1)j (x0).
The substitution Φ = w + v(x0, ξ) = w + ξ · ϕ leads to the following system
w =[I − Γx0(µA0 + F ′
y(x0, y(ε), εk))]−1 (−
∨ξ · ∨ϕ (x0)+
µn∑
r=1
pr∑ρ=1
ξrρϕ(ρ+1)r (x0) + F ′
y(x0, y(ε), εk)v(x0, ξ)),
〈w, γ(σ)s 〉 = 0, s = 1, . . . , n; σ = 1, . . . , ps,
where ϕ(pr+1)r (x0) = ϕ
(p1)r (x0) and the symbol ” ∨ ” means the omission of
summands ξr1, ϕ(1)r (x0). The first equation of this system uniquely determines
w = w(x0, v(x0, ξ), ε), while the second one leads to the BEqR≡LRS of theeigenvalue problem (22)
t(x0, v(x0, ξ), ε) ≡n∑
j=1
tj(x0, v(x0, ξ), ε)ϕj(x0) = Px0w(x0, v(x0, ξ), ε) = 0.
(23)The equation (22) for the bifurcation point (Lgx0, 0) is reduced to the system
_
BLgx0 Lg(x− x0) = F ′Lgy(Lgx0, Lgy(ε), εk)Lg(x− x0) + µA0Lg(x− x0) +n∑
j=1
ξj1z(1)j (Lgx0),
ξs,σ = 〈Lg(x− x0), γ(σ)s (Lgx0)〉.
Analogously, the substitution Lg(x − x0) = w + Lgv(x0, ξ) = w + v(Lgx0, ξ)gives the system
Kg
_
Bx0 (L−1g w + v(x0, ξ)) =
KgF ′y(x0, y(ε), εk)(L−1g w + v(x0, ξ)) + KgµA0(L
−1g w + v(x0, ξ)) +
n∑j=1
ξj1z(1)j (Lgx0),
ξsσ = 〈w, γ(σ)s (Lgx0)〉,
Branching equation in the root-subspace ... 105
whence taking into account the relation_Bx0 v(x0, ξ) =
∨v (x0, ξ), according to
the unique solvability of the first equation, it follows L−1g w = w(x0, v(x0, ξ), ε)
=⇒ w = Lgw(x0, v(x0, ξ), ε) together with the group symmetry inheritancetheorem for BEqR- linear resolving system for (22)
t(Lgx0, Lgv(x0, ξ), ε) ≡ PLgx0w = PLgx0Lgw = LgPx0w = Lgt(x0, Lgv(x0, ξ), ε).(24)
Definition 2.1. [11]-[14] BEqR(≡LRS (23)) is an equation of potential typeif in a neighborhood of the bifurcation point (x0; 0) for the vector t(y, v(y, ξ), ε) =(t11, . . . , t1p1 , . . . , tn1, . . . , tnpn) the equality
t(y, v(y, ξ), ε) = d · gradyU(y, ξ, ε) (25)
holds with invertible operator d. Then the functional U(y, ξ, ε) is called thepotential of BEqR (23) and the linear by ξ and nonlinear by y operator t asthe pseudogradient of the functional U .
In the article [11] (analogously [12]) the result about the reduction of LRS (23)for the eigenvalue problem (22) is proved.
Theorem 2.2. Let in conditions c1)− c3) the LRS (23) be of potential type,assume that its potential belongs to the class C2 in some neighborhood of thebifurcation point (x0; 0) and is invariant of the representation Lg of the groupGl(a), and let s be the dimension of stationary subgroup of the element x0 andκ = l − s > 0. Then
1 if κ = n then for all (ξ(ε), ε) (v(x0, ξ(ε), ε)) from some neighborhood ofzero in Rn+1, LRS (23) is satisfied identically;
2 if κ < n and n ≥ 2, then the partial reduction of LRS takes place i.e.under accepted stipulation in c2) condition about the enumeration ofbasic elements in EkB
1 the first kκ = p1 + . . . + pκ equations are linearcombinations of the other pκ+1 . . . + pn.
This theorem allows to reduce the order in LRS (23) on the equations quan-tity and by this the order of the determinant (21) as the bifurcation equationof the eigenvalue problem (20).
Remark 2.2. Since the LRS (23) is equivariant only relative to stationarysubgroup Gs, the passage to the basis of irreducible invariant with respect toLg, g ∈ Gs subspaces, analogously to [17], leads to the LRS (23) decompositionon independent systems. Thus the BEq for the finding µ(ε) is decomposed onseparate factors that simplifies the Newton’s diagram application. Accordingto the results of [6] here we have the BEq reduction at once by Jordan chains.
106 Irina V. Konopleva, Boris V. Loginov, Yuri B. Rousak
Remark 2.3. 1. A separate article will be devoted to the application ofTheorem 2.1 to the investigation of bifurcating solutions stability at Poincare-Andronov-Hopf bifurcation.2. By using our previous results [18] the results obtained in this work can begeneralized on the case of Noetherian operators.3. By using the results [19, 20, 6], the stability results for stationary bifurcatingsolutions may be established in the presence of intertwining conditions of theequation (1).
References[1] B.V. Loginov, On the stability of the solutions of differential equations with degenerate
operator at the derivative, Izv. Akad. Nauk Uzbek SSR, fiz-mat., No.1, (1988) 29-32;Letter to the Editor, Izv. Akad. Nauk Uzbek SSR, fiz-mat., 2, (1988) 78. (in Russian)
[2] B.V. Loginov, Yu.B. Rousak, Generalized Jordan structure in the problem of stabilityof bifurcation equations, Nonlinear Analysis. TMA, 17, 3 (1991) 219-231.
[3] L.R. Kim-Tyan, B.V. Loginov, Yu.B. Rousak, On solutions stability in differential equa-tions with degenerate operator at the highest derivative in Banach spaces, Uzbek Math.J., 4(1999) 31-36. (in Russian)
[4] B.V. Loginov, Branching equation in the root subspace, Nonlinear Analysis, TMA, 32,3(1998) 439-448.
[5] B.V. Loginov, I. V. Konopleva, Bifurcation systems in the root-subspace, their relation,symmetry and reduction possibilities, Proceedings of Int. Conf. ”Functional Spaces VI”,Wroclaw, 2001, Poland, M. Dekker, 2003, 160-174.
[6] B. Karasozen, I. Konopleva, B. Loginov, Hereditary symmetry of resolving sys-tems in nonlinear equations with Fredholm operators, Nonl. Anal. and Appl.: ToV.Lakshnikantham on his 80th Birthday (Ravi P. Agarwal, Donal O’Regan-eds.)Kluwer Acad. Publ. Dordrecht, 2(2003) 617-644.
[7] B.V. Loginov, Branching of the solutions of nonlinear equations and their group sym-metry, Vestnik of Samara State Univ., 4(10)(1998), 15-75. (in Russian)
[8] M. M. Vainberg, V.A. Trenogin, Branching theory of solutions of nonlinear equations,Moscow, Nauka, 1969; Wolter Noordorf, Leyden, 1973.
[9] B.V. Loginov, Yu.B. Rousak, Generalized Jordan structure in branching theory, In:”Direct and Inverse Problems for PDEqs and Their Applications (M. Salakhitdinov-ed.) Fan, AN Uzbek SSR, Tashkent, 1978, 113-148. (in Russian)
[10] Yu.B. Rousak, Generalized Jordan structure in branching theory, PhD dissertation V.I.Romanovsky Mathematical University, Acad. Sci. Uzbek SSR, Tashkent, 1979. (in Rus-sian)
[11] I.V. Konopleva, B.V. Loginov, Yu.B. Rousak, Symmetry and Potentiality of branchingequations in the root-subspaces for implicitly given stationary and dynamic bifurca-tion problems, Izv. Severo-Kaukaz. Nauchn. Center Visch. Shkoly, (2009)(in print), (inRussian).
[12] I.V. Konopleva, B.V. Loginov, Yu.B. Rousak, Symmetry and Potentiality in generalproblem of branching theory, Izvestiya VUZ, Mathematics, 4(527) (2006), 30–40.
[13] B.V. Loginov, I.V. Konopleva, Yu.B. Rousak, Bifurcation and symmetry in differentialequations nonresolved with respect to derivative, ROMAI Journal 3, 1(2007), 151–173.
Branching equation in the root-subspace ... 107
[14] I.V. Konopleva, B.V. Loginov, Bifurcation, Symmetry and Cosymmetry in Differen-tial Equations Unresolved with Respect to the Derivative With Variational BranchingEquations, Doklady Mathematics, 80, 1(2009) 541–546.
[15] N.I. Makarenko, On solutions branching for invariant variational equations, RussianAcad. Sci., Doklady Mathematics, 348, 3(1996), 302–304.
[16] N.I. Makarenko,Symmetry and cosymmetry of variational problems in waves theory,Proc. Int. School-Seminar ”Applications of symmetry and cosymmetry in theories ofbifurcations and phase transitions”, Sochi (August 2001), Rostov- on Don Univ., (2001)109–120. (in Russian)
[17] B.V. Loginov, Branching Theory of Solutions of Nonlinear Equations Under GroupInvariance Conditions, Tashkent, Fan, 1985. (in Russian)
[18] B.V. Loginov,L.R. Kim-Tyan, Yu.B. Rousak, Modification of the Lyapounov-Schmidtmethod and the stability of solutions of differential equations with a singular operatorof finite index multiplying the derivative, Russian Acad. Sci., Doklady Mathematics,47, 3, (1993) 599–603. (in Russian)
[19] B.V. Loginov, I.V. Konopleva, Symmetry of resolving systems in degenerated functionalequations, Proc. Int. Conf. ”Symmetry and Differential Equations” (V.K. Andreev,V.V. Vasiliev - eds.), Krasnoyarsk, Inst. Math. Model. Siberian Branch. of RAS, (2000),42–46.
[20] V.R. Abdullin, N.A. Sidorov, Interlaced equations in branching theory, Russian Acad.Sci., Doklady Mathematics, 377, 3, (2001) 295–297.
SOLVABILITY OF HAMMERSTEIN EQUATIONSWITH ANGLE-BOUNDED KERNEL
ROMAI J., 5, 2(2009), 109–118
Irina A. LecaNaval Academy “Mircea cel Batran” Constanta, [email protected]
Abstract The Browder-Gupta splitting of angle-bounded operators is interplayedwith the elliptic super-regularization imbedding to study the solvabilityof Hammerstein operator equations.
Keywords: monotone operators, Hammerstein equations.
2000 MSC: 47H05, 47H30, 46B26.
1. INTRODUCTIONA nonlinear integral equation of Hammerstein type is one of the form:
(1) u (x) +∫Ω K (x, y) f (y, u (y)) dy = h (x)
where Ω is a measure space with a σ-finite measure dy, the given function h (x)and the unknown function u (x) are defined on Ω. In the operator-theoreticterms, the problem of determining a solutions of equation (1), with u and hlying in a given Banach space E of functions over Ω can be rewritten in theform of nonlinear operator equation(2) u + KNu = hwith a linear integral operator (generalizing the Green function)(3) Kv (x) =
∫Ω K (x, y) v (y) dy
and a nonlinear superposition (or Nemıtskii) mapping(4) Nu (x) = (Nfu) (x) = f (x, u (x)).
In the sequel, we consider the homogeneous equation attached to (2) andE = X∗, the dual space of the real Banach separable space X. It is well-knownthat a Banach space is reflexive iff every bounded sequence has a weakly con-vergent subsequence. The weak convergence can be replaced by the weak∗convergence of subsequences of bounded sequences on X∗ when X is a sep-arable Banach space. Another reason for the choice of X as separable is tohave conditions for the use of the Browder-Ton elliptic super-regularizationargument.
Let X be a real Banach space with its dual space X∗. Consider the op-erator Hammerstein equation:(5) u + KNu = 0, u ∈ X∗,
109
110 Irina A. Leca
where the operators K : X → X∗ and N : X∗ → X are of monotone type.We start with the basic definitions: Let X be a real Banach space, X∗ its
dual space and denote by (x∗, x) the duality pairing between x∗ ∈ X∗ andx ∈ X. A mapping T : D (T ) ⊆ X → X∗ is said to be monotone if
(Tx− Ty, x− y) ≥ 0, ∀x, y ∈ D (T ).More restrictive, T : D (T ) ⊆ X → X∗ is strongly monotone if there is m > 0such that:
(Tx− Ty, x− y) ≥ m ‖x− y‖2X , ∀x, y ∈ D (T ).
2. ANGLE-BOUNDED OPERATORSA significant concept introduced in the study of Hammerstein equations is
the angle-bounded operators as a subclass of monotone operators.
Definition 2.1. A linear monotone operator K : D (K) ⊆ X → X∗ is said tobe angle-bounded with the constant a ≥ 0, if(6) |(Kx, y)− (Ky, x)| ≤ 2a (Kx, x)
12 · (Ky, y)
12 , for all x, y ∈ D (K).
For the sake of simplicity, we consider further that D (K) = X.Angle-bounded mappings generalize symmetric mappings. Indeed, the angle-
boundedness of K with a = 0 corresponds to the symmetry of K, i.e. (Kx, y) =(Ky, x) for all x, y ∈ X.
As above, a linear mapping K : X → X∗ is said to be strongly monotoneoperator if there is a constant m > 0 such that (Kx, x) ≥ m ‖x‖2
X for allx ∈ X. It follows that every strongly monotone mapping is angle-bounded
with a =‖K‖m
since
|(Kx, y)− (Ky, x)| ≤ 2 ‖K‖ · ‖x‖ · ‖y‖ ≤ 2a (Kx, x)12 (Ky, y)
12 , ∀x, y ∈ X.
We now relate the angle-boundedness to the cyclic monotonicity. An oper-ator T : D (T ) ⊆ X → X∗ is cyclic monotone if
(Tx1, x1 − x2) + (Tx2, x2 − x3) + . . . + (Txn, xn − xn+1) ≥ 0,for all xi ∈ D (T ), i = 1, 2, . . . , n and all n ∈ N, where we set xn+1 = x1. Forn = 2 it reduces to usual monotonicity. A typical example of a cyclic monotonemultivalued mapping is the subdifferential of a proper convex semicontinuousfunction (see [9], pp. 124).
We are interested, in particular, when T : D (T ) ⊆ X → X∗ is 3-monotone,that is,
(Ty − Tx, y − z) ≥ (Tx− Tz, z − x), ∀x, y, z ∈ D (T ).We proved in [7] that the Nemıtskii operators are 3-monotone.In a more general way, we can consider
Solvability of Hammerstein equations with angle-bounded kernel 111
Definition 2.2. An operator T : D (T ) ⊆ X → X∗ is called 3-C-monotone ifthere is a constant C > 0 such that
(Ty − Tx, y − z) ≥ C (Tx− Tz, z − x), ∀x, y, z ∈ D (T ).
We have the following equivalences:
Theorem 2.1. Let K : X → X∗ be a linear monotone operator on the realBanach space X. The following three statements are equivalent:
(a) K is 3-C-monotone, i.e. there is a constant C > 0 such that(7) (Kx−Ky, z − x) ≤ C (Ky −Kz, y − z), for all x, y, z ∈ X;
(b) K satisfies the discriminant inequality(8) (Kv, w)2 ≤ 4C (Kv, v) · (Kw,w), for all v, w ∈ X;
(c) K is angle-bounded, i.e. there is a constant a > 0 such that(9) |K (x, y)−K (y, x)|2 ≤ 4a2K (x, x) K (y, y), for all x, y ∈ X.
Proof. Substituting in (7) v = y − z and w = x− z, by the linearity of K,we obtain
(Kv, w)− (Kw, w) ≤ C (Kv, v), for all v, w ∈ X.Replacing now v by tv, we obtain the inequality
C (Kv, v) t2 − (Kv,w) t + (Kw, w) ≥ 0, for all v, w ∈ X,which is equivalent with the non-negativity of its discriminant, that is (a) ⇔(b).
To establish the other equivalences, we now introduce
[x, y]± =12
((Kx, y)± (Ky, x)), for all x, y ∈ X.
Since K is monotone, we have [x, x]+ ≥ 0 for all x ∈ X and the generalizedSchwarz inequality ensures that(10) [x, y]2+ ≤ [x, x]+ [y, y]+, for all x, y ∈ X.
On the other hand, we have(11) (Kx, y) = [x, y]+ + [x, y]−, for all x, y ∈ X.
Moreover, the inequality (9), namely the angle-boundedness of K, can bewritten in the form:(12) [x, y]2− ≤ a2 [x, x]+ [y, y]+, for all x, y ∈ X.
From (11) and the inequality (A + B)2 ≤ 2A2+2B2 for real A, B, we derive(Kx, y)2 ≤ 2
(1 + a2
)[x, x]+ [y, y]+, for all x, y ∈ X,
that is (8) which is equivalent with the 3-C-monotonicity of K, as we provedearly. Therefore, (c) ⇒ (a).
Finally, if K is 3-C-monotone and [x, y]− = (Kx, y)− [x, y]+ then we obtainthat
[x, y]2− ≤ 2 (4C + 1) [x, x]2+ [y, y]2+, for all x, y ∈ X,which is (9). Thus (a) ⇒ (c), and the equivalences stated above are proved.2
The equivalence (a) ⇔ (c) suggests us an extension of angle-boundedness.
112 Irina A. Leca
Definition 2.3. A nonlinear mapping T : X → X∗ is said to be angle-boundedwith a constant C > 0, if
(Tx− Tz, z − y) ≤ C (Tx− Ty, x− y), for all x, y, z ∈ X.
If y = z then an angle-bounded operator T is monotone.
The angle-bounded operators were introduced by H. Amann [1] and appliedto the investigation of Hammerstein equations especially by F.E. Browder -C.P. Gupta [4] and F.E. Browder [3]. We mention the detailed approaches ofHammerstein equations in the monographs [9],[11] and [12].
3. THE SPLITTING OF ANGLE-BOUNDEDOPERATORS
We will develop the technique used in the proof of Theorem 2.1 to establisha splitting of linear angle-bounded operators, due to Browder-Gupta [4]. Ourapproach combines this splitting with the elliptic super-regularization method[5], recently improved in a general and simpler form by Berkovits [2].
Imbedding theorem of Browder and Ton. Let E be a real separableBanach space. Then there exists a separable Hilbert space H and a compactlinear injection Ψ : H → E such that Ψ (H) is dense in E.
We define further the adjoint operator Ψ∗ : E∗ → H by setting〈Ψ∗w, v〉 = 〈w,Ψv〉, ∀v ∈ H, w ∈ E∗,
where 〈·, ·〉 stands for the inner product in H. Since Ψ (H) is dense in E, Ψ∗is also a linear compact injection.
We describe the Browder-Gupta theorem for splitting the angle-boundedoperators.
Theorem 3.1. Let X be a real separable Banach space and K : X → X∗ alinear monotone operator which is angle-bounded with constant a ≥ 0. Then
(A) there exist a separable Hilbert space H, a linear injection S : X → Hwith its adjoint map S∗ : H → X∗ and a skew-adjoint bounded operatorB : H → H such that K = S∗ (I + B) S.
(B) S : X → H is compact, its range S (X) is dense in H, S∗ : H → X∗ isalso a compact injection and ‖S‖2 ≤ ‖K‖;
(C) B : H → H is a linear skew-adjoint operator, i.e., B∗ = −B, ‖B‖ ≤ aand I + B is bijective.
Proof. It is sufficient to take E = X∗ and change the names of the compactinjections:
S = Ψ∗ : X → H and S∗ = Ψ : H → X∗where the existence of the Hilbert space (H, (·, ·)) is ensured by the Browder-Ton theorem. We carry out the proof of Theorem 2.1 and consider the null-
Solvability of Hammerstein equations with angle-bounded kernel 113
spaceZ =
x ∈ X| [x, x]+ = 0
.
Taking into account (11) and (12), we derive that [u, v]± = 0 for all u ∈ Zand v ∈ Z and thereby [x + u, y + v]± = 0 for all x, y ∈ X and all u, v ∈ Z.Consequently, the equivalence relation x ∼ y iff x − y ∈ Z holds. Denote byH0 the quotient (pre-Hilbert) space X/Z with the equivalence classes definedabove. We denote by U, V, W . . . the classes of H0 and consider the Hilbertspace H as the completion of H0 with respect to the norm ‖U‖2 = [u, u]+ =(Ku, u) for all U ∈ H0 and u ∈ U . Moreover, on H the scalar product〈U, V 〉 = [u, v]+ is defined, whatever are the representatives u ∈ U and v ∈ V .Indeed, from 〈U,U〉 = 0 it follows that [u, u]+ = 0 for every u ∈ U , henceu ∈ Z, therefore U = 0.
The injection S : X → H defined bySu = U , u ∈ U ,
is the natural projection that assigns to each u ∈ X∗ the corresponding equiv-alence class U in H0. Therefore
〈Su, Sv〉 = [u, v]+ =12
((Ku, v) + (Kv, u)).The map S : X → H is linear and continuous and by‖Su‖2
H = 〈Su, Su〉 = (Ku, u) ≤ ‖Ku‖ · ‖u‖ ≤ ‖K‖ · ‖u‖2,it follows that ‖S‖2 ≤ ‖K‖. In addition, since the range of S coincides withH0, the range of S is dense in H and the adjoint map S∗ : H → X∗ is injective.
For U, V ∈ H0, let us consider now the skew-symmetric part:
b (U, V ) = [u, v]− =12
((Ku, v)− (Kv, u)) where u ∈ U and v ∈ V .
Taking into account of the angle-boundedness (12) of K , we have(13) |b (U, V )| ≤ a ‖U‖H ‖V ‖H , for all U, V ∈ H0.
As H0 is dense in H, the bilinear form b : H × H → R is well-definedand induce a linear operator B : H → H such that b (U, V ) = 〈BU, V 〉for all U, V ∈ H. Likewise, by (13) we have ‖B‖ ≤ a. Besides, becauseb (U, V ) = −b (U, V ) it follows that B∗ = −B, i.e., B is a skew-adjoint opera-tor.
The splitting K = S∗ (I + B) S follows from(Ku, v) = [u, v]+ + [u, v]− = 〈Su, Sv〉+ 〈BSu, Sv〉 == 〈(I + B) Su, Sv〉 = (S∗ (I + B) Su, v), ∀u, v ∈ X. 2
For simplcity, we denote further A = I + B : H → H and prove:
Propozitia 3.1. The operators A and A−1 are strongly monotone. Moreover,A : H → H is a bijection.
Proof. Since B is skew-adjoint, we have 〈BU,U〉 = b (U,U) = 0 and(14) 〈AU,U〉 = 〈(I + B) U,U〉 = ‖U‖2
H , for all U ∈ H,which means the strong monotonicity of A. By ‖B‖ ≤ a, it follows that
114 Irina A. Leca
(15) ‖AU‖2H = 〈(I + B) U, (I + B) U〉 = ‖U‖2
H + ‖BU‖2H ≤ (
1 + a2) ‖U‖2
H ,for all U ∈ H. Combine (14) and (15) to derive(16)
⟨U,A−1U
⟩=
⟨A
(A−1U
), A−1U
⟩=
∥∥A−1U∥∥2
H≥ (
1 + a2)−1 ‖U‖2
H ,for all U ∈ H, that is, the strong monotonicity of A−1.
Finally, from ‖AU‖H ≥ ‖U‖H it follows that A is injective and A (H) isclosed. We prove also that A is surjective. Indeed, assuming A (H) 6= H, thereexists 0 6= V ∈ H with 〈AU, V 〉 = 0 for U ∈ H. In particular, 〈AV, V 〉 = 0and (14) implies V = 0, hence A (H) = H. 2
At the end of this section, we list some additional properties and commentsrelated to the previous maps, required further down.
Since we assume that the real separable Banach space is not generally re-flexive, for the monotonicity of nonlinearity N : X∗ → X we will consider Xas a subset of the bidual space X∗∗ and identify the Hilbert space H with itsdual H∗.
4. SOLVABILITY OF HAMMERSTEINEQUATION
We deal with the existence and uniqueness of solutions of Hammersteinequations. Recall that an operator T : D (T ) ⊆ X → X∗ is hemicontinuous ifit is continuous from each line segment of D (T ) to the weak topology of X∗.Further, T is called coercive if
lim‖x‖→∞
(Tx, x)‖x‖ = +∞.
The following basic criterion is used.Surjectivity theorem of Minty-Browder. An operator T : D (T ) ⊆
X → X∗ hemicontinuous, monotone and coercive on a real reflexive Banachspace X is surjective.
Using above notations and the splitting of the angle-bounded operator K,obtained in the factorization Theorem 3.1, the nonlinear Hammerstein equa-tion (5) becomes(17) u + S∗ASNu = 0, u ∈ X∗.
Since S∗ : H → X∗ is an injection, by u = S∗W, eq. (17) is equivalent toS∗W + S∗ASNS∗W = 0, W ∈ H,
that is, S∗ (I + ASNS∗) W = 0. Taking again into account that S∗ is injec-tive, the equation (5) is equivalent to(18) W + ASNS∗W = 0, W ∈ H.
Solvability of Hammerstein equations with angle-bounded kernel 115
Since A : H → H is a bijection, we consider the equivalent form of (17)(19) A−1W + SNS∗W = 0, W ∈ H.for which we establish the following existence and uniqueness result.
Theorem 4.1. Let X be a real separable Banach space, K : X → X∗ a linear,monotone, angle-bounded operator and N : X∗ → X a nonlinear hemicontin-uous monotone mapping. Then the Hammerstein equation u + KNu = 0 hasexactly one solution u ∈ X∗.
Proof. As it has been explained above, taking account of the splitting ofthe angle-bounded operator K, given by Theorem 3.1, the equation (19) onthe associated Hilbert space H to the operator K is equivalent with the initialequation (5) for u = S∗W .
We define F = A−1+SNS∗ and show that F : H → H is a hemicontinuous,strongly monotone operator. Indeed, for U, V ∈ H,〈FU − FV,U − V 〉 =
⟨A−1 (U − V ) , U − V
⟩+ (S∗U − S∗V,NS∗U −NS∗V )
holds and, due to the monotonicity of N and (16), we obtain(20) 〈FU − FV, U − V 〉 =
⟨A−1 (U − V ) , U − V
⟩ ≥ (1 + a2
)−1 ‖U − V ‖2H ,
i.e., F is strongly monotone. Moreover, F is hemicontinuous like N and Fis coercive since it is strongly monotone. By the Minty-Browder surjectivitytheorem of monotone mappings, the operator F : H → H is bijective. Inparticular, there exists a unique solution U ∈ H of the equation FU = 0. 2
To specify for later use some determinations of mappings above introduced,we mention that A∗ : H → H and S∗∗w = Sw for all w ∈ X. Furthermore,for the adjoint operator K∗ = S∗A∗S∗∗ we have the representation(21) K∗x = S∗A∗Sx for all x ∈ X.
Later on, an approximation-solvability in Petryshyn’s sense [6], for theHammerstein equations will be outlined. We confine ourselves to a simpleprojection-solvability [8].
Let Xn be a monotone increasing sequence of finite-dimensional subspacesof X such that
⋃Xn is dense in X. Such a sequence Xn is called a projec-
tional system in X. Defining now the Hilbert subspaces Hn by(22) Hn = (S∗)−1 K∗ (Xn), n = 1, 2, . . ..we will prove that Hn is also a projectional system in H. Indeed, fromXn ⊆ Xn+1 it follows that Hn ⊆ Hn+1 ⊆ H and we have to show that
⋃Hn
is dense in H, that is, 〈U, V 〉 = 0 for all V ∈ ⋃Hn implies U = 0. In fact, for
x ∈ ⋃Xn, we have
⟨U, (S∗)−1 K∗x
⟩= 0 and, by (21),
〈AU,Sx〉 = 〈U,A∗SX〉 =⟨U, (S∗)−1 K∗x
⟩= 0.
Because we assumed that⋃
Xn is dense in X and S : X → H is continuous,we obtain 〈AU,Sx〉 = 0 for all x ∈ X. By assertion (B) in Theorem 3.1, the
116 Irina A. Leca
range S (X) is dense in H and hence AU = 0. As A : H → H is bijective, weconclude that U = 0. 2
Concerning the operator equation(23) Tu = f , u ∈ X,where T : X → X∗ is a (generally, nonlinear) hemicontinuous and mono-tone mapping, X a real separable Banach space and f ∈ X∗, we introduce aGalerkin method. Let Xn be a projectional system in X. With the equa-tion (23), we associate the sequence of finite-dimensional equations or Galerkinequations(24) (Tun, x) = (f, x), ∀x ∈ Xn.
Definition 4.1. For a given f ∈ X∗, equation (23) is projectionally-solvableif the following conditions hold:
(i) The original equation (23) has a unique solution;(ii) There is a number N such that for n ≥ N , the Galerkin equation (24)
has a unique solution un ∈ Xn;(iii) The Galerkin solutions converges, i.e., un → u in X as n →∞.
In a general framework, the approximation-solvability of (23) is related tothe A-properness of mapping T , (see e.g. [10]). However, one of the simplestexample of the projectionally-solvability of equation (23) holds in the casewhen T : X → X∗ is a strongly monotone operator.
Returning to the original Hammerstein nonlinear equation(5) u + KNu = 0, u ∈ X∗,with the operators K : X → X∗ and N : X∗ → X of monotone type, weconsider a Galerkin method, i.e. we study the approximate equations(25) (un + KNun, x) = 0, ∀x ∈ Xn,where Xn is a projection system in X.
Theorem 4.2. The Galerkin equation (25) has exactly one solution un ∈ Xn
for every n ≥ N and the sequence un converges in the norm topology of X∗to the solution of equation (5).
Proof. The conclusion follows from the equivalence between the Galerkinsolution of (25) and the construction of approximate solution Un ∈ Hn of theequation:(26) 〈FUn, V 〉 = 0, ∀V ∈ Hn
where Hn is the associate projection system given by (22).By Theorem 4.1, the Galerkin equation (26) has exactly one solution Un ∈
Hn and Un → U in H as n →∞, where U ∈ H be the unique solution of theequation FU = 0.
To complete the proof, we show that un = S∗Un is a solution of the originalGalerkin equation (25). More precisely, we look for un ∈ K∗ (Xn) and provethe equivalence between the equations (25) and (26).
Solvability of Hammerstein equations with angle-bounded kernel 117
As above in (26), we have〈FUn,W 〉 =
⟨A−1Un,W
⟩+ (NS∗Un, S∗W ), ∀W ∈ Hn.
By (21), for every x ∈ Xn we set K∗x = S∗A∗Sx in X∗. Moreover, to everyx ∈ Xn corresponds W = (S∗)−1 K∗x in H, that is, W = A∗Sx. We compute⟨
A−1Un, W⟩
=⟨A−1 (S∗)−1 un,W
⟩=
⟨A−1 (S∗)−1 un, A∗Sx
⟩=
=⟨(S∗)−1 un, Sx
⟩= (un, x)
and(NS∗Un, S∗W ) = (Nun, S∗A∗Sx) = 〈SNun, A∗Sx〉 == 〈ASNun, Sx〉 = (S∗ASNun, x) = (KNun, x).
Therefore〈FUn,W 〉 = (un + KNun, x).
From Hn = (S∗)−1 K∗ (Xn) and un = S∗Un with Un ∈ Hn it follows thatun ∈ K∗ (Xn) and the equivalence between (25) and (26) is established.
In the end, we prove the strong convergence of the sequence of Galerkinsolutions. Let Pn : H → Hn be the projection. Since Hn is a projectionalsystem in H, we have:(27) ‖PnV − V ‖H → 0 as n →∞,for every V ∈ H. The restriction Fn = PnF|Hn
has the same properties asF , from which the unique solvability of the equation FnU = 0 holds, as well.As above, this equation can be rewritten 〈FnU, V 〉 = 0 for all V ∈ Hn. Wedenote later by Un ∈ Hn the solutions of this Galerkin method. Moreover, ifU ∈ H is a solution of the initial equation FU = 0, then, by (27), we have∥∥F
(PnU
)∥∥H→ 0 as n →∞.
Taking into account the strong monotonicity (20) and continuity of opera-tor F : H → H, we derive for approximate solutions the following estimate:∥∥Un − PnU
∥∥H≤ (
1 + a2) ∥∥Fn
(Un
)− Fn
(PnU
)∥∥H≤
≤ (1 + a2
) ∥∥F(PnU
)∥∥H→ 0
as n →∞ and thereby,∥∥Un − U∥∥
H≤ ‖S∗‖ (∥∥U − PnU
∥∥H
+∥∥PnU − U
∥∥H
) → 0 as n →∞(the norm ‖S∗‖ is taken in the space L (H, X∗)), meaning that Un → U in H.
We set, as above, un = S∗Un and u = S∗U . Then u ∈ X∗ is a unique solu-tion of Hammerstein equation u + KNu = 0 and the continuity of S∗ impliesthat un → u in X∗ as n →∞. 2
Acknowledgements.The author wants to express the gratitude to her Ph.D. advisorDan Pascali for the valuable support in writting this paper.
References
[1] H. Amann, Zum Galerkin-Verfahren fur die Hammerstein Gleichungen, Arch. RationalMech. Anal. 35 (1969), 114-121; MR 42#8345.
118 Irina A. Leca
[2] J. Berkovits, A note on the imbedding theorem of Browder and Ton, Proc. Amer. Math.Soc. 131 (2008), 2963-2966; MR 2004b:46021.
[3] F.E. Browder, Nonlinear functional analysis and nonlinear integral equa-tions of Ham-merstein and Urysohn type, Contributions to nonlinear functional analysis (E.H. Zaran-tonello), 425-500, Academic Press, New York, 1971; MR 52#15143.
[4] F.E. Browder, C.P. Gupta, Monotone operators and nonlinear integral equations ofHammerstein type, Bull. Amer. Math. Soc. 75 (1969), 1347-1353; MR 40#3381.
[5] F.E. Browder, B.A. Ton, Nonlinear functional equations in Banach spaces and ellipticsuper-regularization, Math. Z. 105 (1968) 177-195; MR 38#582.
[6] P. M. Fitzpatrick, W. V. Petryshyn, Galerkin methods in the constructive solvabilityof nonlinear Hammerstein equations with applications in differential equations, Trans.AMS 238 (1978), 321-340; MR 58#47H15.
[7] I. A. Leca, Monotonicity of the Nemitsky operator, A XX-a Sesiune de ComunicariStiintifice cu Participare Internationala NAV-MAR-EDU 2007, Constanta, 2007, 277-283.
[8] D. Pascali, An introduction to numerical functional analysis, Petryshyn’s A-propermapping theory, Lect. Notes Math. Ovidius Univ., Constanta, 2001.
[9] D. Pascali, S. Sburlan, Nonlinear mappings of monotone type, Edit. Academiei, Bu-curesti and Sijthoff & Noordhoff Intern. Publ., Alphen aan den Rijn, 1978; MR80g:47056.
[10] W.V. Petryshyn, Approximation-solvability of nonlinear functional and differentialequations, Marcel Dekker, New York, 1991; MR 94f: 4781.
[11] S.F. Sburlan, Gradul topologic, Edit. Academiei, Bucuresti, 1983; MR 86b:47106.
[12] E. Zeidler, Nonlinear functional analysis and its applications, Part II/A: Linear mono-tone operators; II/B: Nonlinear monotone operators, Spriger-Verlag, New York, 1990;MR 91b:47001, 47002.
NUMERICAL K - BUSY PERIODS ALGORITHMSFOR POLLING SYSTEMS WITH SEMI - MARKOVSWITCHING
ROMAI J., 5, 2(2009), 119–126
Gheorghe Mishkoy, Diana BejenariFree International University of Moldova, Chisinau, Republic of [email protected], [email protected]
Abstract The queueing systems of Polling type are widely used in wireless networks withbroadband centralized management (see, e.g., [1]). One of the important char-acteristics of these systems is the k - busy period [2]. In [3] it is shown thatanalytical results for k - busy periods can be viewed as generalizations of classi-cal Kendall functional equation [4]. Unfortunately, analytical solution for suchtype of generalized equations does not exist. However, using the methodologyof generalized priority systems and generalized algorithms elaborated in [3],numerical solutions with necessary required accuracy can be obtained. Someexamples and numerical results are presented here.Acknowledgement. This work is supported partially by Russian Foundationfor Basic Research (RFFI) grant 08.820.08.09RF and grant 09.820.08.01GF ofthe Federal Ministry of Education and Research (BMBF) of Germany.
Keywords: k - busy period, numerical algorithms. system, LT-algebra.
2000 MSC: 60K25, 68M20, 90B22.
1. INTRODUCTIONIt is known that wireless networks have developed rapidly last years. For
planning regional wireless networks, models and research methods of Pollingsystems are used. In this systems there is a common server for all users whichserves proposed messages by users according to given rules. The main purposeof research of Polling systems is to determine the characteristics of systemsdevelopment. But not always analytical formulas can be used directly, sogreat care is offered to numerical algorithms. In this paper, some examples ofdetermining the k - busy period, that are obtained from numerical algorithmsare presented.
2. THE K - BUSY PERIODDefinition 2.1. The period beginning with the exchange of user and endingwhen the system becomes free from the requirements of class k (messages fromuser k) is named a k - busy period [3].
119
120 Gheorghe Mishkoy, Diana Bejenari
By Πδk is denoted the length of the k period, and by
Πδk(x) = PΠδ
k < x,its distribution function. Further we will consider that
πδk(s) =
∞∫
0
e−sxdΠδk(x)
is the Laplace - Stieltjes transform of distribution function of k -period.
The following result is known [3].
Theorem 2.1. The function πδk(s) is determined from equation
πδk(s) = ck(s + λk − λkπk(s))πk(s), (1)
whereπk(s) = βk(s + λk − λkπk(s)), (2)
and by ck(s) and βk(s) are denoted the Laplace - Stieltjes transforms of dis-tribution functions Ck(x) and Bk(x)
ck(s) =
∞∫
0
e−sxdCk(x),
βk(s) =
∞∫
0
e−sxdBk(x).
Functional equation (2) has no analytical solution, but it can be solvednumerically with some required accuracy. Several numerical algorithms inC++ for solving functional equation (2) and (1), were created, depending onthe type of distribution function taken by Bk(x) and Ck(x).
3. EXAMPLESIn our examples we use the well known [5] types of distribution function:
1 Uniform distribution on [a, b], U(a, b).
F (x) =
0, x < a,(x− a)/(b− a), a ≤ x ≤ b,
1, x > b,
Numerical k - busy periods algorithms for Polling systems with semi - Markov switching 121
E(x) =a + b
2,
f(s) =1
b− a
1s(e−sa − e−sb).
2 Exponential distribution, Exp(λ).
F (x) = 1− e−λx, x > 0,
E(x) =1λ
,
f(s) =λ
s + λ.
3 Erlang distribution, Erl(λ, k).
F (x) =
0, x < 0,x∫0
λ (λu)k−1
(k−1)! e−λudu, x ≥ 0,
E(x) =k
λ,
f(s) =(
λ
λ + s
)k
.
4 Gamma distribution, Ga(λ,a).
α > 0, λ > 0,
F (x) =
0, x < 0,
λα
Γ(α)
∞∫0
xα−1e−λxdx, x ≥ 0,
Γ(α) =
∞∫
0
xα−1e−λxdx, Γ(α) = (k − 1)! for α = k,
E(x) =α
x,
122 Gheorghe Mishkoy, Diana Bejenari
f(s) =(
λ
λ + s
)α
.
Remark 1. For α = 1, Ga(λ, 1) = Exp(λ).
Remark 2. For α = k,Ga(λ, 1) = Erl(λ, k). For this reason we choseonly Erlang distribution.
5 Normal distribution, N(a, σ2).
F (x) =1
σ√
2π
x∫
−∞e−
(u−a)2
2σ2 du, −∞ < x < ∞,
E(x) = a, V (x) = σ2,
f(s) = e(σ2s2−sa).
Remark 3. Normal standard distribution N(0, 1).
Φ(x) =1√2π
x∫
−∞e−
t2
2 dt, −∞ < x < ∞,
E(x) = a = 0, V (x) = σ2 = 1,
f(s) = es2 .
Example 1. The type of distribution function taken by Bk(x) and Ck(x)is the Exponential distribution, so
Bk(x) = 1− ebkx, x > 0,
andCk(x) = 1− eckx, x > 0,
with following parameters:λk =2, 3, 5, 6, 3, 8, 5, 4, 7, 3, 4, 2, 9, 7, 6, 4, 6, 3, 5, 2,bk =7, 5, 9, 4, 6, 2, 5, 4, 8, 6, 5, 3, 4, 5, 7, 6, 2, 4, 8, 6,
Numerical k - busy periods algorithms for Polling systems with semi - Markov switching 123
ck =7, 5, 9, 4, 6, 2, 5, 4, 8, 6, 5, 3, 4, 5, 7, 6, 2, 4, 8, 6.The results of the program are presented in Table 1.
k πk(s) πδk(s) k πk(s) πδ
k(s)
1 0.499831 0.181751 11 0.369369 0.096169
2 0.389415 0.151739 12 0.287740 0.096169
3 0.515323 0.098878 13 0.236751 0.075282
4 0.278846 0.121721 14 0.316054 0.085492
5 0.438095 0.149959 15 0.424283 0.180499
6 0.133333 0.042254 16 0.418269 0.110781
7 0.350425 0.122974 17 0.152174 0.047297
8 0.313589 0.058366 18 0.333156 0.155468
9 0.445795 0.199467 19 0.480989 0.197734
10 0.438095 0.172673 20 0.458472 0.165468
Table 1.
Example 2. The type of distribution function taken by Bk(x) is the Erlangdistribution and by Ck(x) is the Exponential distribution, so
Bk(x) =
0, x < 0,x∫0
λk(λku)k−1
(k−1)! e−λkudu, x ≥ 0,
and
Ck(x) = 1− eckx, x > 0,
with following parameters:λk =3, 5, 7, 6, 8, 4, 5, 7, 3, 6, 8, 5, 9, 7, 6, 2, 4, 1, 1, 1,bk =4, 4, 6, 7, 4, 8, 8, 3, 2, 5, 5, 7, 6, 4, 8, 6, 4, 2, 4, 6,ck =6, 7, 4, 3, 7, 6, 9, 7, 4, 5, 3, 2, 5, 6, 7, 5, 4, 3, 2, 8,pk =7, 5, 4, 3, 6, 5, 2, 5, 5, 7, 6, 5, 9, 8, 6, 5, 4, 6, 7, 6.
The results of the program are presented in Table 2.
124 Gheorghe Mishkoy, Diana Bejenari
k πk(s) πδk(s) k πk(s) πδ
k(s)
1 0.000155 5.828645e-05 11 0.000244 4.069452e-05
2 0.000977 0.000360 12 0.006788 0.000972
3 0.008100 0.001806 13 8.347513e-06 1.98751e-06
4 0.044573 0.008499 14 5.947027e-06 1.784112e-06
5 8.706395e-05 2.770304e-05 15 0.424283 0.001071
6 0.013420 0.004751 16 0.010310 0.003688
7 0.174848 0.078190 17 0.0050 0.001350
8 0.000171 5.705125e-05 18 6.4e-057 1.745465e-05
9 0.000129 3.67441e-05 19 0.000457 9.145366e-05
10 0.000128 3.544855e-05 20 0.006196 0.003099
Table 2.
Example 3. The type of distribution function taken by Bk(x) is theErlangdistribution and by Ck(x) is Normal distribution, so
Bk(x) =
0, x < 0,x∫0
λk(λku)k−1
(k−1)! e−λkudu, x ≥ 0,
and
Ck(x) =1σk
√2π
x∫
−∞e− (u−a)2
2σk2 du, −∞ < x < ∞,
with following parameters:λk =0.6, 0.4, 0.6, 0.7, 0.3, 0.4, 0.6, 0.2, 0.6, 0.4, 0.7, 0.1, 0.3, 0.5, 0.7, 0.5,bk =0.2, 0.4, 0.7, 0.4, 0.6, 0.8, 0.6, 0.3, 0.4, 0.5, 0.3, 0.2, 0.5, 0.4, 0.6, 0.7,ck =0.5, 0.3, 0.5, 0.7, 0.5, 0.3, 0.4, 0.8, 0.5, 0.4, 0.2, 0.3, 0.4, 0.5, 0.3, 0.5,pk =2, 4, 3, 5, 4, 6, 5, 4, 3, 2, 4, 6, 5, 4, 7, 6,σk =2, 4, 3, 5, 4, 6, 5, 4, 3, 2, 4, 6, 5, 4, 7, 6.
The results are in the Table 3.
Numerical k - busy periods algorithms for Polling systems with semi - Markov switching 125
k πk(s) πδk(s) k πk(s) πδ
k(s)
1 0.028566 0.019760 9 0.024037 0.015351
2 0.012551 0.011409 10 0.163897 0.154544
3 0.075241 0.058980 11 0.002108 0.001942
4 0.001348 0.000656 12 0.000544 0.000473
5 0.047328 0.034053 13 0.012758 0.010210
6 0.015996 0.015313 14 0.008963 0.007629
7 0.007416 0.00564 15 0.000682 0.000563
8 0.012482 0.008618 16 0.007012 0.005264
Table 3.
Example 4. The types of distribution function for Bk(x) and Ck(x) aregiven in the Table 6. We used the following notations:
If the distribution function is Uniform we denote by the letter U,
If the distribution function is Erlang – I,
If the distribution function is Exponential – E,
If the distribution function is Normal– N.
λk =0.2, 0.4, 0.7, 0.5, 0.6, 0.3, 0.4, 0.5, 0.6, 0.9, 0.2, 0.3Required parameters for each distribution function are in the Table 4. andTable 5.
U N U I N U I N I
a = 4 σ = 0.6 a = 5 k = 2 σ = 0.9 a = 10 k = 3 σ = 0.4 k = 4b = 1 a = 0.2 b = 2 a = 0.2 b = 2 a = 0.1
Table 4.
U I U I U I N U I
a = 14 k = 1 a = 9 k = 2 a = 3 k = 2 σ = 0.5 a = 4 k = 2b = 7 b = 2 b = 1 a = 0.3 b = 1
Table 5.
126 Gheorghe Mishkoy, Diana Bejenari
The results of the program are given in Table 6.
k Bk(x) Ck(x) πk(s) πδk(s) k Bk(x) Ck(x) πk(s) πδ
k(s)
1 U N 0.367026 0.363908 7 E I 0.704387 0.496661
2 E U 0.846724 0.250955 8 E U 0.920868 0.178205
3 I N 0.599745 0.675492 9 I E 0.624615 0.565239
4 E U 0.894586 0.147271 10 U I 0.139467 0.059018
5 I N 0.608076 0.601602 11 N U 0.947065 0.451609
6 I U 0.68603 0.014749 12 I E 0.699749 0.657033
Table 6.
References[1] V. M. Vishnevsky, O. V. Semenova, Polling Systems: The theory and applications in
the broadband wireless networks, Moscow, Texnocfera, 2007 ( in Russian).
[2] V. V. Rycov, Gh. K. Mishkoy, A new approach for analysis of polling systems. //Pro-ceedings of the International Conference on Control Problems, Moscow, (2009), 1749 -1758.
[3] Gh. K. Mishkoy, Generalized Priority Systems, Chisinau: Acad. of Sc. of Moldova,Stiinta, 2009, (in Russian).
[4] H. Takagi, Queueing Analysis: Vacation and Priority Systems, North-Hodland, Else-vier Science Publ, vol. 1, 1991.
[5] N. A. J. Hastings, J. B. Peacock, Statistical Distribution, Moskow, Statistics, 1980, (inRussian).
‘FRAGILE BITS’ VS. MULTI-ENROLLMENT -A CASE STUDY OF IRIS RECOGNITION ONBATH UNIVERSITY IRIS DATABASE
ROMAI J., 5, 2(2009), 127–144
Nicolaie Popescu-Bodorin“ Spiru Haret” University, Bucharest, Romaniahttp://fmi.spiruharet.ro/bodorin/
Abstract This paper explores the use of fragile bits in the context of a recently proposediris recognition methodology based on Circular Fuzzy Iris Segmentation andGabor Analytic Iris Texture Binary Encoder. Iris images from Bath UniversityIris Database are encoded as iris codes at three different lengths (192, 512,768 Bytes) and used to test the concept of fragile bits. Six iris recognitiontests are presented in order to illustrate the efficiency of the recently proposediris recognition methodology in both single-enrollment and multi-enrollmentiris recognition scenarios. Three additional tests show that at least in therecognition scenarios presented here, using a certain type of fragile bits willdetermine a split of the set of genuine scores into two distributions havingtwo completely different statistics, case in which apparent improvements ofthe recognition performance measured through decidability index and throughFisher’s ratio are irrelevant. Also, some important differences between differenttypes of fragile bits are stated here for the first time.
Keywords: iris recognition, iris segmentation, circular fuzzy iris segmentation, Gabor an-
alytic iris texture binary encoder, fragile bits;.
2000 MSC: 68T10, 68U10, 68N99, 44A15.
1. INTRODUCTIONFrom the early stages of our PhD study we found that an open problem in
iris recognition is the fact that we can’t say for sure if a given iris databaseis or isn’t a representative sample. To be more precise, if the length of theiris binary code is assumed to be 1024, then the numerical space represent-ing the iris population counts more than 1.7E + 308 elements. Now, let’simagine a huge iris database containing, let’s say, 1E + 12 images and as-sume that extraordinary iris recognition results have been proved using thisdatabase. We should use some sampling techniques enabling us to extrapolatethese results to entire iris code population (and to other similar databases,in particular), despite the fact that representativity rate of our hypotheticaldatabase is nearly null (1E− 296). Unfortunately, such techniques don’t existand consequently, in these circumstances, explaining the differences betweentheoretical and experimental results could prove to be difficult and misleading.
127
128 Nicolaie Popescu-Bodorin
A possible example is the concept of fragile bits, initially introduced to explainthe difference between experimentally determined False Reject Rates and thetheoretically predicted values.
As an alternative, we consider that the set of all available iris-codes isso sparse and scattered within the iris-code population that the matchingbetween the bits of any two different irides happens only by chance. This isthe first major hypothesis in our approach to iris recognition. It was initiallyformulated by Daugman in the early 90s but never fully exploited ever since.Present paper shows that accepting and following this hypothesis leads totheoretical and experimental iris recognition results agreeing each other. Inthis scenario we will give here some examples showing that masking the socalled fragile bits don’t necessary improve the iris recognition performance.
The concept of fragile bits was proposed by Bolle and all [1] but the firstwork truly investigating this subject is very recent indeed (Hollingsworth andall [4]). The cited paper introduces fragile bits in two ways: first, as a maskof unstable bits that are flipping from 0 to 1 or vice versa in a chosen numberof observations, and second, as a mask of the bits that are more susceptible toflip because of a phase instability phenomenon: in one particular observation,a bit is considered to be fragile if the corresponding complex value encodingthe corresponding iris pixel is located close to the real or imaginary axis. But,in fact, the two cases describe two different concepts:- In the first case, fragile bits are defined as an a priori knowledge inferred froma given number of previous observation as a mask containing the pixels thatare proven to be unstable by their values, but the causality of this instabilityis not evident. In this context, it is very important to answer the followingquestions: is this mask a feature of the iris or it is just a feature of the givenset of observations? Does it depend on the iris texture or on the variability ofthe acquisition conditions?- As defined in the second case, the concept of fragile bits represents, in fact,an a posteriori knowledge about the current observation(s), a mask defined bya fuzzy membership assignment conditioned by the degree of closeness to thereal/imaginary axis or, as mentioned in [4], by an adaptive thresholding of thelowest quartile calculated for the histogram of the set containing the absolutevalues corresponding to the complex representation of the unwrapped iris.- To test the concept of fragile bits in the first scenario, the current iris codeswill be compared to the stored template gallery using the mask of fragile bitscomputed for the enrolled identities. The results of the tests could give theanswers to the questions formulated above. But, the most important outcomeof such a test will be to find out if the use of fragile bits is or isn’t compatiblewith the hypothesis that the matching between the bits of any two differentirides happens only by chance.- In the second scenario, testing the concept of fragile bits means to assume
‘Fragile Bits’ vs. Multi-Enrollment - a case study of iris recognition... 129
that any two iris codes will be compared using a mask computed at run-time.The best possible outcome of such a test would be an improvement in thesystem performance quantified as a narrowing of the genuine or /and imposterscore distributions, an increase of the distance between the two classes ofscores, or an improvement of other values calculated as evaluation criteria(decidability index, Fisher’s ratio, [17])
We must clarify that we call a priori knowledge any piece of knowledge/dataused to bring the biometric system in a fully functional ready state able totreat the client recognition request. One notable difference between the abovetwo cases is that in the former, a priori knowledge includes the (stored) masksof fragile bits for all enrolled identities, while in the latter, a priori knowledgeincludes a method to produce (at run-time) a mask for any given pair of iriscodes. From the beginning, it can be remarked that if the mask of fragile bitsmainly depends on other things than the iris texture, at least one from theabove tests will fail to improve recognition performance.
2. CIRCULAR FUZZY IRIS SEGMENTATIONThe segmentation algorithm presented here was proposed in [12] as an al-
ternative to the currently available segmentation procedures that use integro-differential Daugman operator [2], or Hough Transform [16], or active contours[3]. The most important difference between previous segmentation methodsand CFIS is the dimension of the parameter space needed to be searched in or-der to find pupillary and limbic boundary: in CFIS procedure iris boundariesare found by solving one dimensional optimization problems.
CFIS procedure consists in the following operations: pupil finding and lim-bic boundary circular approximation. Pupillary and limbic boundaries areassumed to be concentric circles.
An anatomic argument for using this hypothesis is that since the pupilis nearly circular, there must be a circular concentric iris ring controllingthe pupil movements. Such a circular iris ring is expected to play the mostimportant role in iris recognition, despite the fact that it appears to be a roughapproximation of the actual iris.
A system requirement sustaining the above formulated hypothesis is thatthe segmentation routine must be fast and energy-efficient. Nearly losslessunwrapping of the iris can be computed using a polar / bipolar coordinatetransform depending on the type assumed for the iris: concentric / eccentriccircular ring. The latter is computationally more expensive than the formerbecause the eccentricity varies from a sample to another and consequently, onebipolar mapping must be (re)computed for each sample (eye image). Whenthe iris ring is assumed to be concentric, the polar mapping is computed oncefor all samples, during program initialization.
130 Nicolaie Popescu-Bodorin
2.1. FAST PUPIL FINDER ALGORITHMThe proposed pupil finder algorithm can be stated as follows:
Fast Pupil Finder Algorithm (N. Popescu-Bodorin): INPUT: the eye
image IM; 1.Extract the pupil cluster (Fig.1.a):
PC = fkmq(IM,16);
PC = (PC == min(PC(:)));
2.Compute horizontal and vertical Run Length
quantization of PC (Fig.2.a-b):
RLV(:,,j) = vrleq(PC);
RLH(j,:) = hrleq(PC);
3.Compute the pupil indicator PI (Fig.2.c):
[k, PI] = getpi(RLH, RLV);
PI = find(PI == 1);
PI = PI(1);
4.Extract available pupil segment through a flood-fill
operation (Fig.3):
P = imfill(PC, PI);
5.Fill the specular lights:
P = rlefillsl(P);
6.Approximate the pupil by an ellipse; OUTPUT: The ellipse
approximating the pupil; END.
An example of finding the pupil indicator is stated as follows:
function [k,PI] = getpi(RLH, RLV);
k=16; PI = 0*RLH;
While PI is the null matrix do:
Compute the k-means quantization of RLV and RLH:
RLHQ = fkmq(RLH, k);
RLVQ = fkmq(RLV, k);
Select the logical index of the highest cluster
within RLHQ and RLVQ, respectively:
LIH = ( RLHQ == max(RLHQ(:)) );
LIV = ( RLVQ == max(RLVQ(:)) );
Compute the binary matrix PI as logical
conjunction of LIH and LIV:
PI = LIH & LIV;
k = k -1;
EndWhile;
END.
We must clarify that the pupil indicator is found as a preimage correspond-ing to the maximum value of a fuzzy membership assignment describing theactual pupil as a subset of the pupil cluster: for each pixel within the pupilcluster, directional run-length coefficients encode the degree of membershipof that pixel to the actual pupil. The argument is the fact that being (orcontaining) the most circular solid object within the pupil cluster, the actual
‘Fragile Bits’ vs. Multi-Enrollment - a case study of iris recognition... 131
pupil is the most resilient set to erosion [9] to be found in the pupil cluster.More details on this topic can be found in [12].
Fig. 1. Original eye image (a) and its 8-means quantization (b); The pupil cluster PC (c);
Fig. 2. Vertical run-length uint8 quantization of the pupil cluster (a); Horizontal run-length quantization of the pupil cluster (b); The pupil indicator PI (c). Images are presentedin binary or 8-bit complement.
Fig. 3. Extracting available pupil segment through a flood-fill operation started from anypixel within the pupil indicator (a); Approximating the pupil by an ellipse (b)
Also, the computation of the pupil indicator depends on a single parameter:a threshold for the (uint8) requantized horizontal and vertical run-length coef-ficients computed for the pupil cluster, threshold above which the membershipof a pixel to the actual pupil is guaranteed. For these two reasons explainedabove, the proposed pupil finder procedure is a fuzzy approach that solves aone-dimensional optimization problem.
2.2. LIMBIC BOUNDARY APPROXIMATIONThe Fast Pupil Finder procedure presented above guarantees accurate pupil
localization and enables us to unwrap the eye image (Fig.4.a - image from [15])in polar coordinates (Fig.4.b) and also to practice the localization of the limbicboundary in the rectangular unwrapped eye image (Fig.4.c), obtaining an irissegment as in Fig.4.e.
132 Nicolaie Popescu-Bodorin
Fig. 4. Iris segmentation stages
Circular Fuzzy Iris Segmentation Procedure (N. Popescu-Bodorin):
INPUT: the eye image IM; 1.Apply the Fast Pupil Finder procedure
to find pupil radius and pupil center;
2.Unwrap the eye image in polar coordinates (UI - Fig.4.b)
through a lossless pixel-to-pixel transcoding from the circles
within the original image to the lines within the unwrapped
iris area UI;
3.Strech the unwrapped eye image UI
to a rectangle (RUI - Fig.4.c);
4.Compute three column vectors: A, B, C, where:
A contains the means of the lines within UI matrix;
B contains the means of the lines within RUI matrix;
C contains the means of the lines within [A B] matrix;
5.Compute P, Q, R as being 3-means
quantization of A, B, C, respectively (Fig.5);
6.For each line of the unwrapped eye image, count
the votes given by P,Q and R. All the lines receiving
at least two positive votes is assumed to belong
to the actual iris segment;
7.Find limbic boundary and extract the iris segment
(Fig.5, Fig.4.d, Fig.4.e);
OUTPUT: pupil center, pupil radius, index of the line
representing limbic boundary and the final iris segment;
END.
‘Fragile Bits’ vs. Multi-Enrollment - a case study of iris recognition... 133
Line (in unwrapped eye image)Mem
ber
ship
ass
ign
men
t (t
o a
ctu
al p
up
il / i
ris
/ no
n ir
is s
egm
ent)
20 40 60 80 100 120 140 160 180 200 220
0
20
40
60
80
100
120
140
PQR
PUPIL
IRIS SEGMENT
LINES OUTSIDE THE IRIS
Fig. 5. Iris segmentation procedure: Line assignment (step 5 of the CFIS procedure)
10 20 30 40 50 60 70 80 90 100 1100
0.5
1
1.5
2
2.5
3
Lines within unwrapped iris area
fuzzy assignment to the pupilfuzzy assignment to the irisfuzzy indicator of the lines outside the iriscombined fuzzy indicator for the above three zonescombined crisp indicator for the above three zonesfuzzy indicator for the iris boundaries
Lines within unwrapped iris area (transposed)
PU
PIL
PUPIL
IRIS
IRIS
LIN
ES
OU
TS
IDE
TH
E IR
IS
LINES OUTSIDE THE IRIS
Fuzzy Boundaries
Fig. 6. Fuzzy iris segment and fuzzy iris boundaries
134 Nicolaie Popescu-Bodorin
Fig. 7. Circular Fuzzy Iris Segmentation Demo Program
2.3. EXAMPLES OF K-MEANS BASED FUZZYCLUSTERS AND FUZZY BOUNDARIES:FUZZY IRIS BAND, FUZZY IRISBOUNDARIES
The following two questions are among the most frequent questions askedby the readers of the previously published papers treating the above describedsegmentation procedure: why should we think that CFIS is a fuzzy procedure?what is fuzzy in CFIS procedure?
This section is meant to answer these questions.Let us comment Fig.6 which shows what is happening with the vector B
at the steps 4-5 of the CFIS procedure: hidden behind the combined crispindicator function (crisp membership assignment) of three clusters like thosemarked in Fig.5 (pupil, iris and the area outside the iris) there are fuzzy mem-bership assignment functions defined from the set of lines within rectangularunwrapped iris area (RUI) to each of the above mentioned regions and even tothe iris boundaries. Hence, there is no doubt that area delimited between thefuzzy iris boundaries is the fuzzy iris band (within the rectangular unwrappediris area RUI). Its preimage through the polar mapping is a circular fuzzy irisring.
Three fuzzy iris bands are determined using the vectors A, B, C. The finalresult is computed evaluating the chances that the lines within the unwrappediris area to belong to the actual iris segment. This is done in the step 6 ofthe CFIS procedure by counting the votes received for each line within theunwrapped iris area as a member of a fuzzy iris band. More details regardingthe fuzzification of the iris segment and boundaries can be found in [11].
‘Fragile Bits’ vs. Multi-Enrollment - a case study of iris recognition... 135
On the other hand, there is no doubt that after the pupil is found, the limbicboundary is determined searching for a line number.
For all these reasons explained above, the proposed CFIS procedure is afuzzy approach that solves a one-dimensional optimization problem.
A demo application illustrating CFIS (Fig.7) is available for download [14].It can be tested using entire Bath University Iris Database [15] (the free ver-sion).
3. GABOR ANALYTIC IRIS TEXTURE BINARYENCODER
Gabor Analytic Iris Texture Binary Encoder was initially introduced in [13].It transforms the output of CFIS procedure (unwrapped circular iris ring) intoa binary iris code as follows:
Gabor Analytic Iris Texture Binary Encoder (N. Popescu-Bodorin):INPUT: unwrapped iris segment IM (Fig.4.e), window dimenssion s, desired dimenssiond = [d1, d2] for iris codes;
1 Compute I as being the resized version of unwrapped iris segment to desired dimens-sion d;
2 For each line of I compute the complex matrix AS as being the strong analyticrepresentation of I using window size s;
3 Compute the binary iris code IC (Fig.8)as the sign of the phase for all componentswithin AS;
4 Compute the binary iris mask M corresponding to the various iris occlusions (specularlights, diffuse reflections, eyelashes, eyelids, etc), if any;
OUTPUT: The binary iris code IC and the binary mask M ;
END.
Fig. 8. Two similar iris codes obtained for similar iris images ([15], 0001-L-0001.j2c, 0001-L-0003.j2c) using Gabor Analytic Iris Texture Binary Encoder
136 Nicolaie Popescu-Bodorin
In the above encoding procedure, the term ‘strong analytic representation ofI’ refers to the sum of the iris segment I and its Hilbert transform calculatedusing blocks of dimension 1× s.
Some useful details regarding the strong analytic signal, the Hilbert Trans-form and a synthetic example on generating iris binary code can be found in[13].
4. EXPERIMENTAL RESULTSEleven recognition tests were performed using the iris images from [15] in
order to prove the efficiency of the recently proposed iris recognition approach([12], [13]) and to quantify the changes in recognition performance when us-ing fragile bits. In two of these tests we have worked around two types offragile bits: those caused by the phase instability ([4]) and those suspectedto occur at the boundaries between regions of zeros and regions of ones ([4]).Unfortunately, the results obtained by masking these two types of fragile bitswere weaker than the results of the tests T7, T8 and T9 (commented below)in which the fragile bits are assumed to be inferred from a given number ofprevious observation as a mask containing the pixels that are proven to be un-stable by their values in 30% of the observed iris codes ([4]). For this reason,only nine of these tests are discussed here.
4.1. TERMINOLOGYIn a single-eye / single-enrollment scenario, each identity is enrolled through
a single template of one eye, while in a single-eye/multi-enrollment scenario,each identity is enrolled through a given number of templates sampled for thesame eye.
In an authentication scenario the user exposes its iris to the system andclaim an identity; the iris code extracted from user input is compared withthe gallery templates stored under the claimed identity; the system accepts orrejects the claim depending on a similarity score computed by comparing theclaming iris code to all iris codes enrolled under the claimed identity.
In a recognition (identification) scenario the user exposes its iris to thesystem and wait for the system to recognize him as an (un)enrolled identity;the system response is depending on the biggest similarity score computed bycomparing the input iris code to the entire available gallery.
Hamming Similarity of two iris codes is the ratio between the number ofbits that agree and the number of compared bits.
Non-match ( or imposter, or inter-class) score distribution is the distribu-tion of scores computed for iris codes sampled from different eyes.
Match (or genuine, or intra-class) score distribution is the distribution ofscores computed for pairs of iris codes sampled from the same eye.
‘Fragile Bits’ vs. Multi-Enrollment - a case study of iris recognition... 137
For a given threshold, the False Accept Rate is experimentally determinedas the ratio between the number of imposter scores exceeding the thresholdand the total number of imposter scores. The False Reject Rate (FRR) is theratio between the number of genuine scores not exceeding the threshold andthe total number of genuine scores.
The Odds of False Accept (OFA) is computed as cumulative of the theo-retical imposter distribution above the given threshold and approximates thedefinite integral
∫ 1t Ipdf (τ)dτ, where Ipdf is the theoretical probability density
function of the imposter distribution and t is the threshold.The Odds of False Reject (OFR) is computed as cumulative of the theo-
retical genuine distribution below the given threshold and approximates thedefinite integral:
∫ t0 Gpdf (τ)dτ, where Gpdf is the theoretical probability den-
sity function of the genuine distribution and t is the threshold.Equal Error Rate (EER) is the common value of FAR and FRR at that
thresold value for which they equal each other.
4.2. A CUSTOM MEAN-DEVIATIONSIMILARITY SCORE FORMULTI-ENROLLMENT SCENARIOS
Suppose that the value s of the standard deviation is known for the imposterscore distribution. Let C be the current input iris code which must be com-pared to a set of templates E = E1, E2, ..., En enrolled under an arbitraryidentity. Consider the set of Hamming similarities between C and each ofthe enrolled templates: S = HS(C,E1), HS(C, E2), ..., HS(C, En). Thenthe Mean-Deviation Similarity Score (N. Popescu-Bodorin) between the inputtemplate C and the arbitrary identity E is defined here as follows:
MDSS(C, E) = mean(S) + std(S)− s/2.
The above formula was heuristically determined by running a lot of tests andusing neural network support to minimize the FAR and the OFA while keepingthe FRR and the OFR at reasonable values (1-3%) within a given thresholdrange.
4.3. THE TESTSThe tables (1,2) present the results of nine tests undertaken to illustrate
the efficiency of the recently proposed iris recognition methodology (CircularFuzzy Iris Segmentation, Gabor Analytic Iris Texture Binary Encoder, Mean-Deviation Similarity Score f or Multi-Enrollment Iris Recognition Scenarios).All of them are recognition tests. The differences between the tests are givenby changing enrollment scenarios, the lengths of the resulting iris codes, the
138 Nicolaie Popescu-Bodorin
dimension of the Hilbert filter and the type of similarity score used to compareiris codes.
The acronyms used in the Tables 1, 2 have the following meanings:- DF - (Number of) Degrees-of-Freedom for a given score distribution;- EER - Equal Error Rate;- FAR - False Accept Rate;- FB - (indicates the use of) Fragile Bits;- H - Hamming Similarity Measure;- MDSS - Mean-Deviation Similarity Score for single-eye/multi-enrollment
scenarios;- OFA - Odds of False Accept;- RSESE - Recognition in Single-Eye/Single-Enrollment scenario.- RSEME - Recognition in Single-Eye/Multiple-Enrollment scenario.- STD - Standard Deviation.
In the tests (T7),(T8),(T9) the fragile bits are assumed to be inferred from agiven number of previous observation as a mask containing the pixels that areproven to be unstable by their values in 30% of the observed iris codes, evenif the causality of this instability is not evident.
For each eye, from all images available in the database, 5 images wereselected as previously known observation used to compute the mask of fragilebits. Hence, the set of all available iris codes is spilted into two parts:
- S9, the subset containing those 125 iris codes based on which the masksof fragile bits are computed;
- S7, the complement of S9.
There are three regular single-enrollment tests (T1,T3,T5), three regular multi-enrollment tests (T2,T4,T6) and three single-enrolment tests involving fragilebits: (T7,T8,T9).
In the test T7, all iris codes are compared, except those 125 images from thesubset S9 (5 images for each eye) while in the test T8 there is no exception.The test T9 takes into account only those 125 iris codes excepted in T7.
The purpose of these three tests (T7-9) is to show that using the masksof fragile bits improves recognition scores only for that set of 125 images ini-tially used to compute the masks. For the other iris codes the results do notdemonstrate a net improvement. Consequently, for the iris codes generatedby GAITBE, each mask of this type depends mainly on the given set of ob-servations and it is not a true feature of the iris. Consequently, the set ofgenuine scores is splitted into two distributions having two completely differ-ent statistics (see Fig.10) and the apparent improvements of the recognitionperformance measured through decidability index and through Fisher’s ratioare, in fact, irrelevant.
‘Fragile Bits’ vs. Multi-Enrollment - a case study of iris recognition... 139
Tab
le1
Sta
tist
ics
ofex
per
imen
taldata
TE
STID
(T1)
(T2)
(T3)
(T4)
(T5)
(T6)
(T7)
(T8)
(T9)
Cod
ele
ngth
:19
219
276
876
851
251
251
251
251
2
Win
dow
size
:1x
81x
81x
161x
161x
81x
81x
81x
81x
8
Sim
ilari
tyH
MD
SSH
MD
SSH
MD
SSH
HH
Com
pari
son
type
1-1
1-5
1-1
1-5
1-1
1-5
1-1F
B1-
1FB
1-1F
B
Tes
t/Sy
stem
type
RSE
SER
SEM
ER
SESE
RSE
ME
RSE
SER
SEM
ER
SESE
RSE
SER
SESE
INT
ER
-CLA
SSM
ean
0.50
380.
5049
0.50
380.
5031
0.50
300.
5041
0.50
590.
5062
0.50
72M
edia
n0.
5039
0.50
490.
5037
0.50
300.
5030
0.50
410.
5058
0.50
610.
5067
STD
0.01
930.
0169
0.01
660.
0152
0.01
560.
0135
0.02
560.
0264
0.02
84D
F66
987
590
910
8910
3013
7438
036
030
9Sk
ewne
ss-0
.007
9-0
.006
60.
0024
0.02
660.
0115
0.04
700.
0213
0.03
720.
0823
Kur
tosi
s0.
0366
0.03
920.
0488
0.04
640.
1454
0.18
780.
0648
0.06
24-0
.004
0M
inim
0.41
670.
4367
0.43
510.
4452
0.42
730.
4406
0.38
960.
3837
0.39
46M
axim
0.59
180.
5716
0.57
540.
5639
0.57
990.
5632
0.62
790.
6279
0.61
13W
idth
0.17
510.
1349
0.14
030.
1187
0.15
260.
1226
0.23
830.
2442
0.21
67
INT
RA
-CLA
SSM
ean
0.67
180.
7039
0.66
780.
6993
0.66
840.
6988
0.72
070.
7363
0.79
93M
edia
n0.
6706
0.70
190.
6659
0.69
620.
6666
0.69
530.
7209
0.73
670.
8005
STD
0.05
500.
0464
0.05
330.
0467
0.05
020.
0440
0.04
970.
0526
0.03
70D
F73
9778
9688
109
8170
117
Skew
ness
0.35
280.
2200
0.45
470.
2851
0.42
530.
1883
-0.0
051
-0.0
563
-0.2
802
Kur
tosi
s1.
0995
0.12
461.
3840
0.16
271.
5520
0.18
081.
1405
0.61
180.
3234
Min
im0.
4909
0.54
540.
4884
0.54
890.
4952
0.54
240.
5152
0.51
520.
6547
Max
im1
0.84
501
0.83
531
0.83
031
10.
9072
Wid
th0.
5091
0.29
960.
5116
0.28
640.
5048
0.28
800.
4848
0.48
480.
2525
Ove
rlap
0.10
090.
0262
0.08
690.
0150
0.08
470.
0208
0.11
270.
1127
0
140 Nicolaie Popescu-Bodorin
Tab
le2
Sta
tist
ics
ofE
xper
imen
talD
ata
(conti
nued
)
TE
STID
(T1)
(T2)
(T3)
(T4)
(T5)
(T6)
(T7)
(T8)
(T9)
Ove
rlap
0.10
090.
0262
0.08
690.
0150
0.08
470.
0208
0.11
270.
1127
0
Dec
idab
ility
4.07
945.
6943
4.15
555.
6495
4.44
925.
9865
5.42
725.
5336
8.84
88
Fis
her
rati
o8.
3208
16.2
128.
6340
15.9
589.
8975
17.9
1914
.727
15.3
1039
.150
EE
R0.
0126
1.3E
-39.
3E-3
1.3E
-37.
5E-
31.
3E-3
4.4E
-33.
9E-3
0
At
aFA
Rof
:0.
001
FR
R0.
0222
0.00
130.
0159
0.00
130.
0115
0.00
130.
0087
0.00
640
Thr
esho
ld0.
5635
0.55
700.
5555
0.55
030.
5531
0.54
88.5
8718
.589
88.5
9668
At
aFR
Rof
0.01
:FA
R0.
0243
00.
0072
00.
0020
06.
29E
-41.
43E
-40
OFA
0.02
322.
1E-0
90.
0068
1.5E
-10
0.00
153.
2E-1
34.
54E
-41.
16E
-42.
3E-1
2O
FR
0.00
920.
0160
0.01
050.
0155
0.00
880.
0131
0.00
460.
0057
0.00
50T
hres
hold
.542
31.6
0429
.544
75.5
9847
.549
24.6
0112
.590
97.6
0319
.703
99
At
ath
resh
old
of:
0.59
0.59
0.59
0.59
0.59
0.59
0.63
0.63
0.63
FR
R0.
0564
0.00
270.
0587
0.00
540.
0510
0.00
810.
0380
0.02
510
OFR
0.06
820.
0071
0.07
210.
0097
0.05
920.
0067
0.03
420.
0216
2.4E
-6FA
R2.
1E-6
00
00
00
00
OFA
4.1E
-62.
4E-7
1.0E
-74.
9E-9
1.2E
-89.
7E-1
16.
5E-7
1.3E
-67.
9E-6
At
ath
resh
old
of:
0.6
0.6
0.6
0.6
0.6
0.6
0.64
0.64
0.64
FR
R0.
0806
0.00
810.
0853
0.01
070.
0717
0.00
940.
0504
0.03
350
OFR
0.09
550.
0126
0.10
160.
0168
0.08
650.
0123
0.05
240.
0335
8.4E
-6FA
R0
00
00
00
00
OFA
3.2E
-79.
3E-9
3.3E
-98E
-11
2.4E
-10
5.9E
-13
8.5E
-81.
9E-7
1.5E
-6
‘Fragile Bits’ vs. Multi-Enrollment - a case study of iris recognition... 141
0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
10−4
10−2
0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
10−4
10−2
0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
10−4
10−2
(T5): inter−class similarity scores (log scale)(T5): intra−class similarity scores (log scale)
(T6): inter−class similarity scores (log scale)(T6): intra−class similarity scores (log scale)
(T7): inter−class similarity scores (log scale)(T7): intra−class similarity scores (log scale)
Fig. 9. Inter/intra-class distributions for the tests T5, T6 and T7 (from top to buttom)
Using the same 125 images from the subset S9 as enrollment samples andusing the Mean-Deviation Similarity Score, the tests T2, T4 and T6 give us ameasure about what an improvement really is.
Also, by comparing the tests (T1, T2), (T3, T4) and (T5, T6), it can beseen that increasing the quantity of a priori knowledge (T2, T4, T6) decreasesthe imposter-genuine decision uncertainty in terms of:
- narrowing the ranges of both distributions and their overlapping interval;
- decreasing the standard deviation in each class of scores;- better encoding of the statistical independence between iris codes repre-
senting different eyes;- decreasing the Odds of False Accept and the actual False Accept Rates
computed for threshold values within the neighborhood of the observedmaximum imposter score;
- decreasing the tail thickness for both distributions of scores;- decreasing the Odds of False Accept and the actual False Accept Rates
at a given False Reject Rate of 1%;
142 Nicolaie Popescu-Bodorin
0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.950
0.01
0.02
0.03
0.04
0.05
0.06
0.07Genuine distributions computed for the subsets S7 and S9 using fragile bits
0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.950
0.01
0.02
0.03
0.04
0.05Genuine distributions computed without using fragile bits
T7 − S7T8 − All iris codesT9 − S9
T5 − S7T5 − All iris codesT5 − S9
Fig. 10. Analyzing the effects of using fragile bits
- decreasing the Odds of False Reject and the actual False Reject Ratesat a given False Accept Rate of 0.1%.
Also, the way in which a priori knowledge is added to a biometric systemproved to be very important:
- when defining identities through a collection of enrolled templates (T2,T4, T6) each identity includes a little variability and the intra-classdistribution becomes narrower and so does the inter-class distribution(the fact that iris codes of different irides match each other by chancebecomes more obvious in the experimental data);
- by contrast, inferring the existence of a mask of fragile bits defined asin T7, T8 and T9 produces a weaker (and contestable) improvement inrecognition performance. By comparing the tests (T5,T6) and (T5,T7)it can be seen that, in the test T7, the bigger distance between themeans of the genuine and imposter score distributions is compensatedby an increase in their standard deviations;
‘Fragile Bits’ vs. Multi-Enrollment - a case study of iris recognition... 143
- the tests T5 and T7 show that the mean of the intra-class distributionincreases with 8% and the width of intra-class distribution decreases with4%, but unfortunately, the width of inter-class distribution increases with26%;
- a totally different situation is revealed by comparing the tests T5 andT6; the mean of the intra-class distribution increases with 4.5% and thewidths of both inter/intra-class distributions decrease with 19.6% and43%, respectively.
All remarks from above are illustrated in Fig.9 and Fig.10. For comparison,the Tables 1, 2 and the following resources will certainly be useful:
- Fig.3 in [10], where similar results (FRR values at a FAR of 0.001) arepresented for three state-of-the-art iris recognition algorithms evaluatedin ICE 2006 [8]: CAM-2 (Cambridge), IrTch-2 (IriTech), SI-2 (Sagem-Iridian);
- Table 1 in [17], Fig.5 in [7], Fig.5 in [6], Fig.9 in [5].
5. CONCLUSION AND FUTURE WORKSThis paper proved that using the proposed methodology and implement-
ing iris recognition in multiple-enrollment scenario lead to very good perfor-mances.
Only minor (and contestable) improvements had been obtained using theconcept of fragile bits. For the iris codes generated with Gabor Analytic IrisTexture Binary Encoder, the mask of fragile bits proved to be a strong featureof a given set of observations (mainly influenced by the acquisition process),but a weak feature of the observed iris.
Future works will explore the possibilities of reducing the number of com-parisons which are needed in order to obtain a recognition decision in a multi-enrollment scenario.
AcknowledgmentThe author wishes to thank Professor Luminita State (University of Pitesti,
RO) for the comments, criticism, and constant moral and scientific supportduring the last two years. The author would also like to thank ProfessorDonald Monro (University of Bath, UK) for granting the access to the BathUniversity Iris Database.
References
[1] R.M. Bolle, S. Pankanti, J.H. Connell, N. Ratha, Iris Individuality: A Partial IrisModel, Proc. of the 17th Int. Conf. on Pattern Recognition, II, 927-930, 2004.
144 Nicolaie Popescu-Bodorin
[2] J.G. Daugman, High Confidence Visual Recognition of Persons by a Test of StatisticalIndependence, IEEE TPAMI, 15, 11(1993).
[3] J.G. Daugman, New Methods in Iris Recognition, IEEE TSMC, 37, 5(2007).
[4] K.P. Hollingsworth, K.W. Bowyer, P.J. Flynn, The Best Bits in an Iris Code, IEEETPAMI, 31, 6(2009).
[5] X. Liu, K. W. Bowyer, P. J. Flynn, Experiments with an Improved Iris SegmentationAlgorithm, Proc. of the Fourth IEEE Workshop on Automatic Identification AdvancedTechnologies, 118-123, 2005.
[6] L. Ma, T. Tan, Y. Wang, D. Zhang, Efficient Iris Recognition by Characterizing KeyLocal Variations, IEEE Trans. on Image Processing, 13, 6(2004).
[7] D. M. Monro, S. Rakshit, Rotation Compensated Human Iris Matching, IEEE Work-shop on Signal Processing Applications for Public Security and Forensics, 2007.
[8] National Institute of Standards and Technology, Iris Challenge Evaluation,http://iris.nist.gov/ice/.
[9] W. Pegden, Sets resilient to erosion, web reference at Department of Mathematics,Rutgers University, June 2008, http://people.cs.uchicago.edu/˜wes/erosion.pdf.
[10] P. J. Phillips, W. T. Scruggs, A. J. O’Toole, P. J. Flynn, K. W. Bowyer, C. L. Schott,M. Sharpe, Face Recognition Vendor Test 2006 and Iris Challenge Evaluation 2006Large-Scale Results, National Institute of Standards and Technology, March 2007.http://iris.nist.gov/ice/FRVT2006andICE2006LargeScaleReport.pdf.
[11] N. Popescu-Bodorin, A Fuzzy View on k-Means Based Signal Quantization with Ap-plication in Iris Segmentation, 17th Telecommunications Forum - TELFOR 2009, Bel-grade, November 2009,http://fmi.spiruharet.ro/bodorin/articles/telfor09.pdf.
[12] N. Popescu-Bodorin, Circular Fuzzy Iris Segmentation, Proc. of the 4th Annual SouthEast European Doctoral Student Conference, vol. 1, 471-479, South-East EuropeanResearch Centre, Thessaloniki, July 2009.
[13] N. Popescu-Bodorin, Gabor Analytic Iris Texture Binary Encoder, Proceedings of the4th Annual South East European Doctoral Student Conference, vol. 1, 505-513, South-East European Research Centre, Thessaloniki, July 2009.
[14] N. Popescu-Bodorin, Circular Fast Fuzzy Iris Segmentation and Fast k-Means Quanti-zation Demo Programs, June 2009,http://fmi.spiruharet.ro/bodorin/arch/cffis.zip, fkmq.zip.
[15] University of Bath Iris Database,http://www.bath.ac.uk/elec-eng/research/sipg/irisweb/.
[16] R.P. Wildes, Iris Recognition - An Emerging Biometric Technology, Proceedings of theIEEE, 85, 9(1997).
[17] S. Ziauddin, M. N. Dailey, Iris Recognition Performance Enhacement using WeightedMajority Voting, ICIP, 2008,http://www.cs.ait.ac.th/ mdailey/papers/Ziauddin-WeightedMajority.pdf.
BIFURCATION IN A NONLINEAR BUSINESSCYCLE MODEL
ROMAI J., 5, 2(2009), 145–152
Carmen Rocsoreanu, Mihaela SterpuUniversity of Craiova, [email protected], [email protected]
Abstract We consider a business cycle model consisting of three time-delay differentialequations. The system is obtained by combining a nonlinear IS-LM model witha Kaldor-Kalecki model for the business cycle with delayed investment. Thevariables stand for the gross product, the capital stock and the interest rate.It is assumed that the capital growth is due to the past investment decisionsand there is a time delay after which the capital equipment is available forproduction. A qualitative analysis of this model in the particular case whenthe investment function (I), the saving function (S) and the demand for money(L) are linear functions on their arguments, while the money supply (M) isconstant was performed in [2]. Our paper considers the case when the functionsI, S, L and M are nonlinear, but their derivatives satisfy some conditions. Weanalyze the stability of the equilibrium point and we obtain Hopf bifurcationconditions. The theoretical results are illustrated by numerical simulations forparticular cases.
Keywords: business cycle, generalized IS-LM model, Hopf bifurcation, delay differential
equations.
2000 MSC: 34D, 37G, 37N.
1. THE GENERALIZED IS-LM MODEL
Consider the following generalized IS-LM model for the business cycle withdelayed investment
·Y = α (I (Y,K, r)− S (Y, r)) (0)·r = β (L (Y, r)−M)·
K = I (Y (t− θ) ,K, r)− δK
obtained by combining the standard IS-LM model
·Y = α (I (Y, K, r)− S (Y, r)) (1)·r = β (L (Y, r)−M)
145
146 Carmen Rocsoreanu, Mihaela Sterpu
with the Kaldor Kalecki model of trade cycle with delayed investment
·Y = α (I (Y, K)− S (Y, K)) (2)K = I (Y (t− θ) ,K)− δK
where the variables are:Y - the gross product,r - the interest rate,K - the capital stock,
the functions in the right hand side are:I - the investment functionS - the savings function,L - the demand of money
and the parameters are:M - the constant money supplyα - the adjustment coefficient in the goods market, α > 0β - the adjustment coefficient in the money market, β > 0δ - the depreciation rate of the capital stock, 0 ≤ δ ≤ 1.The time-delay θ appears in the gross product function Y. The initial con-
ditions for the model (1) are Y (t) = ϕ (t) , t ∈ [−θ, 0] , r (0) = r0, K (0) = K0.The Kaldor-Kalecki model as the time-delay differential equation system
(3) was introduced in [6]. The influence of the time delay on the stability ofsystem (3) was investigated in [8], [10].
The IS-LM model for the business cycle without time delay was analyzedin [1], [9].
More general IS-LM models including delayed taxes revenues are analyzedin [3], [4].
A qualitative analysis of model (1) in the particular case when the invest-ment function (I), the saving function (S) and the demand for money (L) arelinear functions on their arguments, while the money supply (M) is constantwas done in [2].
Economic considerations lead to the following constrains on the economicvariables:
∂I
∂Y> 0,
∂I
∂r< 0,
∂I
∂K< 0, (3)
∂S
∂Y> 0,
∂S
∂r> 0,
∂L
∂Y> 0,
∂L
∂r< 0.
Bifurcation in a nonlinear business cycle model 147
Our paper considers the case when the functions I, S, and L are nonlinear,but their derivatives satisfy the preceding conditions. We analyze the stabil-ity of the equilibrium point and we obtain Hopf bifurcation conditions. Thetheoretical results are illustrated by numerical simulations for particular cases.
In the following analysis we consider the investment, the savings and theliquidity functions of the form inspired by [3], [4] :
I (Y (t) , r (t) ,K (t)) = AY a (t)rb (t)
− cK (t) , A > 0, 0 ≤ c ≤ 1, a > 0, b > 0,
S (Y (t) , r (t)) = sY a (t) rb (t) , 0 < s < 1, (4)
L (Y (t) , r (t)) = gY (t) +h
r (t)− r, g, h, r > 0,
where r is a very small rate of interest generating the liquidity trap as r (t)fall to the level r.
2. ECONOMIC EQUILIBRIUM AND ITSSTABILITY
The linearization of the system (1) in a neighborhood of an equilibriumpoint (Y ∗, r∗,K∗) reads:
x (t) = A [x (t)− x∗] + B [x (t− θ)− x∗] (6)
wherex (t) = (Y (t) , r (t) ,K (t))T , x∗ = (Y ∗, r∗, K∗)T ,
A =
α (IY − SY ) α (Ir − Sr) αIK
βLY βLr 00 Ir IK − δ
, B =
0 0 00 0 0IY 0 0
, (7)
and
IY =∂I
∂Y(Y ∗, r∗,K∗) , Ir =
∂I
∂r(Y ∗, r∗,K∗) , IK =
∂I
∂K(Y ∗, r∗,K∗) ,
SY =∂S
∂Y(Y ∗, r∗,K∗) , Sr =
∂S
∂r(Y ∗, r∗,K∗) ,
LY =∂L
∂Y(Y ∗, r∗,K∗) , Lr =
∂L
∂r(Y ∗, r∗,K∗) .
The characteristic equation of system (6)
det(λI3 −A−Be−λθ
)= 0
can be put in the form
P (λ) + Q (λ) e−λθ = 0, (8)
148 Carmen Rocsoreanu, Mihaela Sterpu
where P (λ) = λ3 + p2λ2 + p1λ + p0, Q (λ) = q1λ + q0 and
p0 = − det(A),p1 = αβ[Lr(IY − SY )− LY (Ir − Sr)] + α(IY − SY ) · (IK − δ) + βLr(IK − δ),p2 = −tr (A) = −α (IY − SY )− βLr − IK + δ,
q0 = −αIY IK > 0,
q1 = αβIY IKLr > 0.
Without delay, i.e. for θ = 0, system (1) with I, S, L given by (5), possessesa unique equilibrium (Y ∗, r∗, K∗) , satisfying Y ∗ > 0, r∗ > 0,K∗ > 0, givenby
Y ∗ =1g
(M − h
r∗ − r
), r∗ =
(δA
s (c + δ)
) 12b
, K∗ =s
δ(Y ∗)a (r∗)b . (9)
Using to the Hurwitz criterion, we obtain the following result :Proposition 1. For θ = 0, the equilibrium (Y ∗, r∗,K∗) of (1) in the
settings (5) is locally asymptotically stable if and only if
p2 > 0,
p2 (p1 + q1)− (p0 + q0) > 0.
In addition, at θ = 0, the equilibrium is a Hopf singularity if
p2 (p1 + q1)− (p0 + q0) = 0,
3. HOPF BIFURCATION WITH POSITIVE TIMEDELAY
For θ > 0 the equilibrium point is stable if and only if all the eigenvalues ofthe linearized system have negative real parts.
For small time delay, the Hopf bifurcation point can be found using thelinear stability analysis. Putting e−λθ ≈ 1 − λθ, the characteristic equation(8) reads
λ3 + (p2 − θq1) λ2 + (p1 + q1 − θq0) λ + p0 + q0 = 0.
By the Hopf bifurcation theorem and the Hurwitz criterion we get the followingresult:
Proposition 2. For small θ, a Hopf bifurcation occurs at θ = θ0 if thefollowing conditions are fulfilled:
(p2 − θ0q1) (p1 + q1 − θ0q0) = p0 + q0
(p2 − θ0q1) (p0 + q0) > 0, (p1 + q1 − θ0q0) > 0,
Bifurcation in a nonlinear business cycle model 149
andq1 (p1 + q1 − θ0q0) + q0 (p2 − θ0q1) 6= 0.
The last condition in Proposition 2 is equivalent with Re dλdθ (θ0) 6= 0.
For larger time delay, in order to obtain values of the time delay θ such thatthe equilibrium point (Y ∗, r∗, K∗) is a Hopf singularity, we have to determinethe eigenvalues of the form λ = iω, with ω > 0.
From the characteristic equation we get:
sinωθ = ω−ω2 (q0 − q1p2) + q0p1 − p0q1
ω2q21 + q2
0
, (9)
cosωθ = C (ω) ≡ ω4q1 + ω2 (q0p2 − q1p1)− p0q0
ω2q21 + q2
0
.
Taking into account that sin2 ωθ+cos2 ωθ = 1, a necessary condition to have ωas a solution of (10) is that ω must be a positive root of the following equation:
ω6 + aF ω4 + bF ω2 + cF = 0 (11)
where:aF = p2
2 − 2p1, bF = p11 − q1
1 − 2p0p2, cF = p20 − q2
0.
Denote by ∆ the discriminant of the cubic equation obtained from (11).The following result holds for equation (11).Lemma 3. Equation (11) possesses at least one positive solution in one of
the situations given in the table below:
Number of positive solutions ω Conditions
1
cF < 0 and (bF ≤ 0 or aF ≥ 0)cF < 0 , bF > 0 , aF < 0, ∆ > 0cF = 0, bF < 0cF = 0, bF = 0, aF < 0
2 cF > 0, bF < 0 or aF < 0, ∆ ≤ 0cF = 0, bF > 0, aF < 0
3 cF < 0, bF > 0, aF < 0,∆ ≤ 0
Let ω∗ be a positive root of (11). Then λ∗ = iω∗ is an eigenvalue corre-sponding to the equilibrium, if and only if
θ∗ =
1ω∗ [2 (j + 1) π −Arc cosC (ω∗)] , for C (ω∗) > 0
1ω∗ [2jπ + Arc cosC (ω∗)] , for C (ω∗) < 0
, (12)
where j = 0, 1, 2, ..... The values θ∗ such that the d Re λdθ (θ∗) 6= 0 are Hopf
bifurcation values. Taking into account the characteristic equation (8) we
150 Carmen Rocsoreanu, Mihaela Sterpu
Fig. 1. Figure 1. Variation of θ with respect to M for δ = 0.1, A := 0.1, a := 1, b := 0.5,s := 1, α := 1, β := 1, g := 0.05, h := 0.2, r := 0.001, c := 0.9,
obtain
d Reλ
dθ(θ∗) =
(ω∗)2[3 (ω∗)4 + 2 (ω∗)2
(p22 − 2p1
)+ p2
2 − 2p2p0 − q21
]
d,
where d is a positive term. Thus a Hopf bifurcation takes place at θ∗, provided
3 (ω∗)4 + 2 (ω∗)2(p22 − 2p1
)+ p2
2 − 2p2p0 − q21 6= 0. (13)
We can formulate the following concluding results:Theorem 4. If the equilibrium point (Y ∗, r∗,K∗) is locally asymptotically
stable without time delay, then in conditions of Lemma 3 there is at least onestability switch.
Theorem 5. Assume that conditions in Lemma 3 are satisfied and letωbif = ω∗ be the smallest positive solution of (11) and θbif = θ∗ given by (12)with j = 0. Then a Hopf bifurcation occurs at θbif , provided condition (13) isfulfilled.
In order to obtain the type of the Hopf bifurcation (supercritical or subcrit-ical), the normal form of system (1) around the equilibrium must be derived[7], [5].
The variation of θ∗ given by (12) with respect to M is illustrated in thefollowing figures for several values of δ, namely 0.1, 0.15, and 0.08, respectively.We considered the parameters A := 0.1, a := 1, b := 0.5, s := 1, α := 1, β := 1,g := 0.05, h := 0.2, r := 0.001, c := 0.9, and we plotted the curves given by(12) for j = 0, 1, 2. The stability of the equilibrium switches to instabilitythrough a Hopf bifurcation when the curve θ = θ1,j=0 is crossed transversally.
Bifurcation in a nonlinear business cycle model 151
Fig. 2. Figure 2. Variation of θ with respect to M for δ = 0.15, A := 0.1, a := 1, b := 0.5,s := 1, α := 1, β := 1, g := 0.05, h := 0.2, r := 0.001, c := 0.9,
Fig. 3. Figure 3. Variation of θ with respect to M for δ = 0.08, A := 0.1, a := 1, b := 0.5,s := 1, α := 1, β := 1, g := 0.05, h := 0.2, r := 0.001, c := 0.9,
152 Carmen Rocsoreanu, Mihaela Sterpu
4. CONCLUSIONSThis paper is focused on the mathematical analysis of a generalized IS-LM
model for the business cycle with delay investment and nonlinear investment,savings and demand of money functions. Conditions for the stability of theeconomic equilibrium are determined. We gave necessary conditions for thedelay parameter which ensure stability switches and Hopf bifurcation occur-rence.
References[1] G. Gabisch and H.W. Lorenz, Business cycle theory-A survey of methods and concepts,
Springer, New York, 1989.
[2] J. Cai, Hopf bifurcation in the IS-LM business cycle model with time delay, ElectronicJournal of Differential Equations, vol. 2005, 15 (2005), 1-6.
[3] L. Cesare and M. Sportelli, A dynamic IS-LM model with delayed taxation revenues,Chaos, Solitons and Fractals 25 (2005), 233-244.
[4] M. Neamtu, D. Opris and C. Chilarescu, Hopf bifurcation in a dynamic IS-LM modelwith time delay, Chaos, Solitons and Fractals 34 (2007), 519-530.
[5] A.V. Ion, On the Bautin bifurcation for systems of delay differential equations, ActaUniv. Apulensis Math. Inform. 8 (2004), 235-246.
[6] A. Krawiec, M. SzydÃlowski, The Kaldor Kalecki business cycle model, Annals of Oper-ations Research 89 (1999), 89-100.
[7] G. Mircea, M. Neamtu and D. Opris, Hopf bifurcation for dynamical system with timedelay and applications, Ed. Mirton, Timisoara , 2004.(in Romanian)
[8] M. Szydlowski and A. Krawiec, The Kaldor-Kalecki model of business cycle as a two-dimensional dynamical system, J Nonlinear Math Phys. 8 (2001), 266-271.
[9] W.B. Zhang, Capital and knowledge, Springer, Berlin, 2005.
[10] C. Zhang and J. Wei Stability and bifurcation analysis in a kind of business cycle modelwith delay, Chaos, Solitons and Fractals 22 (2004), 883-896.
EXISTENCE AND UNIQUENESS OF FUZZYSOLUTION FOR LINEAR VOLTERRA FUZZYINTEGRAL EQUATIONS, PROVED BYADOMIAN DECOMPOSITION METHOD
ROMAI J., 5, 2(2009), 153–161
Hamid Rouhparvar, Tofigh Allahviranloo, Saeid AbbasbandyDepartment of Mathematics, Science and Research Branch,Islamic Azad University, Tehran, [email protected]
Abstract In the present paper, the existence and uniqueness of fuzzy solution for a thelinear Volterra fuzzy integral equation is proved via the Adomian decomposi-tion method.
Keywords: linear Volterra fuzzy integral equation, Adomian decomposition method.
2000 MSC: 97U99.
1. INTRODUCTIONThe concept of integration of fuzzy functions has been introduced by Dubois
and Prade [1], Goetschel and Voxman [2], Kaleva [3] and others. However, ifthe fuzzy function is continuous, all the various procedures yield the sameresult. The fuzzy integral is applied in fuzzy integral equations, such thatthere is a growing interest in fuzzy integral equations particularly in the pastdecade. The fuzzy integral equations have been studied by [4, 5, 6] and otherauthors.
Several criteria of existence of solutions of the Volterra fuzzy integral equa-tion are given under the compactness-type conditions, by applying the em-bedding theorem in [7] and the Darbo fixed-points theorem in [8], and theLipschitz condition, by applying the successive iterations of Picard method in[5, 9, 10]. In this paper, the existence theorems are proved for linear Voltterafuzzy integral equation under the Lipschitz condition and arbitrary kernelsby means of the successive iterations of the Adomian decomposition method(ADM) [11, 12, 13] involving fuzzy set-valued function of a real variable wherevalues are normal, convex, upper semicontinuous and compactly supportedfuzzy sets in Rn.
153
154 Hamid Rouhparvar, Tofigh Allahviranloo, Saeid Abbasbandy
2. PRELIMINARIESBy PK(Rn), we denote the family of all nonempty compact convex subsets
of Rn. Let T = [a, b] ⊂ R be a compact interval and denote [3]
En = u : Rn → [0, 1]|u satisfies (i)− (iv) below,where
i) u is normal i.e. there exists an x0 ∈ Rn such that u(x0) = 1,
ii) u is fuzzy convex,
iii) u is upper semicontinuous,
iv) [u]0 = x ∈ R|u(x) > 0 is compact.
For 0 < α ≤ 1 denote [u]α = x ∈ Rn|u(x) ≥ α. Then from (i)-(iv), it followsthat the α-level set [u]α ∈ PK(Rn) for all 0 ≤ α ≤ 1.
If g : Rn × Rn → Rn is a function, then using Zadeh’s extension principlewe can extend g to En ×En → En by the relation
g(u, v)(z) = supz=g(x,y)
minu(x), v(y).
It is well known that [g(u, v)]α = g([u]α, [v]α) for each u, v ∈ En, 0 ≤ α ≤ 1 andcontinuous function g. Moreover, we have [u+v]α = [u]α +[v]α, [ku]α = k[u]α,where k ∈ R.
Define D : En×En → R+ by the relation D(u, v) = sup d([u]α, [v]α), whered is the Hausdorff metric defined in PK(Rn). Then D is a metric in En.Furthermore, we know that [14]
• (En, D) is a complete metric space,
• D(θu, θv) = |θ|D(u, v) for every u, v ∈ En and θ ∈ R,
• D(u + w, v + w) = D(u, v) for all u, v, w ∈ En.
It can be proved that D(u+v, w+z) ≤ D(u,w)+D(v, z) for u, v, w and z ∈ En.Recall that the real numbers can be embedded to En by the correspondence
c(t) =
1 for t = c,0 elsewhere.
Definition 2.1. [3] We say that a function F : T → En is strongly measurableif for all α ∈ [0, 1] the set-valued function Fα : T → PK(Rn) defined by
Fα(t) = [F (t)]α
Existence and uniqueness of fuzzy solution for linear Volterra fuzzy integral equations... 155
is (Lebesgue) measurable, where PK(Rn) is endowed with the topology gener-ated by the Hausdorff metric d.
A function F : T → En is called integrable bounded if there exists anintegrable function h such that ||x|| < h(t) for all x ∈ F0(t).
Definition 2.2. [3] Let F : T → En. The integral of F over T , denoted by∫T F (t) dt or
∫ ba F (t) dt, is defined levelwise by
[∫T F (t) dt]α =
∫T Fα(t) dt = ∫T f(t) dt|f : T → Rn is ameasurable section for Fα
for all 0 < α ≤ 1.
Proposition 2.1. [3] Let F, G : T → En be integrable and θ ∈ R. Then
i)∫
(F + G) =∫
F +∫
G,
ii)∫
θF = θ∫
F ,
iii) D(F, G) is integrable,
iv) D(∫
F,∫
G) ≤ ∫D(F,G).
Definition 2.3. [15] A function F : T → En is bounded if there exists aconstant M > 0 such that D(F (t), 0) ≤ M for all t ∈ T .
3. EXISTENCE AND UNIQUENESS OF FUZZYSOLUTION
We consider the linear Voltera fuzzy integral equation
x(t) = f(t) +∫ t
0k(t, s)x(s) ds, t ≥ 0, (1)
where Ω = (t, s)|0 ≤ s ≤ t ≤ γ.Theorem 3.1. Assume the following conditions are satisfied
i) f : [0, γ] → En is continuous and bounded,
ii) k : Ω → R is a continuous function,
iii) if x, y : [0, γ] → En are continuous, then the Lipschitz condition
D(k(t, s)x(s), k(t, s)y(s)) ≤ LD(x(t), y(t)), (2)
is satisfied, with 0 < L < 1γ .
156 Hamid Rouhparvar, Tofigh Allahviranloo, Saeid Abbasbandy
Then there exists a unique fuzzy solution x(t) of (1) and the successive itera-tions (see Appendix)
ϕ0(t) = f(t),
ϕn+1(t) = f(t) +∑n+1
i=1
∫ t0 k(t, s) xi−1(s) ds, (n ≥ 0),
(3)
are uniformly convergent to x(t) on [0, γ], where
x0(t) = f(t),
xn(t) =∫ t0 k(t, s)xn−1(s) ds, (n ≥ 1).
(4)
First we prove the following Lemma.
Lemma 3.1. If the conditions of Theorem 3.1 hold and xn is given by (4)then
I) xn(t) is bounded,
II) xn(t) is continuous.
Proof. I) Clearly x0(t) = f(t) is bounded by the assumption. Assumexn−1(t) is bounded. From (2) and (4) we have
D(xn(t), 0) = D(∫ t0 k(t, s) xn−1(s) ds, 0) ≤ ∫ t
0 D(k(t, s) xn−1(s), 0) ds
≤ L∫ t0 D(xn−1(s), 0)ds ≤ γL supt∈[0,γ] D(xn−1(t), 0),
hence by induction xn(t) is bounded.II) To prove continuity, we suppose 0 ≤ t ≤ t ≤ γ, hence
D(xn(t), xn(t))
= D(∫ t0 k(t, s)xn−1(s) ds,
∫ t0 k(t, s)xn−1(s) ds)
= D(∫ t0 k(t, s)xn−1(s) ds,
∫ t0 k(t, s)xn−1(s) ds +
∫ tt k(t, s)xn−1(s) ds)
≤ D(∫ t0 k(t, s)xn−1(s) ds,
∫ t0 k(t, s)xn−1(s) ds) + D(
∫ tt k(t, s)xn−1(s) ds, 0)
≤ γ sups∈[0,γ] D(k(t, s)xn−1(s), k(t, s)xn−1(s))+
+(t− t) sups∈[0,γ] D(xn−1(s), 0).
As a result we obtain
D(xn(t), xn(t)) → 0 as t → t.
Thus xn(t) is continuous on [0, γ].2
Existence and uniqueness of fuzzy solution for linear Volterra fuzzy integral equations... 157
Proof of Theorem 3.1. We assert that all ϕn(t) are bounded on [0, γ]. Infact, ϕ0(t) = f(t) is bounded by the assumption. Suppose ϕn−1(t) is bounded.From (3) we have
D(ϕn(t), 0) = D(f(t) +∑n
i=1
∫ t0 k(t, s)xi−1(s) ds, 0)
= D(f(t) +∑n−1
i=1
∫ t0 k(t, s)xi−1(s) ds +
∫ t0 k(t, s)xn−1(s) ds, 0)
= D(ϕn−1(t) +∫ t0 k(t, s)xn−1(s) ds, 0)
≤ D(ϕn−1(t), 0) + D(∫ t0 k(t, s)xn−1(s) ds, 0)
≤ D(ϕn−1(t), 0) + D(xn(t), 0),
from induction and Lemma 3.1 part (I) we have that ϕn(t) is bounded. Con-sequently, ϕn(t) is a sequence of bounded functions on [0, γ].
In the following, we prove that ϕn(t) are continuous on [0, γ]. By Lemma3.1 part (II) for 0 ≤ t ≤ t ≤ β, we have
D(ϕn(t), ϕn(t))
≤ D(f(t), f(t)) + D(∑n
i=1
∫ t0 k(t, s)xi−1(s) ds,
∑ni=1
∫ t0 k(t, s)xi−1(s) ds)
≤ D(f(t), f(t)) + D(∑n
i=1
∫ t0 k(t, s)xi−1(s) ds,
∑ni=1
∫ t0 k(t, s)xi−1(s) ds)+
+D(∑n
i=1
∫ tt k(t, s)xi−1(s) ds, 0)
≤ D(f(t), f(t)) +∫ t0 D(
∑ni=1 k(t, s)xi−1(s),
∑ni=1 k(t, s)xi−1(s)) ds+
+∫ tt D(
∑ni=1 k(t, s)xi−1(s), 0) ds
≤ D(f(t), f(t)) + γ sups∈[0,γ] D(∑n
i=1 k(t, s)xi−1(s),∑n
i=1 k(t, s)xi−1(s))+
+(t− t) sups∈[0,γ] D(∑n
i=1 k(t, s)xi−1(s), 0).
Finally we obtain
D(ϕn(t), ϕn(t)) → 0 as t → t.
Therefore the sequence ϕn(t) is continuous on [0, γ].To prove uniform convergence of the sequence ϕn(t), for n ≥ 1 we have
D(ϕn+1(t), ϕn(t))
= D(f(t) +∑n+1
i=1
∫ t0 k(t, s)xi−1(s) ds, ϕn(t))
= D(ϕn(t) +∫ t0 k(t, s)xn(s) ds, ϕn(t))
= D(∫ t0 k(t, s)xn(s) ds, 0)
≤ ∫ t0 D(k(t, s)xn(s), 0) ds
≤ γL supt∈[0,γ] D(xn(t), 0).
158 Hamid Rouhparvar, Tofigh Allahviranloo, Saeid Abbasbandy
Hence we obtain
supt∈[0,γ]
D(ϕn+1(t), ϕn(t)) ≤ γL supt∈[0,γ]
D(xn(t), 0). (5)
From another point of view, by (12) we can obtain for n ≥ 1,
D(xn(t), 0) = D(∫ t0 k(t, s)xn−1(s) ds, 0)
≤ ∫ t0 D(k(t, s)xn−1(s), 0) ds
≤ γL supt∈[0,γ] D(xn−1(t), 0)
...
≤ (γL)n supt∈[0,γ] D(x0(t), 0) = (γL)n supt∈[0,γ] D(f(t), 0),
that is,sup
t∈[0,γ]D(xn(t), 0) ≤ Q(γL)n, (6)
where Q = supt∈[0,γ] D(f(t), 0). For n ≥ 0, from (5) and (6) we obtain
supt∈[0,γ]
D(ϕn+1(t), ϕn(t)) ≤ Q(γL)n+1.
The series QγL∑∞
n=0(γL)n is convergent, hence the series∑∞
n=0 D(ϕn+1(t), ϕn(t))is controlled uniformly on [0, γ] this implying the uniform convergence of thesequence ϕn(t). If we denote x(t) = limn→∞ ϕn(t), then x(t) satisfies (1).It is obviously continuous and bounded on [0, γ].
At last, we prove the uniqueness of solution. Let x(t) and y(t) be twocontinuous solutions of (1) on [0, γ]. Then
0 ≤ D(x(t), y(t)) = D(x(t) + ϕn(t), y(t) + ϕn(t))
≤ D(x(t), ϕn(t)) + D(y(t), ϕn(t)),
and since ϕn(t) is convergent to solution of (1),
D(x(t), ϕn(t)) → 0,
D(y(t), ϕn(t)) → 0,
when n → ∞, then D(x(t), y(t)) = 0 that is x(t) = y(t). This finishes theproof of Theorem 3.1.2
Theorem 3.2. Let 0 < M < 1. Suppose that the following conditions aresatisfied:
i) f : [0, γ] → En is continuous and bounded,
Existence and uniqueness of fuzzy solution for linear Volterra fuzzy integral equations... 159
ii) k : Ω → R is a continuous function and∫ t0 |k(t, s)| ds ≤ M ,
then there exists a unique fuzzy solution x(t) : [0, γ] → En of (1) and thesuccessive iterations
ϕ0(t) = f(t),
ϕn+1(t) = f(t) +∑n+1
i=1
∫ t0 k(t, s)xi−1(s) ds, (n ≥ 0),
(7)
are uniformly convergent to x(t) on [0, γ].
Proof. Proving of uniqueness of solution, and that ϕn(t) is bounded andcontinuous, is similar proving of Theorem 3.1 and is omitted.
We only prove the uniform convergence of the sequence ϕn(t).For n ≥ 1 we have
D(ϕn+1(t), ϕn(t))
= D(f(t) +∑n+1
i=1
∫ t0 k(t, s)xi−1(s) ds, ϕn(t))
= D(ϕn(t) +∫ t0 k(t, s)xn(s) ds, ϕn(t))
= D(∫ t0 k(t, s)xn(s) ds, 0)
≤ supt∈[0,γ] D(xn(t), 0)∫ t0 |k(t, s)| ds
≤ M supt∈[0,γ] D(xn(t), 0),
hence we obtain
supt∈[0,γ]
D(ϕn+1(t), ϕn(t)) ≤ M supt∈[0,γ]
D(xn(t), 0). (8)
In the same way from (6) we have
supt∈[0,γ]
D(xn(t), 0) ≤ QMn,
which Q = supt∈[0,γ] D(f(t), 0), in result, from (8) we have
supt∈[0,γ]
D(ϕn+1(t), ϕn(t)) ≤ QMn+1,
Since the series QM∑∞
n=0 Mn is convergent, the series∑∞
n=0 D(ϕn+1(t), ϕn(t))is uniformly controlled on [0, γ] this implying the uniform convergence of thesequence ϕn(t). If we denote x(t) = limn→∞ ϕn(t), then x(t) satisfies (1).It is obviously continuous and bounded on [0, γ]. This finishes the proof ofTheorem 3.2.2
160 Hamid Rouhparvar, Tofigh Allahviranloo, Saeid Abbasbandy
4. CONCLUSIONIn this paper we proved, by using the ADM, the existence and uniqueness of
fuzzy solution for the linear Voltera fuzzy integral equations with an arbitrarycontinuous kernel. Also, we represented uniform convergence to the exactunique fuzzy solution in the theorems.
Acknowledgements: The authors would like to express their thanks tothe Science and Research Branch Islamic Azad University for the financialsupport and also to the referees for their valuable suggestions.
5. APPENDIXThe Adomian decomposition methodConsider the linear Voltera crisp integral equation as
x(t) = f(t) +∫ t
0k(t, s)x(s) ds, (9)
where f and k are known functions and x is to be determined. The Adomiandecomposition method consists of representing x as a series
x(t) =∞∑
n=0
xn(t). (10)
Because in eq. (9) there are no nonlinear terms, in the infinite series do notappear the so-called Adomian polynomials. Now by replacing (10) in (9), wewill have
∞∑
n=0
xn(t) = f(t) +∫ t
0k(t, s)
∞∑
n=0
xn(t) ds. (11)
Following Adomian analysis, Adomian decomposition method uses the recur-sive relations
x0(t) = f(t),
xn+1(t) =∫ t0 k(t, s)xn(s) ds, n ≥ 0.
(12)
We assume ϕn(t) =∑n
i=0 xi(t), obviously we have
x(t) = limn→∞ϕn(t),
hence we rewrite successive iterations (12) as follows
ϕ0(t) = f(t),
ϕn+1(t) = f(t) +∑n+1
i=1
∫ t0 k(t, s)xi−1(s) ds, n ≥ 0.
(13)
Existence and uniqueness of fuzzy solution for linear Volterra fuzzy integral equations... 161
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[10] J. Y. Park, H. K. Han, Existence and uniqueness theorem for a solution of fuzzy Volterraintegral equations, Fuzzy Sets and Systems, 105(1999), 481-488.
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[12] E. Babolian, A. Davari, Numerical implementation of Adomian decomposition methodfor linear Volterra integral equations of the second kind, Applied Mathematics andComputation, 165(2005), 223-227.
[13] A.M. Wazwaz, The existence of noise terms for systems of inhomogeneous differentialand integral equations, Appl. Math. and Comput., 146(2003), 81-92.
[14] M.L. Puri, D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl., 114(1986),409-422.
[15] P. Diamond, P.E. Kloeden, Metric spaces of fuzzy sets: theory and applications, World
Scienific, Singapore, 1994.
ON A CERTAIN DIFFERENTIAL INEQUALITY
ROMAI J., 5, 2(2009), 163–167
Roxana SendrutiuUniversity of Oradea, [email protected]
Abstract This work presents certain conditions on the functions A, B : U → C, whereU = z ∈ C : |z| < 1, such that the differential inequality
Re [A(z)p2(z) + B(z)p(z) + α(zp′(z)− a)3−
−3aβ(zp′(z))2
+ 3a2γ(zp′(z)) + δ] > 0
implies Re p(z) > 0, where p ∈ H[1, n], a ≥ 0, α, β, γ ∈ C. The aboveinequality is a generalization of a certain inequality obtained earlier by B.A.Frasin. Some related results are also provided.
Keywords: differential inequality, functions with positive real part.
2000 MSC: 30C80.
1. INTRODUCTIONWe denote by H[U ] the class of holomorphic functions in the open unit disc.
For a ∈ C and n ∈ N∗ we set
H[a, n] = f ∈ H[U ], f(z) = a + anzn + an+1zn+1 + ..., z ∈ U.
In order to prove the new results we use the following lemma, which is aparticular form of Theorem 2.3.i [3, p.35].
Lemma 1.1. [3] Let ψ : C2 × U → C be a function satisfying
Re ψ(ρi, σ; z) ≤ 0, (1)
where ρ, σ ∈ R, σ ≤ −n2 (1 + ρ2), z ∈ U and n ≥ 1.
If p ∈ H[1, n] andRe ψ
(p(z), zp′(z); z
)> 0 (2)
thenRe p(z) > 0.
2. MAIN RESULTSFollowing the results in [1] (see also [4,5,6,7]) we obtain the next theorem.
163
164 Roxana Sendrutiu
Theorem 2.1. Consider α ∈ C, Re α ≥ 0, β, γ ∈ C, α+β ∈ R+, α+γ ∈ R+,
a ≥ 0, δ <
(n3
8+ a3
)Re α+
3an2
4(α + β)+
3a2n
2(α + γ) where n is a positive
integer. Assume that the functions A,B : U → C satisfy
(i) Re A(z) > −3n3
8Re α− 3an2
2(α + β)− 3a2n
2(α + γ);
(ii) Im 2B(z) ≤ 4[3n3
8Re α +
3an2
2(α + β) +
3a2n
2(α + γ) + Re A(z)
]·
·[(
n3
8+ a3
)Re α +
3an2
4(α + β) +
3an2
2(α + γ)− δ
].
(3)If p ∈ H[1, n] and
Re [A(z)p2(z)+B(z)p(z)+α(zp′(z)−a)3−3aβ(zp′(z))2+3a2γ(zp′(z))+δ] > 0(4)
thenRe p(z) > 0.
Proof. Let ψ : C2 × U → C be defined by
ψ(p(z), zp′(z); z) = A(z)p2(z) + B(z)p(z)+ (5)
+α(zp′(z)− a)3 − 3aβ(zp′(z))2 + 3a2γ(zp′(z)) + δ.
From (4) we get
Re ψ(p(z), zp′(z); z) > 0, z ∈ U. (6)
For σ, ρ ∈ R satisfying σ ≤ −n2 (1 + ρ2), hence −σ2 ≤ −n2
4 (1 + ρ2)2,σ3 ≤ −n3
8 (1 + ρ2)3 and z ∈ U , by using (3) we obtain
Re ψ(ρi, σ; z) = Re [A(z)(ρi)2 + B(z)ρi + α(σ − a)3 − 3aβσ2 + 3a2γσ + δ] =
−ρ2Re A(z)− ρImB(z) + (σ3 − a3)Re α− 3a(α + β)σ2 + 3a2(α + γ)σ + δ ≤
−ρ2ReA(z)− ρImB(z)− n3
8(1 + ρ2)3Reα− a3Reα− 3an2
4(α + β)(1 + ρ2)2−
−3a2n
2(α + γ)(1 + ρ2) + δ =
−n3
8ρ6Re α−
[3n3
8Reα +
3an2
4(α + β)
]ρ4−
On a certain differential inequality 165
−[(
3n3
8Re α +
3an2
2(α + β) +
3an2
2(α + γ) + Re A(z)
)ρ2+
+ρIm B(z) +(
n3
8+ a3
)Reα +
3an2
4(α + β) +
3an2
2(α + γ)− δ
]≤ 0.
By using Lemma 1.1 we have Rep(z) > 0.
Remark 2.1. For a = 1 similar results were obtained in [2].
If δ =(
n3
8+ a3
)Reα+
3an2
4(α+β)+
3an2
2(α+γ), then Theorem 2.1 can
be rewritten as follows.
Corollary 2.1. Consider α ∈ C, Reα ≥ 0, β, γ ∈ C, α+β ∈ R+, α+γ ∈ R+,a ≥ 0 and let n be a positive integer. Assume that the functions A,B : U → Cwith ImB(z) = 0 satisfy
ReA(z) ≥ −3n3
8Reα− 3an2
2(α + β)− 3a2n
2(α + γ). (7)
If p ∈ H[1, n] and
Re[A(z)p2(z) + B(z)p(z) + α(zp′(z)− a)3 − 3aβ(zp′(z))2 + 3a2γ(zp′(z)+
+(
n3
8+ a3
)Reα +
3an2
4(α + β) +
3a2n
2(α + γ)] > 0
then Rep(z) > 0.
Taking β = γ = α in the Theorem 2.1, we have
Corollary 2.2. Consider α ∈ C, Reα ≥ 0, a ≥ 0, n a positive integer, and
δ such that δ <
(n3
8+ a3 +
3an2
2+ 3a2n
)Reα. Assume that the functions
A, B : U → C satisfy
(i) ReA(z) >
(−3n3
8− 3an2 − 3a2n
)· Reα;
(ii) Im2B(z) ≤ 4 ·[(
3n3
8+ 3an2 + 3a2n
)Reα + ReA(z)
]·
·[(
n3
8+ a3 +
3an2
2+ 3a2n
)Reα− δ
].
(8)
If p ∈ H[1, n] and
Re[A(z)p2(z)+B(z)p(z)+α(zp′(z)−a)3−3aα(zp′(z))2+3a2α(zp′(z))+δ] > 0(9)
166 Roxana Sendrutiu
thenRep(z) > 0.
Taking α + β = α + γ = 1 in Theorem 2.1, we obtain
Corollary 2.3. Consider α ∈ C, Re α ≥ 0, a ≥ 0, n a positive integer, and
δ ∈ R such that δ <
(n3
8+ a3
)Reα+
3an2
4+
3a2n
2. Assume that the functions
A, B : U → C satisfy
(i) ReA(z) > −3n3
8Reα− 3an2
2− 3a2n
2;
(ii) Im2B(z) ≤ 4(
3n3
8Reα +
3an2
2+
3a2n
2+ ReA(z)
)·
·[(
n3
8+ a3
)Reα +
3an2
4+
3a2n
2− δ
].
(10)
If p ∈ H[1, n] and
Re[A(z)p2(z) + B(z)p(z) + α(zp′(z)− a)3 − 3a(1− α)(zp′(z))2+ (11)
+3a2(1− α)(zp′(z)) + δ] > 0
thenRep(z) > 0.
Taking α = 0 in the Theorem 2.1, we have
Corollary 2.4. Consider β, γ > 0, a ≥ 0, δ <3an2
4β +
3a2n
2γ where n is a
positive integer. Assume that the functions A, B : U → C satisfy
(i) ReA(z) > −3an2
2β − 3a2n
2γ;
(ii) Im2B(z) ≤ 4(
3an2
2β +
3a2n
2γ + ReA(z)
)·
·(
3an2
4β +
3a2n
2γ − δ
).
(12)
If p ∈ H[1, n] and
Re[A(z)p2(z) + B(z)p(z)− 3aβ(zp′(z))2 + 3a2γ(zp′(z)) + δ] > 0 (13)
thenRep(z) > 0.
Taking β = γ = 0 in the Theorem 2.1, we obtain
On a certain differential inequality 167
Corollary 2.5. Consider α > 0, a ≥ 0, δ <
(n3
8+ a3 +
3an2
4+
3a2n
2
)· α
where n is a positive integer. Assume that the functions A,B : U → C satisfy
(i) ReA(z) > −(
3n3
8− 3an2
2− 3a2n
2
)· α;
(ii) Im2B(z) ≤ 4[(
3n3
8+
3an2
2+
3a2n
2
)α + ReA(z)
]·
·[(
n3
8+ a3 +
3an2
4+
3a2n
2
)α− δ
].
(14)
If p ∈ H[1, n] and
Re[A(z)p2(z) + B(z)p(z) + α(zp′(z)− a)3 + δ] > 0 (15)
thenRep(z) > 0.
References[1] A. Catas, On certain analytic functions with positive real part, (to appear).
[2] B.A. Frasin, On a differential inequality, An. Univ. Oradea Fasc. Math. 14(2007), 81-87.
[3] S.S. Miller, P.T. Mocanu, Differential subordinations. Theory and applications, Pureand Applied Mathematics, Marcel Dekker, Inc., New York, 2000.
[4] G.I. Oros, A. Catas, A new differential inequality I, Gen. Math. 11(2003), No. 1-2,47-52.
[5] Gh. Oros, G.I. Oros, A. Catas, A new differential inequality II, Studia Univ. Babes-Bolyai Math. 49(2004), No.4, 85-90.
[6] Gh. Oros, G.I. Oros, A. Catas, On a special differential inequality II, An. Univ. OradeaFasc. Math. 11(2004), 119-122.
[7] Gh. Oros, G.I. Oros, A. Catas, On a special differential inequality I, An. Univ. OradeaFasc. Math. 10(2003), 109-113.
NOTE ON A PAPER ON NONLINEAR INVERSETIME HEAT EQUATION IN THE UNBOUNDEDREGION
ROMAI J., 5, 2(2009), 169–180
Nguyen Huy Tuan, Dang Duc Trong, Pham Hoang QuanDepartment of Mathematics, Sai Gon University, Viet NamDepartment of Mathematics, HoChiMinh City National University, Viet NamDepartment of Mathematics, Sai Gon University, Viet Namtuanhuy [email protected]
Abstract The nonlinear inverse time heat problem
ut − uxx = f(x, t, u(x, t)), u(x, T ) = ϕ(x), (x, t) ∈ R× (0, T )
in the unbounded region is regularized by Fourier transform method. Some newconvergence rates are obtained. Meanwhile, some quite sharp error estimatesbetween the approximate solution and exact solution are provided and theconvergence of the approximate solution to the exact one at t = 0 is alsoproved. This work extends some previous results in [3, 4, 6, 11, 12, 13, 15, 16].
Keywords: nonlinear heat problem, ill-posed problem, Fourier transform, contraction prin-
ciple.
2000 MSC: 35K05, 35K99, 47J06, 47H10.
1. INTRODUCTIONIn this paper we consider the nonlinear heat backward problem in the un-
bounded region
ut − uxx = f(x, t, u(x, t)), (x, t) ∈ R× (0, T ),u(x, T ) = ϕ(x), x ∈ R,
(1)
where ϕ(x), and f(x, t, z) are given. This problem is called in the literaturebackward heat problem, backward Cauchy problem, or final value problem.It is well known that (1) is severely ill-posed i.e., solutions do not alwaysexist, and in the case of existence, these do not depend continuously on thegiven data, and regularization methods for this problem are required. Infact, from small noise contaminated physical measurements, the correspondingsolutions have large errors and this makes the numerical calculations difficult.In the mathematical literature various methods have been proposed for solvingbackward Cauchy problems. We can mention the method of quasi-solution
169
170 Nguyen Huy Tuan, Dang Duc Trong, Pham Hoang Quan
(QS-method) by Tikhonov, the method of quasi-reversibility (QR method) byLattes and Lions, the quasi boundary value method (Q.B.V. method) and theC-regularized semigroups technique.
In the case f = 0, the problem (1) becomes
ut − uxx = 0, (x, t) ∈ R× (0, T ),u(x, T ) = ϕ(x), x ∈ R,
(2)
The problem (2) was recently investigated by many authors such as Chu LiFu et al [16, 3, 4], Trong and Lien [15], Dinh Nho Hao and Nguyen Van Duc[6]. Although there are many works on the linear homogeneous case of thebackward heat problem (2), the literature on the nonlinear case (the problem(1)) is quite scarce.Recently, in [11], Trong and Quan have established, under the hypothesis thatf is a global Lipschitzian function, the existence of a unique solution for someapproximate well-posed problem as follows
uε(x, t) =1√2π
+∞∫
−∞
e−tp2
+e−Tp2 ϕ(p)eipxdp (3)
− 1√2π
+∞∫
−∞
T∫
t
e−tp2
εsT + e−sp2
f(p, s, uε)eipxdpds,
where ε is a positive parameter and
g(ξ) =1√2π
∫ +∞
−∞g(x)e−iξxdx
is the Fourier transform of g. Under a strong smoothness assumption on theoriginal solution namely
T∫
0
∞∫
−∞
∣∣∣∣∂
∂t
(esp2
u(p, t))∣∣∣∣
2
dpdt < ∞
and ∞∫
−∞
∣∣∣eTp2ϕ(p)
∣∣∣2dp < ∞,
they obtained the following error estimate
‖u(., t)− uε(., t)‖ ≤√
M exp(3k2T (T − t)
2)ε
tT , (4)
Note on a paper on nonlinear inverse time heat equation in the unbounded region 171
where ‖ · ‖ is the norm in L2(R) and
M = 3
∞∫
−∞
∣∣∣eTp2ϕ(p)
∣∣∣2dp + 3T
T∫
0
∞∫
−∞
∣∣∣∣∂
∂s
(esp2
u(p, s))∣∣∣∣
2
dpds. (5)
The right hand side of (4) is not close to zero if ε is fixed and t tends to zero.Hence, the convergence of the approximate solution is very slow when t is ina neighborhood of zero. This is a weak point of the paper [11].
In this paper, we use new methods in order to improve the results given in[11]. The major aim of this paper is to provide a quite simple and convenientnew regularization method. We construct an approximate problem which,under some assumptions on the solution of the initial problem, leads to fasterconvergence and stronger error estimates. Moreover, the convergence of theapproximate solution at t = 0 is also proved.
This paper is organized as follows. In the following section we outline themain results while the proofs will be given in section 3.
2. THE MAIN RESULTSLet f : R× [0, T ]× R→ R. The function f is said to be a global Lipschitz
function if there exists a constant K > 0 (independent of x, t, u, v) such that
|f(x, t, u)− f(x, t, v)| ≤ K|u− v|, (6)
for all (x, t) ∈ R× [0, T ].Let
g(ξ) =1√2π
∫ +∞
−∞g(x)e−iξxdx
be the Fourier transform of the function g(x) ∈ L2(R). The unique solutionof (1) is easily seen to satisfy
u(ξ, t) = e(T−t)ξ2
ϕ(ξ)−T∫
t
e−(t−s)ξ2
f(ξ, s, u)ds. (7)
In the present paper, we approximate problem (7) by the following problem
wβ(ξ, t) =e−tξ2
βξ2 + e−Tξ2 ϕ(ξ)−T∫
t
e(s−t−T )ξ2
βξ2 + e−Tξ2 f(ξ, s, w)ds, (8)
172 Nguyen Huy Tuan, Dang Duc Trong, Pham Hoang Quan
or
wβ(x, t) = 1√2π
+∞∫−∞
e−tξ2
βξ2+e−Tξ2ϕ(ξ)eiξxdξ − 1√
2π
+∞∫−∞
T∫t
e(s−t−T )ξ2
βξ2+e−Tξ2f(ξ, s, wβ)eiξxdξds
(9)where β is a positive number such that 0 < β < eT .
Theorem 1. Assume that f : R× [0, T ]×R→ R satisfies f(x, t, 0) = 0 andthat (6) holds. Let ϕ ∈ L2(R). Then the problem (9) has a unique solution
u ∈ C([0, T ];H1(R)) ∩ L2(0, T ;H2(R)) ∩ C1((0, T );H1(R)).
Furthermore, if w, v are the solutions of the problem (9) corresponding to thefinal values ϕ and φ, then
‖w(., t)− v(., t)‖ ≤ βtT−1
(T
1 + ln(Tβ )
)1− tT
eK2(T−t)2‖ϕ− φ‖. (10)
Remark 1. In [1], the stability of magnitude is of order eTε . In [3, 11, 14],
the stability estimate is of order εtT−1.
In our paper, we give a better estimation of the stability order, which is
CβtT−1
(T
1+ln(Tβ
)
)1− tT
. This proves the advantages of our method.
Theorem 2. Let f, ϕ, u be as in Theorem 1. Suppose that ϕ ∈ L2(R) satisfieseTξ2
ϕ(ξ) ∈ L2(R) and problem (7) has an unique solution u ∈ C([0, T ];H1(R))such that
+∞∫
−∞|(ξ2etξ2
u(ξ, t))|2dξdt < ∞.
Consider ϕβ ∈ L2(R) such that ‖ϕβ − ϕ‖ ≤ β. Let wβ be the unique solutionof problem (9) corresponding to ϕβ. Then, we have
‖u(., t)− wβ(., t)‖ ≤ Cβt/T
(T
1 + ln(Tβ )
)1− tT
, ∀t ∈ (0, T ],
where
B = 2
+∞∫
−∞
∣∣∣ξ2etξ2
u(ξ)∣∣∣2dξ, C =
√2eK2(T−t)2 +
√Be
3T2K2
2 ,
for all 0 < β < eT .Remark 2. 1. In [12], Tautenhahn and his group established the followingerror
‖u(., t)− uβ(., t)‖ ≤ 2E1− tT β
tT
Note on a paper on nonlinear inverse time heat equation in the unbounded region 173
and they proved that it is an order optimal stability estimate in L2(R). Thus,in this case, our method is also of optimal order.
2. Notice that the convergence estimate in [11, 14] also given by a functionof the form C
tT does not give any useful information on the continuous de-
pendence of the solution at t = 0. Actually, when t → 0+, the accuracy of theregularized solution becomes progressively lower.
3. In our method, we give a new error estimation in the original time t = 0,which is not in [11]. Thus, we estimate the error as follows
‖u(., 0)− uβ(., 0)‖ ≤ CT
(1 + ln(
T
β))−1
. (11)
Moreover, comparing (11) with the result obtained in [6, 11, 14] we see thatestimate (11) is sharp and the best known estimate.
3. PROOF OF THE MAIN RESULTSFor 0 ≤ t ≤ s ≤ T , denote
A(ξ, t) =exp−tξ2
βξ2 + exp−Tξ2 , B(ξ, s, t) =exp(s− t− T )ξ2βξ2 + exp−Tξ2 , C(β) =
T
1 + ln(Tβ )
.
It is easy to prove that
A(ξ, t) ≤ βtT−1
(T
1 + ln(Tβ )
)1− tT
= A(β, t), (12)
B(ξ, s, t) ≤ βtT− s
T
(T
1 + ln(Tβ )
) sT− t
T
= A(β, t)B(β, s), (13)
where
B(β, s) = β1− sT
(T
1 + ln(Tβ )
) sT−1
.
Proof of Theorem 1. a) Existence and uniqueness of the solution.Denote
G(w)(x, t) =1√2π
ψ(x, t)− 1√2π
+∞∫
−∞
T∫
t
e(s−t−T )ξ2
βξ2 + e−Tξ2 f(ξ, s, w)eiξxdξds
for all w ∈ C([0, T ]; L2(R)) and
ψ(x, t) =
+∞∫
−∞
e−tξ2
βξ2 + e−Tξ2 ϕ(ξ)eiξxdξ.
174 Nguyen Huy Tuan, Dang Duc Trong, Pham Hoang Quan
Due to f(x, y, 0) = 0, and to the Lipschitzian property of f(x, y, w) withrespect to w, we get G(w) ∈ C([0, T ]; L2(R)) for every w ∈ C([0, T ];L2(R)).We claim that , for every w, v ∈ C([0, T ];L2(R)),m ≥ 1, we have
‖Gm(w)(., t)−Gm(v)(., t)‖2 ≤(
K
β
)2m (T − t)mCm
m!|||w − v|||2, (14)
where C = maxT, 1 and |||.||| is the sup norm in C([0, T ];L2(R)). We shallprove the latter inequality by induction.When m = 1, we have
‖G(w)(., t)−G(v)(., t)‖2 = ‖G(w)(., t)− G(v)(., t)‖2
=
+∞∫
−∞
∣∣∣∣∣∣
T∫
t
e(s−t−T )ξ2
βξ2 + e−Tξ2
(f(ξ, s, w)− f(ξ, s, v)
)ds
∣∣∣∣∣∣
2
dξ
≤+∞∫
−∞
T∫
t
(e(s−t−T )ξ2
βξ2 + e−Tξ2
)2
ds
∫ T
t
∣∣∣f(ξ, s, w)− f(ξ, s, v)∣∣∣2ds
2
dξ
≤ 1β2 (T − t)
T∫
t
‖f(., s, w(., s))− f(., s, v(., s))‖2ds =
=1β2 (T − t)
T∫
t
‖f(., s, w(., s))− f(., s, v(., s))‖2ds
≤ K2
β2 (T − t)
T∫
t
‖w(., s)− v(., s)‖2ds ≤ CK2
β2 (T − t)|||w − v|||2.
Therefore (14) holds for m = 1. Assume that (14) holds for m = p. We provethat it also holds for m = p + 1. We have
‖Gp+1(w)(., t)−Gp+1(v)(., t)‖2 = ‖G(Gp(w))(., t)− G(Gp(v))(., t)‖2
=
+∞∫
−∞
∣∣∣∣∣∣
T∫
t
e(s−t−T )ξ2
βξ2 + e−Tξ2
(f(ξ, s, Gp(w))− f(ξ, s, Gp(v))
)ds
∣∣∣∣∣∣
2
dξ
≤+∞∫
−∞
T∫
t
(e(s−t−T )ξ2
βξ2 + e−Tξ2
)2
ds
∫ T
t
∣∣∣f(ξ, s,Gp(w))− f(ξ, s, Gp(v))∣∣∣2ds
2
dξ.
Note on a paper on nonlinear inverse time heat equation in the unbounded region 175
Hence‖Gp+1(w)(., t)−Gp+1(v)(., t)‖2 =
≤ 1β2 (T − t)
T∫
t
‖f(., s,Gp(w)(., s))− f(., s,Gp(v)(., s))‖2ds
≤ K2
β2 (T − t)
T∫
t
‖Gp(w)(., s)−Gp(v)(., s)‖2ds
≤ K2
β2 (T − t)(
K
β
)2pT∫
t
(T − s)p
p!dsCp|||w − v|||2
≤(
K
β
)2(p+1) (T − t)(p+1)C(p+1)
(p + 1)!|||w − v|||2.
Therefore, by the induction principle, (14) holds for every m and, thus,
|||Gm(w)−Gm(v)||| ≤(
K
β
)m Tm/2
√m!
Cm|||w − v|||
for every w, v ∈ C([0, T ];L2(R)).Consider G : C([0, T ]; L2(R)) → C([0, T ]; L2(R)). Since
limm→∞
(K
β
)m Tm/2Cm
√m!
= 0,
there exists a positive integer number m0 such that Gm0 is a contraction. Itfollows that Gm0(w) = w has a unique solution uβ ∈ C([0, T ]; L2(R)).
We claim that G(uβ) = uβ. In fact, one has G(Gm0(uβ)) = G(uβ). HenceGm0(G(uβ)) = G(uβ). By the uniqueness of the fixed point of Gm0 , onehas G(uβ) = uβ, i.e., the equation G(w) = w has a unique solution uβ ∈C([0, T ];L2(R)).
b. Proof of estimate (10). Let w and v be two solution of the problem(9) corresponding to the final values ϕ and φ. We have
‖w(., t)− v(., t)‖2 ≤ 2
+∞∫
−∞
∣∣∣A(ξ, t)(ϕ(ξ)− φ(ξ)
)∣∣∣2dξ
+2
+∞∫
−∞
∣∣∣∣∣∣
T∫
t
B(ξ, s, t)(f(ξ, s, w)− f(ξ, s, v)
)ds
∣∣∣∣∣∣
2
dξ = J1 + J2.
176 Nguyen Huy Tuan, Dang Duc Trong, Pham Hoang Quan
By using (12), the terms J1, J2 can be estimated as follows
J1 = 2
+∞∫
−∞
∣∣∣A(ξ, t)(ϕ(ξ)− φ(ξ)
)∣∣∣2dξ ≤ 2β
2tT−2
(T
1 + ln(Tβ )
)2− 2tT
‖ϕ− φ‖2
≤ 2β2tT−2
(T
1 + ln(Tβ )
)2− 2tT
‖ϕ− φ‖2.
By using (13), we get
J2 = 2
+∞∫
−∞
∣∣∣∣∣∣
T∫
t
B(ξ, s, t)(f(ξ, s, w)− f(ξ, s, v)
)ds
∣∣∣∣∣∣
2
dξ
≤ 2(T − t)(A(β, t)
)2+∞∫
−∞
T∫
t
(B(β, s)
)2 ∣∣∣f(ξ, s, w)− f(ξ, s, v)∣∣∣2dsdξ
≤ 2(T − t)(A(β, t)
)2K2
T∫
t
(B(β, s)
)2‖w(., s)− v(., s)‖2 ds.
It follows that
‖u(., t)− v(., t)‖2 ≤ 2β2tT−2
(T
1 + ln(Tβ )
)2− 2tT
‖ϕ− φ‖2
+2(T − t)(A(β, t)
)2K2 ×
T∫
t
(B(β, s)
)2‖w(., s)− v(., s)‖2 ds.
Hence
β−2tT (C(β))
2tT ‖w(., t)− v(., t)‖2 ≤ 2β−2 (C(β))2 ‖ϕ− φ‖2 +
+2K2(T − t)∫ T
tβ−2sT (C(β))
2sT ‖w(., s)− v(., s)‖2ds.
By using the Gronwall’s inequality, we obtain
‖w(., t)− v(., t)‖ ≤√
2βtT−1
(T
1 + ln(Tβ )
)1− tT
eK2(T−t)2‖ϕ− φ‖,
and this ends the proof of Theorem1.
Note on a paper on nonlinear inverse time heat equation in the unbounded region 177
Proof of Theorem 2. By using (7), we have
‖u(., t)− wβ(., t)‖2 =
+∞∫
−∞|u(ξ, t)− wβ(ξ, t)|2dξ
=
+∞∫
−∞
∣∣∣∣∣∣(e(T−t)ξ2 −A(ξ, t))ϕ(ξ)−
T∫
t
e−(t−s)ξ2
f(ξ, s, u)ds+
T∫
t
B(ξ, s, t)f(ξ, s, wβ)ds
∣∣∣∣∣∣
2
dξ
=
+∞∫
−∞
∣∣∣∣∣∣βξ2e(T−t)ξ2
(βξ2 + e−Tξ2)ϕ(ξ) +
T∫
t
e(s−t−T )ξ2
βξ2 + e−Tξ2 (f(ξ, s, u)− f(ξ, s, wβ))ds
−T∫
t
e(s−t)ξ2βξ2
(βξ2 + e−Tξ2)f(ξ, s, u)ds
∣∣∣∣∣∣
2
dξ
=
+∞∫
−∞
∣∣∣∣∣∣βξ2
(βξ2 + e−Tξ2)u(ξ, t) +
T∫
t
e(s−t−T )ξ2
βξ2 + e−Tξ2 (f(ξ, s, u)− f(ξ, s, wβ))ds
∣∣∣∣∣∣
2
dξ
≤ 2
+∞∫
−∞
∣∣∣∣∣βe−tξ2
(βξ2 + e−Tξ2)ξ2etξ2
u(ξ, t)
∣∣∣∣∣2
dξ + 2
+∞∫
−∞
∣∣∣∣∣∣
T∫
t
B(ξ, s, t)|f(ξ, s, u)− f(ξ, s, wβ)|ds
∣∣∣∣∣∣
2
dξ.
By using (13), we get
‖u(., t)− wβ(., t)‖2 ≤ 2(βA(β, t)
)2+∞∫
−∞
∣∣∣ξ2etξ2
u(ξ, t)∣∣∣2dξ+
+2
+∞∫
−∞
∣∣∣∣∣∣
T∫
t
A(β, t).B(β, s)∣∣∣f(ξ, s, u)− f(ξ, s, wβ)
∣∣∣ds
∣∣∣∣∣∣
2
dξ
= 2A1 + 2A2.
where the term A1 is equal to
A1 = β2t/T
(T
1 + ln(Tβ )
)2− 2tT
+∞∫
−∞
∣∣∣ξ2etξ2
u(ξ, t)∣∣∣2dξ.
178 Nguyen Huy Tuan, Dang Duc Trong, Pham Hoang Quan
We estimate A2 as follows
A2 =
+∞∫
−∞
∣∣∣∣∣∣
T∫
t
A(β, t)B(β, s)∣∣∣f(ξ, s, u)− f(ξ, s, wβ)
∣∣∣ds
∣∣∣∣∣∣
2
dξ
≤(βA(β, t)
)2(T − t)
+∞∫
−∞
T∫
t
(1β
B(β, s))2 ∣∣∣f(ξ, s, u)− f(ξ, s, wβ)ds
∣∣∣2dξ
=(βA(β, t)
)2(T − t)
T∫
t
(1β
B(β, s))2
‖f(., s, u(., s))− f(., s, wβ(., s))‖2 ds
≤ β2tT (C(β))2−
2tT (T − t)
T∫
t
β−2sT (C(β))
2sT−2 K2 ‖u(., s)− wβ(., s)‖2 ds.
Hence
‖u(., t)− wβ(., t)‖2 ≤ 2β2t/T (C(β))2−2tT
+∞∫
−∞
∣∣∣ξ2etξ2
u(ξ, t)∣∣∣2dξ+
+2β2t/T (C(β))2−2tT (T − t)×
T∫
t
β−2sT (C(β))
2sT−2 K2 ‖u(., s)− wβ(., s)‖2 ds
≤ 2β2t/T (C(β))2−2tT
+∞∫
−∞
∣∣∣ξ2etξ2
u(ξ, t)∣∣∣2dξ+
+2K2β2t/T (C(β))2−2tT T ×
T∫
t
β−2s/T (C(β))2sT−2 ‖u(., s)− wβ(., s)‖2 ds.
Hence
β−2t/T
(T
1 + ln(Tβ )
) 2tT−2
‖u(., t)− wβ(., t)‖2 ≤
≤ B + 3K2T
T∫
t
β−2s/T
(T
1 + ln(Tβ )
) 2sT−2
‖u(., s)− wβ(., s)‖2 ds,
where
Note on a paper on nonlinear inverse time heat equation in the unbounded region 179
B = 2
∞∫
−∞
∣∣∣ξ2etξ2
u(ξ, t)∣∣∣2dξ.
By applying the Gronwall’s inequality, we get
β−2t/T
(T
1 + ln(Tβ )
) 2tT−2
‖u(., t)− wβ(., t)‖2 ≤ Be3K2T (T−t).
Hence
‖u(., t)− wβ(., t)‖ ≤√
Be3T2K2
2 βt/T
(T
1 + ln(Tβ )
)1− tT
. (15)
Let uβ be the approximate solution of problem (9) corresponding to thefinal value ϕβ. From the step 2 of Theorem 1 and (15)
‖uβ(., t)− u(., t)‖ ≤ ‖uβ(., t)− wβ(., t)‖+ ‖wβ(., t)− u(., t)‖
≤√
2βtT−1
(T
1 + ln(Tβ )
)1− tT
eK2(T−t)2‖ϕ− ϕβ‖
+βt/T
(T
1 + ln(Tβ )
)1− tT √
Be3T2K2
2 ≤ Cβt/T
(T
1 + ln(Tβ )
)1− tT
where
C =√
2eK2(T−t)2 +√
B exp3T 2K2
2,
for all t ∈ [0, T ].
References[1] G. W. Clark, S. F. Oppenheimer, Quasireversibility methods for non-well posed prob-
lems, Elect. J. Diff. Eqns., 1994, 8(1994), 1-9.
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180 Nguyen Huy Tuan, Dang Duc Trong, Pham Hoang Quan
[7] R. Lattes, J.-L. Lions, Methode de quasi-reversibilite et applications, Dunod, Paris,1967.
[8] R.E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl,47 (1974), 563-572.
[9] I. V. Mel’nikova, Q. Zheng, J. Zheng, Regularization of weakly ill-posed Cauchy problem,J. Inv. Ill-posed Problems, 10, 5(2002), 385-393.
[10] W. B. Muniz, A comparison of some inverse methods for estimating the initial condi-tion of the heat equation, J. Comput. Appl. Math., 103(1999), 145-163.
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[15] D. D. Trong, T. N. Lien, Regularization of a discrete backward problem using coefficientsof truncated Lagrange polynomials, Electron. J. Diff. Eqns., 2007, 51(2007), 1-14.
[16] X.-T. Xiong, C. L. Fu, Z. Qian, X. Gao, Error estimates of a difference approximationmethod for a backward heat conduction problem, Int. J. of Math. and Math. Sci., 2006,Article ID 45489, 1-9.
STABILITY OF ELASTIC ELEMENTSOF WING PROFILE WITH TIME DELAYOF BASES REACTIONS
ROMAI J., 5, 2(2009), 181–192
Petr A. Velmisov, Andrey V. AnkilovUlyanovsk State Technical University, [email protected]
Abstract Lyapunov stability of elastic elements (plates) of wing profile in interactionwith flow of fluid or gas is studied. Subsonic regime is considered. Aerody-namic load is determined by asymptotic aeromechanics equations [1]. The timedelay of the elements bases reactions is taken into account. Statements andinvestigation methods offered for dynamical damping elastic bodies, being incontact with subsonic flow of the fluid or gas, lead to the study of linked initialboundary problems to system of partial differential equations. Being basedon the construction of functionals, corresponding to this system, conditions ofsolutions stability are obtained. Similar problems without time delay of theplates bases reactions were earlier considered in works [2-5].Acknowledgement This work is realized with the support of grant RFBR-RA07-01-91680.
Keywords: aeroelasticity; dynamic stability; time delay; partial integro-differential equa-
tions; retarded argument.
2000 MSC: 76G25.
1. INTRODUCTIONLet us consider the planar problem of aeroelasticity concerning small fluctu-
ations, appearing with non-circulatory flow around thin-shelled construction- the model of wing, whose component parts are n elastic elements-insertions.Assume that in the plane xOy, in which occur joint oscillations of elastic in-sertions and gas, segment [c, d] on axis Ox corresponds to wing, and segments[a2k−1, a2k], k = 1, ..., n, −∞ < c ≤ a2k−1 < a2k ≤ a2k+1 < a2k+2 ≤ d <+∞, k = 1, ..., n− 1, to elastic insertions.
At large distances the velocity of gas is V and has the direction of axis Ox.Let the perturbations of homogeneous gas stream be directed along Ox-
axis and assume that the deviations of the elements are small. We denote byϕ(x, y, t) the potential of gas velocity and wk(x, t) (k = 1, ..., n) the oscillationsfunctions of the elements.
The potential of velocity satisfies the Laplace equation
4ϕ ≡ ϕxx + ϕyy = 0. (x, y) ∈ G = R2\[c, d]. (1)
181
182 Petr A. Velmisov, Andrey V. Ankilov
Fig. 1. Wing profile.
We linearize the boundary condition
ϕ±y (x, 0, t) = wk(x, t) + V w′k(x, t), x ∈ (a2k−1, a2k), k = 1, ..., n, (2)
ϕ±y (x, 0, t) = V f±k+1′(x), x ∈ (a2k, a2k+1), k = 1, ..., n− 1,
ϕ±y (x, 0, t) = V f±1′(x), x ∈ (c, a1), ϕ±y (x, 0, t) = V f±n+1
′(x), x ∈ (a2n, d).(3)
The condition of perturbations absence at large distances is
| 5 ϕ|2∞ = (ϕ2x + ϕ2
y + ϕ2t )∞ = 0. (4)
By linearizing the Lagrange-Cauchy integral, we obtain the equations of smallfluctuations of elastic plates as
Lk(wk) = ρ(ϕ+
t − ϕ−t)
+ ρV(ϕ+
x − ϕ−x), y = 0, (5)
x ∈ (a2k−1, a2k), k = 1, ..., n,
Lk(wk) ≡[Dkw
′′k(x, t) + β2k w ′′
k (x, t)]′′ + Mkwk(x, t)+
+(Nk(t)w′k(x, t)
)′ + β1kwk(x, t) + β0kwk(x, t− τ). (6)
Here a point over letters denotes the time derivative, a prime is used for thederivative with respect to x or x1, subindices x, y, t designate partial deriva-tives with respect to the corresponding variables; ρ is the density of the gas; Vis velocity of the undisturbed homogeneous flow; Dk and Mk are the moduli ofrigidity and the specific masses of the elements; Nk are the forces compressing(spraining) the elements; β1k are the rotational inertia coefficients; β2k aredamping coefficients of the elements material; β0k are the stiffness coefficientsof the bases; τ is a time of the delay of the elements bases reactions; f±k (x)are given functions determining the form of undeformable parts of a wing.
By using methods of the theory of complex variable functions [6], the prob-lem (1) - (6) can be reduced to the following equation for the unknown function
Stability of elastic elements of wing profile with time delay of bases reactions 183
of plate oscillations
Lk(wk) = −ρ
π
n∑
i=1
a2i∫
a2i−1
(wi(x1, t) + V wi′ (x1, t))K(x1, x) dx1−
−V ρ
π
n∑
i=1
a2i∫
a2i−1
(wi(x1, t)+V wi′ (x1, t))
∂K(x1, x)∂x
dx1 +V 2ρ
π
a1∫
c
(f+1′(x1)+ (7)
+f−1′(x1))G(x1, x) dx1 +
V 2ρ
π
d∫
a2n
(f+n+1
′(x1) + f−n+1′(x1))G(x1, x) dx1+
+V 2ρ
π
n−1∑
i=1
a2i+1∫
a2i
(f+i′(x1) + f−i
′(x1))G(x1, x) dx1, x ∈ (a2k−1, a2k),
where
K(x1, x) = 2 ln
√(x− c)(d− x1) +
√(x1 − c)(d− x)
|√
(x− c)(d− x1)−√
(x1 − c)(d− x) |,
G(x1, x) =
√(x− c)(d− x) +
√(x1 − c)(d− x1)√
(x− c)(d− x)(x− x1), x1, x ∈ [c, d], x1 6= x.
2. STABILITY INVESTIGATIONWe want to determine sufficient conditions of stability of solutions of the
integro-differential equations system (7) with respect to perturbations of theinitial conditions. Since the system of equations (7) is linear, it is enough tostudy the stability of trivial solution of the corresponding system of homoge-neous equations
Lk(wk) = −ρ
π
n∑
i=1
a2i∫
a2i−1
(wi(τ , t) + V w ′i (τ , t))K(τ , x) dτ−
−V ρ
π
n∑
i=1
a2i∫
a2i−1
(wi(τ , t) + V w ′i (τ , t))
∂K(τ , x)∂x
dτ. (8)
The dynamic stability of this integro-differential equations system (8) isinvestigated in the hypothesis that at each end of the plates one of the followingsets of boundary conditions is satisfied
wk(x, t) = w′k(x, t) = 0, wk(x, t) = w′′k(x, t) = 0, x = a2k−1 or x = a2k. (9)
184 Petr A. Velmisov, Andrey V. Ankilov
that corresponds to rigid or hinged fastenings. We consider the Lyapunov typefunctional for system (8):
Φ(t) =n∑
k=1
a2k∫
a2k−1
(Mkw
2k + Dkw
′′k2 −Nkw
′k2 + β0kw
2k
)dx+
+n∑
k=1
β0k
a2k∫
a2k−1
t∫
t−τ
dt1
t∫
t1
w2k(x, s)ds
dx + I(t) + J(t), (10)
I(t) =ρ
π
n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
wi(x, t)wj(x1, t)K(x1, x) dx1,
J(t) = −ρV 2
π
n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
w′i(x, t)w′j(x1, t)K(x1, x) dx1.
We compute the derivative of Φ with respect to t
Φ(t) =n∑
k=1
a2k∫
a2k−1
(2Mkwkwk + 2Dkw
′′kw′′k − 2Nkw
′kw
′k + 2β0kwkwk+
+β0k
t∫
t−τ
w2k(x, t)dt1 − β0k
t∫
t−τ
w2k(x, s)ds
dx + I + J .
Considering that wk(x, t − τ) = wk(x, t) −t∫
t−τ
wk(x, s)ds, for functions
wk(x, t) that are solutions of equations system (8), the following equality isobtained
Φ(t) =n∑
k=1
a2k∫
a2k−1
−2wk
Dkw
′′′′k + Nkw
′′k + β0kwk − β0k
t∫
t−τ
wk(x, s)ds+
+β1kwk + β2kw′′′′k +
ρ
π
n∑
i=1
a2i∫
a2i−1
(wi(x1, t) + V w′i(x1, t)
)K(x1, x)dx1+
Stability of elastic elements of wing profile with time delay of bases reactions 185
+V ρ
π
n∑
i=1
a2i∫
a2i−1
(wi(x1, t) + V w′i(x1, t)
) ∂K(x1, x)∂x
dx1
+ 2Dkw
′′kw′′k− (11)
−2Nkw′kw
′k + 2β0kwkwk + β0k
t∫
t−τ
w2k(x, t)dt1 − β0k
t∫
t−τ
w2k(x, s)ds
dx+
+I + J .
Taking into account the conditions (9) and integrating by parts, we get
a2k∫
a2k−1
wkw′′′′k dx =
a2k∫
a2k−1
w′′kw′′kdx,
a2k∫
a2k−1
wkw′′kdx = −
a2k∫
a2k−1
w′kw′kdx,
a2k∫
a2k−1
wkw′′′′k dx =
a2k∫
a2k−1
w′′k2dx.
With these equalities the derivative of Φ becomes
Φ(t) =n∑
k=1
a2k∫
a2k−1
−2β1kw
2k − 2β2kw
′′k
2− 2ρ
πwk
n∑
i=1
a2i∫
a2i−1
(wi(x1, t) + V w′i(x1, t)
)×
×K(x1, x)dx1 − 2V ρ
πwk
n∑
i=1
a2i∫
a2i−1
(wi(x1, t) + V w′i(x1, t)
) ∂K(x1, x)∂x
dx1+
+2β0kwk
t∫
t−τ
wk(x, s)ds + β0kτw2k(x, t)− β0k
t∫
t−τ
w2k(x, s)ds
dx + I + J . (12)
We assume K(x1, x) = K(x, x1) ≥ 0, change the order of integration, usecondition (9), and integrate by parts
n∑
k=1
n∑
i=1
a2k∫
a2k−1
dx
a2i∫
a2i−1
wk(x, t)wi(x1, t)∂K(x1, x)
∂xdx1 =
=n∑
k=1
n∑
i=1
a2i∫
a2i−1
dx1
a2k∫
a2k−1
wk(x, t)wi(x1, t)∂K(x1, x)
∂xdx =
186 Petr A. Velmisov, Andrey V. Ankilov
= −n∑
k=1
n∑
i=1
a2i∫
a2i−1
dx1
a2k∫
a2k−1
w′k(x, t)wi(x1, t)K(x1, x)dx =
= −n∑
k=1
n∑
i=1
a2i∫
a2i−1
dx
a2k∫
a2k−1
w′k(x1, t)wi(x, t)K(x, x1)dx1 =
= −n∑
i=1
n∑
k=1
a2k∫
a2k−1
dx
a2i∫
a2i−1
w′i(x1, t)wk(x, t)K(x1, x)dx1 =
= −n∑
i=1
n∑
k=1
a2k∫
a2k−1
dx
a2i∫
a2i−1
wk(x, t)w′i(x1, t)K(x1, x)dx1 =
= −n∑
k=1
n∑
i=1
a2k∫
a2k−1
dx
a2i∫
a2i−1
wk(x, t)w′i(x1, t)K(x1, x)dx1.
Similarly we get
n∑
k=1
n∑
i=1
a2k∫
a2k−1
dx
a2i∫
a2i−1
wk(x, t)w′i(x1, t)∂K(x1, x)
∂xdx1 =
= −n∑
k=1
n∑
i=1
a2k∫
a2k−1
dx
a2i∫
a2i−1
w′k(x, t)w′i(x1, t)K(x1, x) dx1.
With 2ab ≤ a2 + b2, we have 2wk(x, t)wk(x, s) ≤ w2k(x, t) + w2
k(x, s).By substituting these estimates in (12), we finally find
Φ(t) ≤ −2n∑
k=1
a2k∫
a2k−1
(β1kw
2k + β2kw
′′k
2 − β0kτw2k
)dx−
−2ρ
π
n∑
k=1
a2k∫
a2k−1
wk(x, t)
n∑
i=1
a2i∫
a2i−1
wi(x1, t)K(x1, x) dx1
dx+ (13)
+2ρV 2
π
n∑
k=1
a2k∫
a2k−1
w′k(x, t)
n∑
i=1
a2i∫
a2i−1
w′i(x1, t)K(x1, x) dx1
dx + I + J .
Stability of elastic elements of wing profile with time delay of bases reactions 187
We transform integral I(t), (in the hypothesis K(x1, x) = K(x, x1))
I(t) =d
dt
ρ
π
n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
wi(x, t)wj(x1, t)K(x1, x) dx1 =
=ρ
π
n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
wi(x, t)wj(x1, t)K(x1, x) dx1+
+ρ
π
n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
wi(x, t)wj(x1, t)K(x1, x) dx1 =
=2ρ
π
n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
wi(x, t)wj(x1, t)K(x1, x) dx1.
Similarly
J(t) = −2ρV 2
π
n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
w′i(x, t)w′j(x1, t)K(x1, x) dx1,
and by substituting the obtained forms of I(t), J(t) in (13), we get
Φ(t) ≤ −2n∑
k=1
a2k∫
a2k−1
(β1kw
2k + β2kw
′′k
2 − β0kτw2k
)dx. (14)
Let us consider the boundary problem for equation
ψIV (x) = µψ(x), x ∈ [a2k−1, a2k], k = 1, ..., n
with boundary conditions (9) [7, 8]. This problem is self-adjoint and com-pletely defined. In fact, by integrating by parts, it is not difficult to showthat
a2k∫
a2k−1
u(x)vIV (x) dx =
a2k∫
a2k−1
v(x)uIV (x) dx,
a2k∫
a2k−1
u(x)uIV (x) dx > 0,
188 Petr A. Velmisov, Andrey V. Ankilov
for any functions u (x) and v (x), satisfying considered boundary conditionsand having on [a2k−1, a2k] continuous derivations to the fourth order.
The Raley inequality for the function wk(x, t) isa2k∫
a2k−1
wk(x, t)wIVk (x, t)dx ≥ µ1k
a2k∫
a2k−1
wk(x, t)wk(x, t)dx,
where µ1k is the least eigenvalue of the considered boundary problem. Byintegrating by parts, we obtain
a2k∫
a2k−1
w′′k2(x, t)dx ≥ µ1k
a2k∫
a2k−1
w2k(x, t)dx. (15)
With (15), the inequality (14) takes the following form
Φ(t) ≤ −n∑
k=1
2µ1k
a2k∫
a2k−1
(β1k + µ1kβ2k − β0kτ)w′′k2dx. (16)
If the conditionβ0kτ − β1k − µ1kβ2k ≤ 0, (17)
is fulfilled then Φ(t) ≤ 0. Integrating from 0 to t, we get
Φ(t) ≤ Φ(0). (18)
According to (10)
Φ(0) =n∑
k=1
a2k∫
a2k−1
Mkw
2k(x, 0) + Dkw
′′k2(x, 0)−Nkw
′k2(x, 0) + β0kw
2k(x, 0)
dx+
+ρ
π
n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
wi(x, 0)wj(x1, 0)K(x1, x) dx1− (19)
−ρV 2
π
n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
w′i(x, 0)w′j(x1, 0)K(x1, x) dx1.
On the grounds of an earlier proved theorem [5] it is possible to show
n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
wi(x, t)wj(x1, t)K(x1, x) dx1 ≥ 0,
Stability of elastic elements of wing profile with time delay of bases reactions 189
n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
w′i(x, t)w′j(x1, t)K(x1, x) dx1 ≥ 0. (20)
We estimate the iterated integral, by using inequality 2ab ≤ a2 + b2 andsymmetry of the kernel K(x1, x), as follows
n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
wi(x, 0)wj(x1, 0)K(x, x1)dx1 ≤
≤n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
w2i (x, 0)K(x, x1)dx1. (21)
From (20) and (21) we get the following
Φ(t) ≤n∑
k=1
a2k∫
a2k−1
Mk w0k
2 + Dkw0k′′2 −Nkw0k
′2 + β0kw0k2
dx+
+ρ
π
n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
w2i (x, 0)K(x, x1)dx1,
where w0k = wk(x, 0), w′′0k = w′′k(x, 0), w0k = wk(x, 0), w′0k = w′k(x, 0).Let us denote
Ki = supx∈(a2i−1,a2i)
K1(x), K1(x) =n∑
j=1
a2j∫
a2j−1
K(x1, x)dx1, (22)
then
Φ(t) ≤n∑
k=1
a2k∫
a2k−1
(Mk +
ρKk
π
)w0k
2 + Dkw0k′′2 −Nkw
′0k
2 + β0kw20k
dx.
(23)On the other hand,
Φ(t) ≥n∑
k=1
a2k∫
a2k−1
[Mk wk
2 (x, t) + Dkw′′k2(x, t)−Nkwk
′2(x, t) + β0kw2(x, t)
]dx+
190 Petr A. Velmisov, Andrey V. Ankilov
+ρ
π
n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
wi(x, t)wj(x1, t)K(x1, x) dx1− (24)
−ρV 2
π
n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
w′i(x, t)w′j(x1, t)K(x1, x) dx1.
We estimate the iterated integral, using inequality −2ab ≥ − (a2 + b2
)and
symmetry of the kernel K (τ , x)
−n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
w′i(x, t)w′j(x1, t)K(x, x1)dx1 ≥
≥ −n∑
i=1
n∑
j=1
a2i∫
a2i−1
dx
a2j∫
a2j−1
w′2i (x, t)K(x, x1)dx1. (25)
Estimations (20), (25) allow to write
Φ(t) ≥n∑
k=1
a2k∫
a2k−1
[Mk wk
2 (x, t) + Dkw′′k2(x, t)−Nkw
′k2(x, t)+
+β0kw2k(x, t)
]dx−
n∑
i=1
ρV 2Ki
π
a2i∫
a2i−1
w′i2(x, t)dx. (26)
Thereby, from (23) and (26) we get the following
n∑
k=1
a2k∫
a2k−1
[Mk wk
2 (x, t) + Dkw′′k2(x, t) + β0kw
2k(x, t)
]dx−
−n∑
k=1
(Nk +
ρV 2Kk
π
)w′k
2(x, t)dx ≤ (27)
≤n∑
k=1
a2k∫
a2k−1
(Mk +
ρKk
π
)w0k
2 + Dkw′′0k
2 −Nkw′0k
2 + β0kw0k2
dx.
Consider the boundary problem for equation
ψIV (x) = −λψ′′(x), x ∈ [a2k−1, a2k]
Stability of elastic elements of wing profile with time delay of bases reactions 191
with boundary conditions (9) [7, 8]. This problem is self-adjoint and com-pletely defined. In fact, by integrating by parts, it is not difficult to showthat
a2k∫
a2k−1
u(x)vIV (x) dx =
a2k∫
a2k−1
v(x)uIV (x) dx,
a2k∫
a2k−1
u(x)v′′(x) dx =
a2k∫
a2k−1
v(x)u′′(x) dx,
a2k∫
a2k−1
u(x)uIV (x) dx > 0, −a2k∫
a2k−1
u(x)u′′(x) dx > 0,
for any functions u(x) and v(x), satisfying considered boundary conditionsand having continuous derivatives up to the fourth order on [a2k−1, a2k]. Wewrite the Raley inequality for the function wk(x, t)
a2k∫
a2k−1
wk(x, t)wIVk (x, t)dx ≥ −λ1k
a2k∫
a2k−1
wk(x, t)w′′k(x, t)dx,
where λ1k is the least eigenvalue of the considered boundary problem. Byintegrating by parts, we write this inequality in the form
a2k∫
a2k−1
w′′k2(x, t)dx ≥ λ1k
a2k∫
a2k−1
w′k2(x, t)dx. (28)
By using Bunyakowsky inequality, we have
w2k(x, t) ≤ (a2k − a2k−1)
a2k∫
a2k−1
w′k2(x, t) dx. (29)
Let us assume that
Nk < λ1kDk − ρKkV2
π. (30)
With inequalities (28), (29), we estimate the left hand side of (27) as follows
n∑
k=1
a2k∫
a2k−1
[Mk wk
2 (x, t) + Dkw′′k2(x, t) + β0kw
2k(x, t)
]dx−
−n∑
k=1
a2k∫
a2k−1
(Nk +
ρV 2Kk
π
)w′k
2(x, t)dx ≥
192 Petr A. Velmisov, Andrey V. Ankilov
≥n∑
k=1
(λ1kDk −Nk − ρV 2Kk
π
)w2
k(x, t)a2k − a2k−1
.
Taking into account this estimates and (28), we get the inequality
n∑
k=1
(λ1kDk −Nk − ρV 2Kk
π
)w2
k(x, t)a2k − a2k−1
≤
≤n∑
k=1
a2k∫
a2k−1
(Mk +
ρKk
π
)w0k
2 +(Dk − λ−1
1k Nk
)w0k
′′2 + β0kw0k2
dx,
from where the following theorem.
Theorem 2.1. Let us assume that functions wk(x, t), k = 1, ..., n, satisfy theboundary conditions (9) and let inequalities (17) and (30) be satisfied. Thenthe solutions wk(x, t), k = 1, ..., n, of equations system (8) are stable withrespect to perturbations of the initial values of wk(x, 0), wk(x, 0), w′′k(x, 0).
References[1] P. A. Vel’misov, Asymptotic Equations of Gas Dynamics, Saratov University, 1986.
[2] P. A. Vel’misov, Yu. A. Reshetnikov, Stability of Viscoelastic Plates at Aerohydrody-namic Affect, Saratov University, 1994.
[3] A. V. Ankilov, P. A. Vel’misov, Stability of Viscoelastic Elements of Thin-shelled Con-structions Under Aerohydrodynamic Action, VINITI, Moscow, N2522, 1998.
[4] A. V. Ankilov, P. A. Vel’misov, Stability of viscoelastic elements of channels walls,Ulyanovsk Technical University, 2000.
[5] A. V. Ankilov, P. A. Vel’misov and N. A. Degtyareva, Stability of elastic elementsof wing profile, Applied Mathematics and mechanics, Ulyanovsk Technical University,7(2007), 9-18.
[6] M. A. Lavrentev, B. V. Shabat, Methods of Complex Variable Functions Theory Science,Moscow, 1987.
[7] F. D. Gahov, Boundary Problems, Science, Moscow, 1977.
[8] L. Kollatz, Eigenvalue Problems, Science, Moscow, 1968.
