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Kuwait University Kuwait FoundationDepartment of Mathematics & for the Advancement
Computer Science of Sciences (KFAS)
PROCEEDINGS OF THE
INTERNATIONAL CONFERENCE
ON MATHEMATICS AND ITS
APPLICATIONS
(ICMA 2004)
April 5-7, 2004
State of Kuwait
ORGANIZING COMMITTEE
Dr. Mansour A. Al-Zanaidi, Chairman
Prof. Bader N. Al-Saqabi, Treasurer Prof. Ali Al-Zamel
Prof. Adnan H. Al-Aqeel Prof. Shyam L. Kalla
Prof. Reyadh R. Khazal Prof. Man M. Chawla, Secretary
ii
FOREWORD
This volume contains papers and invited talks presented at the “International
Conference on Mathematics and Its Applications” (ICMA04-Kuwait), which was
held at Kuwait University from April 5 to 7, 2004. All papers contained in this
volume are refereed/peer reviewed.
The Conference covered active research fields of the Department of Mathematics
and Computer Science:
(A) Computational Differential Equations and Linear Algebra.
(B) Integral Transforms and Special Functions with Fractional Calculus.
(C) Groups, Rings & Categories, and Differential Geometry.
The ICMA04-Kuwait was jointly sponsored and organized by the Department of
Mathematics and Computer Science, Kuwait University, and the Kuwait Foundation
for the Advancement of Sciences (KFAS). The Conference was attended by sixty
three participants from twenty two countries.
Keynote speakers at the conference were: G. V. Berghe (Belgium), S. Caenepeel
(Belgium), L. Debnath (USA), D. J. Evans (UK), V. D. Mazurov (Russia), A. J.
Scholl (UK) and B. Wegner (Germany).
The organizers would like to thank all the participants for their cooperation in
preparing their contributions for this Proceedings and their active interest in the
peer reviewing process for the volume. Thanks are also due to all colleagues who
helped in the reviewing process.
All papers in this volume are arranged alphabetically according to the name of
the first author.
We would like to take this opportunity to express our gratitude to the administra-
tion at Kuwait University and Kuwait Foundation for the Advancement of Sciences,
for sponsoring this Conference and publication of the Proceedings.
Finally we offer our sincere thanks to the faculty and staff of the Department of
Mathematics and Computer Science at Kuwait University for taking active interest,
shouldering several responsibilities during the conference and publication of these
proceedings.
Mansour A. Al-Zanaidi, Shyam L. Kalla and Man M. Chawla
Chairman Editors
Kuwait, December 2004.
iii
CONTENTS
Organizing Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
M. Al-Hajri and S. L. Kalla On An Integral Transform Involving Bessel Func-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
N. Alias, M. S. S. Mohamed, A. R. Abdullah PVM-Based Implementation
Of A New ADI Technique (IADEI) In Solving One Dimensional Parabolic Equation
Using PC Cluster System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
H. Alinejad, J. Mahmoodi, S. Sobhanian, M. Momeni A New Nonlinear
Differential Equation For Describing Ion-Acoustic Waves In Plasma . . . . . . . . . . . . . . 27
G. Alobaidi and R. Mallier Using Monte Carlo Methods To Evaluate Sub-
Optimal Exercise Policies For American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
A. Azizi Group Theory And The Capitulation Problem For Certain Biquadratic
Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
P. Balasubramaniam and A. V. A. Kumar Runge-Kutta Neural Networks For
Delay Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
V. Beniash-Kryvets On Rosenberger’s Conjecture For Generalized Triangle Groups
Of Types (2, 10, 2) and (2, 20, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
G. V. Berghe Exponential Fitting: General Approach And Applications For ODE-
Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A. Bounaım, S. Holm, W. Chen, A. Ødegard Mathematics In Medicine: Bioa-
coustic Modeling And Computations For An Ultrasonic Imaging Technique . . . . . . 92
L. Boyadjiev and R. Scherer On The Fractional Heat Equation . . . . . . . . . . . . . . 105
S. Caenepeel and E. De Groot Galois Corings Applied To Partial Galois The-
ory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
iv
S. Caenepeel and M. Iovanov Comodules Over Semiperfect Corings . . . . . . . . . . 135
D. K. Callebaut, G. K. Karugila and A. H. Khater Nonlinear Fourier Analysis
Of Systems Of Partial Differential Equations With Computer Algebra: Survey And
New Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
U. C. De and G. C. Ghosh On Quasi Einstein And Special Quasi Einstein
Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
L. Debnath Four Major Discoveries In Applied Mathematics During The Second
Half Of The Twentieth Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
D. J. Evans The Generalised Coupled Alternating Group Explicit (CAGE)
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
B. I. Golubov Dyadic Fractional Differentiation And Integration Of Walsh Trans-
form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
G. Heinig and K. Rost Fast “Split” Algorithms For Toeplitz And Toeplitz-Plus-
Hankel Matrices With Arbitrary Rank Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
M. N. M. Ibrahim Productivity Of Oil Wells In Arbitrarily Shaped Reservoirs 313
M. Lehn and R. Scherer Mathematical Modeling And Scientific Computing Ex-
emplary For Chemical Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
M. Lenard Weighted (0,2)-Interpolation With Additional Interpolatory Condi-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
V. D. Mazurov Groups With Prescribed Orders Of Elements . . . . . . . . . . . . . . . . . . . 343
P. Miskinis Integrable Nonlocal Burgers Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
S. Naik and S. Ponnusamy Univalency Of A Convolution Operator Concerning
Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
M. A. Pathan On Unified Elliptic-Type Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
M. Savsar Modeling Of An Unreliable Production Merging Process With Interme-
diate Storage Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
v
N. El Saadi, M. Adioui and O. Arino A Mathematical Analysis For An Aggre-
gation Model Of Phytoplankton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
S. M. El-Sayed Iterative Methods For Some Nonlinear Matrix Equations . . . . . . . 406
A. M. A. El-Sayed Fractional Calculus And Intermediate Physical Processes . . . 417
M. Sayed Coset Enumeration Algorithm For Symmetrically Generated Groups . . 424
S. V. Tikhonov and V. I. Yanchevskiı Pythagoras Numbers Of Function Fields
Of Genus Zero Curves Defined Over Hereditarily Pythagorean Fields . . . . . . . . . . . . . 438
B. Wegner Mathematics On Its Way Into Information Society . . . . . . . . . . . . . . . . . . 444
M. Wojtowicz Algebraic Structures Arising From Euler’s And Pythagorean Equa-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
M. A. Al-Zanaidi, C. Grossmann and A. Noack Implicit Taylor Methods For
Parabolic Equations With Jumps In Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
vi
ON AN INTEGRAL TRANSFORM INVOLVING BESSELFUNCTIONS
M. Al-Hajri and S. L. Kalla
Department of Mathematics & Computer Science
Kuwait university, P. O. Box. 5969, Safat 13060, KUWAIT
email: kalla@mcs.sci.kuniv.edu.kw
Abstract
This paper deals with a new integral transform, involving a combination of Bessel
functions as a kernel. The inversion formula is established and some properties
are given. This transform can be used to solve some mixed boundary value prob-
lems. We consider here a problem of heat conduction in an infinite and semi-infinite
cylinder a ≤ r ≤ b, with radiation-type boundary conditions.
AMS Subj. classification: 44A20, 35K20, 80A20.
Keywords: Integral transform, Bessel Functions, Differential equations, Boundary
conditions.
1. INTRODUCTION
Let f(t) be a given function defined on an interval [a, b], that belongs to a certain
class of functions. An integral transform of f(t) is a mapping of the form,
T [f(t); s] = f(s) =
∫ b
a
K(s, t)f(t)dt,
provided that the integral exists. K(s, t) is a prescribed function, called the ker-
nel of the transform [2,5,6,11]. Among the well known transforms are the Laplace,
Fourier, Hankel, Stieltjes and Mellin transforms. The most versatile of these, the
Laplace transform has been widely used to solve differential equations, and par-
ticularly problems related to heat transfer and electrical circuits. On the other
hand for problems in which there is an axial symmetry, the Hankel transforms are
found to be most appropriate. The Mellin transform being closely related to the
Fourier transform, has its own peculiar uses, as for deriving expansion and solving
problems with wedge shape boundaries. In general, the use of an integral transform
often reduces a partial differential equation in n independent variables to (n − 1)
variables, that provides a simplification of the problem.
1
The success of the use of integral transforms to solve boundary value problems
and to exclude a variable with range (0,∞) or (−∞,∞) led investigators to con-
sider finite integral transforms. Doetsch considered finite Fourier transforms, and
Sneddon [11] extended the idea of Bessel function kernel, called ’Finite Hankel
Transforms’. Using the Sturm-Liouville theory [3,], a number of integral transforms
can be implemented, according to the prescribed boundary conditions.
Recently Khajah [9] has considered a modified Hankel transform in the form,
Jµ[f(z); s, λ] =
∫ b
0
zλf(z)Jµ(zs)dz
where f(z) satisfies Dirichlet’s conditions on the interval [a, b]. He has derived
the inversion formula, Parseval-type identities, transform of derivatives, as well as
transforms of products of the form zλf(z).
Using the Sturm-Liouville theory Kalla and Villalobos [7,8] have defined and
studied an integral transform defined as,
T [f(x), a, b, ν; λi] = fν(λi) =
∫ b
a
xf(x)Cν(λix)dx,
where
Cν(λix) = {Yν(λia) + Bν(λib)}Jν(λix)
−{Jν(λia) + Aν(λib)}Yν(λix)
and
Aν(λx) = Jν(λx) + hλJ′ν(λx)
Bν(λx) = Yν(λx) + hλY′ν (λx)
and λi are the positive roots of equation,
Jν(λa)Bν(λb) − Yν(λa)Aν(λb) = 0
This transform has been used to solve a heat conduction problem in an infinite
cylinder bounded by the surface r = a, r = b (b > a).
In this paper we define and study a new integral transform involving Bessel
functions of first and second kind, by invoking the Sturm-Liouville theory. Inversion
formula is established and some properties are mentioned. The transform has been
2
used to solve a heat conduction problem in an infinite and a semi- infinite circular
cylinder, bounded by surfaces r = a and r = b (b > a), with radiation-type boundary
conditions on both surfaces.
2. DEFINITION AND INVERSION FORMULA
Consider Bessel’s differential equation
x2y′′
+ xy′+ (λ2x2 − ν2)y = 0 , x ∈ [a, b] (1)
with homogeneous boundary conditions:
y(a) + h1y′(a) = y(b) + h2y
′(b) = 0 (2)
The general solution of (1) is given by:
y(x) = c1Jν(λx) + c2Yν(λx) (3)
where c1 , c2 are arbitrary constants, and Jν(x), Yν(x) are the Bessel functions of first
and second kind respectively. To obtain a solution of (1) that satisfies conditions
(2), we have
c1
[Jν(λa) + h1λJ
′ν(λa)
]+ c2
[Yν(λa) + h1λY
′ν (λa)
]= 0 (4)
c1
[Jν(λb) + h2λJ
′ν(λb)
]+ c2
[Yν(λb) + h2λY
′ν (λb)
]= 0 (5)
from which we deduce
c1
c2
= −Yν(λa) + h1λY′ν (λa)
Jν(λa) + h1λJ ′ν(λa)
= −Yν(λb) + h2λY′ν (λb)
Jν(λb) + h2λJ ′ν(λb)
(6)
Let
Aν(λx, hk) = Jν(λx) + hkλJ′ν(λx) , k = 1, 2
Bν(λx, hk) = Yν(λx) + hkλY′ν (λx) , k = 1, 2
Then, the function given by (3) is a solution of equation (1), subject to the conditions
(2), if λ is a root of the transcendental equation,
Bν(λa, h1)Aν(λb, h2) − Aν(λa, h1)Bν(λb, h2) = 0 (7)
Henceforth, we take λi (i = 1, 2, . . .) to be the positive roots of equation (7). Then,
from (4–5), we have
yi(x) =c1
Bν(λia, h1)[Jν(λix)Bν(λia, h1) − Aν(λia, h1)Yν(λix)] (8)
=c1
Bν(λib, h2)[Jν(λix)Bν(λib, h2) − Aν(λib, h2)Yν(λix)] (9)
3
If we define
Zi = Bν(λia, h1) + Bν(λib, h2) , Wi = Aν(λia, h1) + Aν(λib, h2)
then the following functions are taken to be solutions of (1–2):
Mν(λix) = Zi Jν(λix) − Wi Yν(λix) (10)
By Sturm-Liouville theory [3], the functions of the system (10) are orthogonal on
the interval [a, b] with weight function x, that is
∫ b
a
xMν(λix) Mν(λjx)dx =
0 , i �= j
Mν(λi) , i = j
(11)
where Mν(λi) = ‖√xMν(λix)‖22 — the weighted L2 norm. If a function f(x) and
its first derivative are piecewise continuous on the interval [a, b], then the relation
T [f(x), a, b, ν; λi] = fν(λi) =
∫ b
a
xf(x)Mν(λix)dx (12)
defines a linear integral transform. To derive the inversion formula for this transform,
given the series expansion,
f(x) =∞∑i=1
aiMν(λix) (13)
we multiply (13) by xMν(λjx) and integrate both sides with respect to x to get the
coefficients:
ai =1
Mν(λi)
∫ b
a
x f(x)Mν(λix) dx =fν(λi)
Mν(λi), i = 1, 2, . . . (14)
and the inversion formula becomes
f(x) =∞∑i=1
fν(λi)
Mν(λi)Mν(λix) (15)
Using some well known properties of Bessel functions [12] we can easily derive the
following relation:
2Mν(λi) = Z2i
[b2P (λi, b, ν) − a2P (λi, a, ν)
] − 2ZiWi
[b2Q(λi, b, ν) − a2Q(λi, a, ν)
]
+W2i
[b2R(λi, b, ν) − a2R(λi, a, ν)
](16)
4
in which
P (λi, µ, ν) = J2ν (λiµ) − Jν−1(λiµ)Jν+1(λiµ)
R(λi, µ, ν) = Y 2ν (λiµ) − Yν−1(λiµ)Yν+1(λiµ)
and
Q(λi, µ, ν) = J′ν(λiµ)Yν−1(λiµ) − 1
λiµJν−1(λiµ)Yν(λiµ) − J
′ν−1(λiµ)Yν(λiµ)
and µ stands for a or b . It is not difficult to verify some properties of the transform
from definition (12). For example,
T [αf(x) + βg(x), a, b, ν; λi] = αf(λi) + βg(λi) (17)
T [f(px), a, b, ν; λi] =
∫ b
a
xf(px)Mν(λix)dx =1
p2T [f(x), pa, pb, ν; λi/p ] (18)
3. TRANSFORM OF A DIFFERENTIAL OPERATOR
We derive the transform of the following operator
Df(x) =d2
dx2f(x) +
1
x
d
dxf(x) − ν2
x2f(x) , a ≤ x ≤ b (19)
Let I be the transform of the first two terms of D , that is
I =
∫ b
a
x[f ′′(x)+
1
xf ′(x)
]Mν(λix) dx =
∫ b
a
xf ′′(x)Mν(λix)dx+
∫ b
a
f ′(x)Mν(λix)dx
Integration by parts of the first integral leads to,
∫ b
a
xf ′′(x)Mν(λix)dx = xMν(λix)f ′(x)∣∣∣ba−
∫ b
a
f ′(x)[xλiM
′ν(λix) + Mν(λix)
]dx,
and hence,
I = xMν(λix)f ′(x)∣∣∣ba− λi
∫ b
a
xf ′(x) M ′ν(λix) dx
Integrating by parts once again leads to,
I = x [f ′(x)Mν(λix)−λif(x)M ′ν(λix)]
∣∣∣ba+
∫ b
a
x−1[λ2
i x2M ′′
ν (λix)+λixM ′ν(λix)
]f(x) dx
Since Mν satisfies (1)we have
λ2i x
2M ′′ν (λix) + λixM ′
ν(λix) = (ν2 − λ2i x
2) Mν(λix)
5
and∫ b
a
x−1[λ2
i x2M ′′
ν (λix) + λixM ′ν(λix)
]f(x) dx =
∫ b
a
x[ν2
x2− λ2
]f(x) Mν(λix) dx
Furthermore, it follows from the boundary conditions (2) that
λiM′ν(λia) =
1
h1
Mν(λia) , λiM′ν(λib) =
1
h2
Mν(λib)
Hence
I =b
h2
Mν(λib)[f(b)+h2f
′(b)]− a
h1
Mν(λia)[f(a)+h1f
′(a)]−λ2
i f(λi)+T
[ν2
x2f(x)
]
and the transform of the operator D in (19) becomes
T [Df(x)] =b
h2
Mν(λib)[f(b) + h2f
′(b)] − a
h1
Mν(λia)[f(a) + h1f
′(a)] − λ2
i f(λi)
(20)
Transform of xν
From definition (12) we have
T [xν , a, b, ν; λi] =
∫ b
a
xν+1Mν(λix) dx
Using a result of [12], namely∫xρ+1Zρ(x)dx = xρ+1Zρ+1(x)
where Zρ(x) is any of the Bessel functions, we obtain
T [xν , a, b, ν; λi] =1
λi
[bν+1 Mν+1(λib) − aν+1 Mν+1(λia)
]
Since
Mν+1(cz) =ν
czMν(cz) − M ′
ν(cz)
the transform becomes,
T [xν , a, b, ν; λi] =bν+1
λi
[ν
λibMν(λib) − M ′
ν(λib)
]− aν+1
λi
[ν
λiaMν(λia) − M ′
ν(λia)
]
Then, from the boundary conditions (2) this reduces to,
T [xν , a, b, ν; λi] =bν+1
λ2i
[ν
b+
1
h2
]Mν(λib) − aν+1
λ2i
[ν
a+
1
h1
]Mν(λia) (21)
6
In particular, the transform of a constant (where ν = 0) is found to be
T [c, a, b, 0; λi] =c
λ2i
[b
h2
M0(λib) − a
h1
M0(λia)
]= c T [1, a, b, 0; λi] (22)
4. HEAT CONDUCTION IN AN INFINITE CYLINDER
Consider a long hollow cylinder of inner radius a and outer radius b , with radiation
type boundary conditions in both outer and inner surface, and a prescribed initial
temperature. The differential equation of the phenomena is:
1
K
∂U
∂t=
∂2U
∂r2+
1
r
∂U
∂r, (23)
where a < r < b , t > 0 and U(r, t) denotes the temperature at any radial position r
at time t; K is a constant that depends on the material of the cylinder. The initial
and boundary conditions are as follows:
U(r, 0) = I(r) , a < r < b (24)
U + h1∂U
∂r
∣∣∣∣r=a
= f(t) , U + h2∂U
∂r
∣∣∣∣r=b
= g(t) , t > 0 (25)
Taking ν = 0 , we consider the transform of U with respect to the radial variable,
that is
U(λi, t) =
∫ b
a
r U(r, t) M0(λir) dr (26)
Referring to (20) and (23), we obtain
1
K
∂U
∂t=
b
h2
M0(λib)
[U + h2
∂U
∂r
]r=b
− a
h1
M0(λia)
[U + h1
∂U
∂r
]r=a
− λ2i U
From the boundary conditions (25) we have this reduced to
1
K
∂U
∂t=
b
h2
M0(λib) g(t) − a
h1
M0(λia) f(t) − λ2i U
and the following ODE is obtained
∂U
∂t+ Kλ2
i U = K
[b
h2
M0(λib) g(t) − a
h1
M0(λia) f(t)
](27)
whose solution is given by
U(λi, t) = exp(−Kλ2i t)
[K
∫ t
0
exp(Kλ2i s)
[ b
h2
M0(λib) g(s) − a
h1
M0(λia) f(s)]ds + C
]
7
Taking the transform of the initial condition (24), namely
U(λi, 0) = I(λi)
leads to C = I(λi) , hence
U s(λi; t) = exp(−Kλ2i t)[
K
∫ t
0
exp(−Kλs[ b
h2
M0(λib) g(s) − a
h1
M0(λia) f(s)]ds + I(λi)
](28)
The solution of (23) follows after applying the inversion formula to the above,
thus
U(r, t) =∞∑i=1
U(λi, t)
Mν(λi)M0(λir) (29)
with the understanding that the summation is taken over all the positive roots of
the equation:
B0(λa, h1)A0(λb, h2) − A0(λa, h1)B0(λb, h2) = 0 (30)
Special cases:
(i) Let us consider the previous problem with the following initial and boundary
conditions;
U(r, 0) = 0 , a < r < b (31)
U + h1∂U
∂r
∣∣∣∣r=a
= U0 (constant), U + h2∂U
∂r
∣∣∣∣r=b
= U1(constant), t > 0
(32)
Then equation (28) will become
U(λi; t) = exp(−Kλ2i t)[
K
∫ t
0
exp(Kλ2i x)
{b
h2
M0(λib)U1 − a
h1
M0(λia)U0
}dx
](33)
=1 − exp(−Kλ2
i t)
λ2i
[bU1
h2
M0(λib) − aU0
h1
M0(λia)
]
And according to (29) the solution will become
U(r, t) =∞∑i=1
1 − exp(−Kλ2i t)
λ2iMν(λi)
[bU1
h2
M0(λib) − aU0
h1
M0(λia)
]M0(λir), i = 1, 2, 3, ...
(34)
where the summation is taken over all positive roots of (30).
8
(ii) If h1 −→ 0 in (25), that is U |r=a = f(t) , our result (29) reduces to a result of
Kalla and Villalobos [8, p.41, eq.(20].
5. HEAT CONDUCTION IN A SEMI-INFINITE CYLINDER
Let us consider the problem of finding the temperature distribution U(r, z, t) in a
hollow semi-infinite cylinder with outer radius b and inner radius a. This problem
is expressed by the differential equation,
1
K
∂U
∂t=
∂2U
∂r2+
1
r
∂U
∂r+
∂2U
∂z2(35)
where a < r < b ; z , t > 0 , and the initial/boundary conditions are taken to be:
U(r, z, 0) = I(r, z) (36)
U(r, 0, t) = V (r, t) , limz→∞
U(r, z, t) = 0 (37)
U + h1∂U
∂r
∣∣∣∣r=a
= f(z, t) , U + h2∂U
∂r
∣∣∣∣r=b
= g(z, t) (38)
Consider the integral transform,
U(λi, z, t) =
∫ b
a
r U(r, z, t)M0(λir) dr (39)
and the Fourier sine transform
U s(λi, p, t) =
∫ ∞
0
U(λi, z, t) sin(p z) dz (40)
Following a similar argument as in the previous section, the transformed equation
of (35) becomes
1
K
∂U
∂t=
b
h2
M0(λib)g(z, t) − a
h1
M0(λia)f(z, t) − λ2i U +
∂2U
∂z2
whose Fourier sine transform is found to be,
1
K
dU s
dt=
b
h2
M0(λib)gs(p, t) − a
h1
M0(λia)fs(p, t) − λ2i U s − p2U s + pV (λi, t)
and which is expressed as,dU s
dt+ K(λ2
i + p2)U s
= K
[b
h2
M0(λib)gs(z, t) − a
h1
M0(λia)fs(z, t) + pV (λi, t)
](41)
9
Now we have the solution of the above ODE given by
U s(λi, p, t) = e−K(λ2i +p2)t
[K
∫ t
0
eK(λ2i +p2) τ
[ b
h2
M0(λib)gs(z, τ)
− a
h1
M0(λia)fs(z, τ) + pV (λi, τ)]dτ + C
]
and from the initial condition, we have C = Is(λi, p) , so that
U s(λi, p, t) = e−K(λ2i +p2)t
[K
∫ t
0
eK(λ2i +p2) τ
[b
h2
M0(λib)gs(z, τ)
− a
h1
M0(λia)fs(z, τ) + pV (λi, τ)]dτ + Is(λi, p)
](42)
Finally, taking respective inverse transforms will lead to the solution:
U(r, z, t) =2
π
∞∑i=1
1
Mν(λi)
[∫ ∞
0
U s(λi, p, t) sin(pt) dp
]M0(λir). (43)
Special cases:
(i) We consider the previous partial differential equation with the following con-
ditions
(U + h1
∂U
∂r
)r=a
= 0, z > 0, t > 0
(U + h2
∂U
∂r
)r=b
=1
z, z > 0, t > 0
U(r, 0, t) = U0, (constant) a < r < b, t > 0
U(r, z, 0) = 0 a < r < b, z > 0
U(r, z, t) → 0 as z → ∞
then (42) will become
Us(λi, p, t) = e−K(λ2i +p2)t
K
t∫0
e−K(λ2i +p2)x
{b
h2
M0(λib)π
2
+pU0
λ2i
[b
h2
M0(λib) − a
h1
M0(λia)
]}dx
]
=1 − e−K(λ2
i +p2)t
λ2i + p2
{p
h2
[π
2+
PU0
λ2i
]M0(λib) − aPU0
λ2i h1
M0(λia)
}
10
and according to (43)
U(r, z, t) =∞∑i=1
2
π
∞∫0
1 − e−K(λ2i +p2)t
λ2i + p2
{p
h2
[π
2+
PU0
λ2i
]M0(λib) − aPU0
λ2i h1
M0(λia)
}
sin(pt)dz} × M0(λir)
Mν(λi).
(ii) For h1 −→ 0 in (38), that U |r=a = f(z, t) , our result (43) reduces to one
considered in [8, p.43; eq. (36)].
6. CONCLUSIONS
Here we have introduced a new finite integral transform (Hankel-type) involving
product of Bessel functions as the Kernel. This transform can be used to solve
certain class of mixed boundary value problems. As indicated in previous sections
4 and 5, this transform is suitable to solve heat conduction problems in hollow
cylinders with radiation (mixed) conditions on both surfaces (r = a and r = b).
The Hankel-type finite transforms considered earlier in [5, 7, 8, 11] were able to
solve problems with both surfaces of the cylinder kept at prescribed temperature or
radiation condition on only one surface, r = b [8].
Numerical treatment of the results obtained here can be done by using “Matem-
atica” and [S. L. Kalla and S. Conde: Tables of Bessel Functions and Roots of
Transcendental Equations, Univ. Zulia, Venezuela, 1978].
ACKNOWLEDGEMENT
The authors are thankful to the referee for some useful comments and suggestions
for a better presentation of this paper.
REFERENCES
[1] Ali, I. and Kalla, S.L., A generalized Hankel transform and its use for solving
certain partial differential equations. Jour. Australian Math. Soc. 41 B
(1999), 105-117.
[2] Andrews, L. C. and Shivamoggi, B. K., Integral Transforms for Engineers,
SPIE, Bellingham, Washington, (1999).
11
[3] Birkhoff, G. and Rota, G. C., Ordinary Differential Equations, Wiley, New
York, (1989).
[4] Churchill, R. V., Operational Mathematics, McGraw-Hill, New York, (1972).
[5] Debnath, L. Integral Transforms and their Applications. CRC Press, Boca
Raton, FL. , (1995).
[6] A. Erdelyi, (Ed.), Tables of Integral Transforms, Vol. 1 &2., McGraw-Hill,
New York. (1954).
[7] Kalla, S.L. and Villalobos, A., On a new integral transform I., Jnanabha, 9-10
(1980), 149-154.
[8] Kalla, S. L. and Villalobos, A., On a new integral transform II., Rev. Tec.
Ing. , Univ. Zulia, 5 (1982), 40-44.
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[10] Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Series and Integrals,
Gordon and Breach, New York.,(1993).
[11] Sneddon, I. N. The Use of Integral Transforms, McGraw-Hill, New Yourk,
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12
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