59
CHAPTER 3 CONVOLUTIONS CONNECTED WITH SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS In contrast to the differentiation operator playing a basic role in ana- lysis, the linear differential operators of the second order are of most essential importance in mathematical physics. Here the classical Sturm-Liou- ville boundary value problem should be mentioned. But in some modern pro- blems the local boundary conditions, as those in the Sturm-Liouville pro- blem, are inadequate and the need of a general treatment of non-local boundary value conditions arises. In this chapter an attempt at extending the convolutional approach to boundary value problems connected with second-order linear differential operators is made. Weare trying to embrace both local anq non-local boundary value conditions. A number of new convolutions is proposed. By specialization, convolutions for some commonly used finite integral trans- formations are found in explicit form, e. g. for Sturm-Liouville an(i for Han- kel finite integral transformations. By forming tensor products of such con- volutions, new Duhamel-type representations of some problems of mathe- matical physics can be found. It is worth mentioning that such Duhamel re- presentations can be used not only for theoretical purposes, but as an al- ternative of the finite difference methods in numerical calculation of the solutions of many well-known problems of mathematical physics. 3.1. CONVOLUTIONS OF RIGHT INVERSE OPERATORS OF THE SQUARE OF THE DIFFERENTIATION The square of the differentiation plays much the same role in the theory of linear differential operators of the second order, as the differen- tiation operator in that of the linear differential operators of the first or- der. But here the sutuation is far more complicated. One reason for such complexity can be seen in the fact that whereas an arbitrary right inverse operator of the differentiation operator can be expressed by a single linear functional, for expressing an arbitrary right inverse operator of the square of the differentiation, two such linear functionals are needed. We still can- not propose convolutions for such general right inverses in expliCit form. But most of the known boundary value problems of mathematical physics can be embraced into a scheme, using only one arbitrary linear functional. 118 I. H. Dimovski, Convolutional Calculus © I. H. Dimovski 1990

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Page 1: Convolutional Calculus || Convolutions Connected with Second-Order Linear Differential Operators

CHAPTER 3

CONVOLUTIONS CONNECTED WITH SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS

In contrast to the differentiation operator playing a basic role in ana­lysis, the linear differential operators of the second order are of most essential importance in mathematical physics. Here the classical Sturm-Liou­ville boundary value problem should be mentioned. But in some modern pro­blems the local boundary conditions, as those in the Sturm-Liouville pro­blem, are inadequate and the need of a general treatment of non-local boundary value conditions arises.

In this chapter an attempt at extending the convolutional approach to boundary value problems connected with second-order linear differential operators is made. Weare trying to embrace both local anq non-local boundary value conditions. A number of new convolutions is proposed. By specialization, convolutions for some commonly used finite integral trans­formations are found in explicit form, e. g. for Sturm-Liouville an(i for Han­kel finite integral transformations. By forming tensor products of such con­volutions, new Duhamel-type representations of some problems of mathe­matical physics can be found. It is worth mentioning that such Duhamel re­presentations can be used not only for theoretical purposes, but as an al­ternative of the finite difference methods in numerical calculation of the solutions of many well-known problems of mathematical physics.

3.1. CONVOLUTIONS OF RIGHT INVERSE OPERATORS OF THE SQUARE OF THE DIFFERENTIATION

The square of the differentiation plays much the same role in the theory of linear differential operators of the second order, as the differen­tiation operator in that of the linear differential operators of the first or­der. But here the sutuation is far more complicated. One reason for such complexity can be seen in the fact that whereas an arbitrary right inverse operator of the differentiation operator can be expressed by a single linear functional, for expressing an arbitrary right inverse operator of the square of the differentiation, two such linear functionals are needed. We still can­not propose convolutions for such general right inverses in expliCit form. But most of the known boundary value problems of mathematical physics can be embraced into a scheme, using only one arbitrary linear functional.

118

I. H. Dimovski, Convolutional Calculus© I. H. Dimovski 1990

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CONVOLUTIONS CONNECTED WITH SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS 119

The right inverses of the square of the differentiation for which we can propose convolutions in explicit form are determined by one Sturm-Liou­ville boundary value condition and one quite general boundary value condi­tion, depending on an arbitrary linear functional.

3.1.1. A convolution connected witft the square of the differentia­tion and depending on an arbitrary linear functional. Here we consider a segment [0, T] and the space <{l([0, TD of the complex-valued continuous functions on [0, T]. For a unification, let us take T= 1, and later on to consider only the space <{l = <{l([0, 1 D.

The 0 rem 3.1.1. Let (p.: <{l -C be a non-zero linear lunctional on <{l. Then the operation

(1)

with (I * g)(t) = ~Ak(x,t)},

x x

k (x t) = {- ji(x+ t-r)g(r)dr-{- jIOx-t-r I)g(1 r I) sgn «x-t-r)r)dr ,t -t

is a bilinear, commutative and associative operation in <(l. It is sepa­rately continuous and annihilators-Ilee too.

Proof. If we introduce the odd continuations lo(t) = /(1 t I) sgn t and go(t) = g(1 t I) sgn t of I(t) and get), the expression k(x, t) under the function­al sign can be written in a simpler form:

x x

(2) k(x, t)= + !I(x+t-r)g(r)dr-{- !Io(x-t-r)go(r)dr. t -t

The bilinearity and commutativity of (1) are evident. Before proving the associativity of (1), let us prove that it is separately continuous. But this is evident from the inequality

(3) II/*glle<1I ~1I.lI/lIe .lIgllc. where II lie denotes the supremum-norm in <{l, and II ~ 11- the norm of ~ in <(l. But for the proof of the associativity of (1) we shall need an esti­mate of the form

(4) 1

in <(l([0, 1]), where 11/111 denotes the integral norm 11/111 = jl/(t)ldt. Here o

we drop the simple and routine calculation leading to (4). Now l'et us prove the associativity of (1). First we verify the associa­

tivity relation for functions of the form I,.(t) = sin It/). with ).=f0. We have

(5) (I,. * I,.)(t) = E(~)fl~~:=~(J.)f,.(t) '.

if ).2=f,u2. Here we have denoted E(z)=~x{coszx}. Then, using (5), we get easily

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120 CHAP1ER3

(6) (/ ) f E(Il)E(v) / ! * fl' * • = (J.2_ 1l2)().2- ,,2) i.

E(v)E(l) E(l)E(Il) + (1l2_12)(IlL,,2) fl'+ (v2-12)("LIl2) f,,·

Due to the symmetry of this expression with respect to 1, p, v, the asso­ciativity relation

(7) ({sin 1t} * { sin pt}) * {sin ,d} = {sin 1t} * ({ sin pt} * {sin vt})

is established. Now we shall differentiate (7) with respect to the parame­ters 1, p, v. This is allowed, since <P is a continuous linear functional. Dif­ferentiating 2l+ 1, 2m+ 1 and 2n+ 1 times on 1, p and v corrt;!spondingly, we get

({(-I )lt21+1 cos 1t} * {( -1 )mt2m+1 cos ,ut}) * {( -1 )nt2n+1 cos vt}

= {( _1)lt21+1 cos 1t} *({( _1)mt2m+l cos ,ut} * {( _1)nt2n+1 cos vt}). For 1, ,u, v--+O we get

(8) ({t21+1} * {t2m+l}) '" {t2n+l} = {t21+1} * ({t2m+l} * {t2n+l})

for l, m, n = 0, 1, 2, .. , . Hence the associativity relation (f * g) * b = f * (g * b) holds for odd polynomials

p q r

f(t)= I ap1t21+1, g(t)= I bqmt2m+1, b(t)= I crni2n+1• 1=0 m=O n=O

By means of the Weierstrass approximation theorem it can easily be seen that each function tp(t)E CC([O, 1 D, with tp(0) = 0, can be uniformly approxi­mated by odd polynomials. Thus, using the separate continuity of (1) with respect to the uniform convergence, we may take it for granted that the associativity relation is proved for functions f, g, bE CC([O, 1]) with f(O)=g(O)=k(O)=O. Then in order to prove the associativity in CC([O, 1]), we should verify the identities

(9) ({ l}*g)*b={1}*(g*b)=({l}*b)*g

and

(10) ({I} * {I}) *b={I} *({1} * b),

assuming that g and b are functions from CC([O, 1]), such that g(O)=h(O)=O. Identity (7) is convenient to be verified at first for functions of the form g(t)=sin 1t and h(t)=sin,ut. Denoting F(z) = <PAsin zx} and E(1)=<PAcos1x}. we get

(11)'

and

(12)

{I} * {sin It}= +[E(l) coslt+F(l) sin It-E(l)]

{coslt} * {sin,ut}= 12~1l2-F(1)Sin,ut

- J.2!:.1l2 [E(,u) cos ,ut+ F(,u) sin ,ut-E(,u) cos 1t].

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CONVOLUTIONS CONNECTED WITH SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS 121

Then, with a direct check, using only formulas (11) and (12), we see that

(13) ({I} * {sin It}) * {sin .ut} = {I} * ({sin It} * {sin.ut})

= ({ I} * {sin.ut}) * {sin lIt},

at least for ).1I~.uIl. Further, by differentiating (13) on ). ane ,u and letting )., .u-O, we get

for l, m=O, 1, 2, .... Now, using again uniform polynomial approximation, we prove (9).

In proving (4) it is easier to use the estimate (4). We shall use the fact that the set of the odd polynomials is dense in 9'([0, 1]) with respect

1

to the integral norm 11/111 = fl/(t) I dt. If I, g and h are arbitrary functions o

from ~({O, 1]), then let In(t), gn(t), hn(t) be odd polynomial sequences con­verging to I(t), g(t), h(t) correspondingly with respect to the integral norm, i. e. such that I!ln-/Il 1-0, II gn-glll-->-O, II hn-h lit -->-0 for n-->- 00. The esti­mate (4) shows that from Un * gn) * hn = In * (gn * hn) it follows (f * Po) * h =1* (g * h), thus proving the associativity in the general case of ~([O, 1 n.

It remains to prove that (1) is an annihilators-free operation in ~([O, 1 D. To this end it is sufficient to show that there exists a non-trivial non-divi­sor of zero in ~([O, 1 D. Since it is assumed that qJ is a non-zero functional, then the function E()') = qJx{ cos ).x} cannot be identically zero. Indeed, if it were qJA cos ).x} = 0 for all ). E C, then by an even number of differentiations on )., we could get qJA(-I)nxlln cos).x} =0, and letting ),-+0, we would get qJAxlln}=O for n=O, 1, .... Hence qJ(f(t»=O for each even polynomial of t. Hence, by approximation, we would get qJ(!) = 0 in ~([O, 1 D. 0

Now, let ). E C be such that E().):j=O. Let us consider the convolutional operator

(15) ) { sin At} ) La/(t = lE(l) * I(t .

Developed by (1) it can be given the form t x

(16) L,J(t) = + [Sin ).(t-T)f(-c)dT- ~~(:: qJx{[COS).(X-T)/(C)d+

Written in this form, it is not difficult to see that Li.. is a right inverse operator of the differentUil '. operator Di.. = DII + ).2, such that y = Li../ is the solution of the elementary boundary value problem

(17)

The condition E().):j= 0 ensures that (17) has a unique solution. Therefore the function sin It/)'E()') is not a divisor of zero of the operation (I), since

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122 CHAPTER 3

if {sin2tj2E(J.)} *1=0, then L,/=O. Applying the operator D!., we get D;.Ld= ° =j.

Now we begin to study the functional properties of (1). To exhibit them, let us transform the expression under the functional sign.

L e m m a 1. For I, g E C(j the identity 1 .... - / 1 ....

(18) k(x, t)= - II(I x-t I-r)g(r)dc++ II(X + t-r)g(r)dr o I

....

++ f 1(lx-t-rl)g(lrl)dr, -I

where k(x, t) is given by (2), holds. Proof. It is easy to see that (18) is equivalent to the equation

.... ....

(19) f 1(1 x-t-r I)g(1 r I)dr+ f 1(1 x-t-r I)g(1 r I) (sgn (x-t-r)r)dr -I -I

1 .... -/1

= 2 f 1(1 x-t I-r)g(r)dr. o

Then we shall prove (19). Let us first consider the case x~ t. Then x 0 x-t x t

f 1(1 x-t-r I)g(l r I)dr= f + f + f = J I(x-t + r)g(r)dr -t -t 0 x-t 0

x-I x

+ I I(x -t-r)g(r)dr + f I(r+ t-x)g(r)dr o x~

and x I

fl(l x-t-r I)g(l r I) sgn «x-t-r)r)dr= - j/(x-t-r)g(r)dr -t 0

x-I x

+ I I(x-t-r)g(r)dr - f I(r+ t-x)g(r)dr, o x-I

and (19) holds. The case x:::;t is treated in the same way. 0 By means of Lemma 1 and using the functional properties of the con­

volution

(20)

proved in sect. 2.1, analogous properties of (1) can be established.

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CONVOLUTIONS CONNECfED WITH SECOND-ORDER UNBAR DIFFERENTIAL OPERATORS 123

The 0 rem 3.1.2. If f, g E ~([O, 1 n, and if f is a function with bound­ed variation on [0, 1], and if f(O) = 0, then f * g E ~1([0, 1 n.

Proof. According to the results of sect. 2.1, the functions x x

h2(t) = cIJx{J f(x+t-r)g(r)dT} hg(t) = cIJx{ f f(1 x-t-r I)g(l r I)dr} t -I

are in ~1([0, 1 D. We shall show that under the assumption f(O) = 0, the function

It-xl

ht(t)=cIJx{ J f(lt-xl-r)g(r)dr} o

belongs to the space ~1([0, 1]) too. Indeed, as we had shown in sect. 1.1, the Duhamel convolution of a function f(t) with a bounded variation and a continuous function get) is a smooth function, and when f(O)=O, the relation

t I

! f f(t-r)g(r)dr= J g(t-r)df(r) o o

holds. If f(O) = 0, then the function h I (t) is differentiable too, and II-x I

h~(t) = cIJi sgn (t -x) f g(U-x I-r)df(r)} o

The expression in the right-hand side is well defined, since the function under the functional sign is a continuous function of the two variables t and x. It is clear that hit) E ~1([0, 1 D. Taking into account the expression for the derivatives of (20), obtained in sect. 2.1, in our case we get

I x-I I

(f * g)'(t) = cIJx{ sgn (t -x) J g(l x-t I-r)df(r) o

(21 )

x x

- / g(x+t-r)df(r)+ -l g(1 x-t-r I)df(l r I)}. 0

A "p1-version of this theorem can be stated: If f, g E £fl([O, 1 D, and if f is equal almost everywhere in [0, 1 J to a

function with bounded variation, then the function f * g, defined by (1), is absolutely continuous on [0, 1].

We do not give a proof of this theorem, since we do not use it later. 3.1.2. Convolutions of the first kind right inverses of d2/dt2. As we

have already said, for unification of the following considerations, we take

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124 CHAPTER 3

as basic the space ~ = (6'([0, 1 J). Here we consider the class of the right inverse operators of the square of the differentiation in ~, defined by a boundary value condition of the first kind in the point t = ° and by an ar­bitrary boundary value condition, determined by a linear functional N in the space ~l([O, 1 D. If L: ~ ->-~2 is such a right inverse operator of (d2jdt2), then the function y = Lf should be the solution of the elementary boundary value problem

(22) Y"=f, y(O) = 0, N(y)=O.

It is assumed that N is a continuous linear functional on ~1([0, 1]), such that the problem (22) has a solution for all f E ~([O, 1 D. fhe continuity of N should be understood with respect to the ~I-norm

(23) llflle! = max {If(t)1 + 1f'(t)I}· IE [0, I)

It is easy to see that for the solvability of (22) it is necessary to assume N{x}=l=O. There is no loss of generality to assume that

(24) N{x}= 1.

If, as earlier, we use I for the Volterra integration operator I

(25) If(t) = ff(-c)dr, o

then the right inverse operator L of d2jdt2, defined by the boundary value problem (22), has a representation of the form

(26) Lf(t) = 12f(t)-N(l2f)t.

The defining projector of L has the form

(27) Ff(t) = f-Lf" = f(O)[l-A{ 1 }t] +N(f)t

and it can be considered as defined on the whole space ~1([0, 1 D. Our main aim here is to find a convolution of L in an explicit form.

This problem is solved in Dim 0 v ski [37]. The 0 rem 3.1.3. The operation

x

(28) U*g)(t)=-(Nol)xg f f(x+t-r)g(r)dr I

x

- ~-ff(lx-t-r,)g('r') sgn «x-i-r)r) dr} -I

is a convolution oj the operator (26) in ~([O, 1 D, and has the represen­tation

(29)

in it.

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CONVOLUTIONS CONNECI'ED wrrn SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS 125

Proof. Since N is a linear functional in ~l, then tjj = - Nol is a linear functional in ~. Then operation (28) has the form (1) and by Theo­rem 3.1.1 it is a bilinear, commutative and associative operation in CC. In order to assert that (28) is a convolution of L in ~, we should prove re­presentation (29). This can be done by a direct check.

Even for smooth functions f, g operation (28) does not always give a differentiable function f * g. In order to ensure f * g to be differentiable, one should assume f(O) = O. Let us introduce the invariant subspace

(30) ~o=~oC[O, 1])={f(t)E~([O, 1]), f(O)=O}

of the operator L. If f, g E~, then it is easy to see that (f * g)(O) = 0, L e. * :CCX~-+~o. Hence * is a convolution of L in ~o too. 0 .

The 0 rem 3.1.4. The space ~6 of twice continuously differentiable functions f(t) in ~([ 0, 1]) with f(O) = 0 is an ideal in the convolutional algebra ~ with multiplication (28), and for fE~6 and gE~ the differentiation for­mula (31) (f * g)" = f" * g + N(f)g

holds. Proof. If f E ~6' then by formula (27) we get

f = Lf" + N(f)t.

Therefore,f*g=(L!")*g+N(f){t}*g=L{j"*g+N(f)g} and hence f*gE~6 and (f*g)"=f"*g+N(f)g. 0

E x amp I e 1. If N(f) = f'(O), then we get Lf = l2f and (28) gives t

(f*g)(t)= f f(t-.,;)g(r)d.,;, o

i. e. the usual Duhamel convolution, which could be expected. Example 2. Let N(f)=f(I). Then y=Lf is the solution of the two­

point boundary value problem

(32) Y"=f, y(O)=y(l)=O.

In this case 1

(33) Lf=l2f(t)-t f (l-.,;)f(.,;)d.,;. o

The defining projector F of L is the operator

(34) Ff(t) = (! -LD)f(t) = (l-t)f(O)+tf(I).

By specialization of operation (28), as a convolution of L in ~ ([0, 1]) we get

(35) 1 [ x

f*g(t)=-f + f f(x+t-r)g(r)dr o t

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126 CHAPTBR3

--} f fClx-t-il)g(lil) sgn{(x-t-i)i)di JdX. -/

The operator (33) presents some interest from the point of view of the analysis, since it is closely connected with Lidstone's polynomials (see W hit ta ke r [106]).

De fin i t ion. The polynomial system {An(t)}:"I' determined recurrent­ly by A: (t) = An-l(t), ~(t) = t and by the boundary value conditions An(O) = An(l) = 0 for n~ 1 is said to be the Lidstone polynomial system.

Due to asymmetry, caused by the choice Ari..t)=t, these polynomials can be named more exactly left Lidstone polynomials. The right Lidstone polynomials can be defined recurrently by 1; (t)= An-l(t), the initial choice Ari..t) = I-t and by the same boundary value conditions An(O) = i;,(I) = 0, n~1.

L e m m a 2. For each integer n~O, identically

(36)

Proof. Let us denote AnCt)=An(l-t), n~O. Obviously, A:(t) =A:(l-t)=An_l(l-t)=An-I(t)and Ari..t) = I-t. The boundary value condi­tions An(O) = An(l) = 0 are fulfilled. Hence An(t) = An(t).

The generalized Taylor formula (see 1.3.4) for the right inverse L of d2jdfJ takes the form

n-I (37) I(t) = I UF/(2k)+Ln 1(2n)

k=lJ

n-I = I (f(2k)(0)Ak(l-t)+j<2k)(I)Ak(t)]+Rn(f; t). 0

k=lJ

By means of convolution (35) we shall find a simpler expression for the remainder term RII(f; t).

The 0 rem 3.1.5. II n ~ 1, then tke remainder term in (37) kas tke representation

I

1 J' (38) Rn(f; t)= 2"" j<211-1)(i)[An-l(1-t-i)-An_l(1 +t-i)]di /

t

+-} J 1(2n-I)(i)[AII_ I(1-t-i)+An_1(1-t+i)]dr. o

Proof. By Theorem 3.1.3 the representation LI = {t} * I holds. Then

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CONVOLUTIONS CONNECfED WTI1I SECOND-ORDER UNBAR DIFFERENTIAL OPERATORS 127

Now, let us sil1)plify this expression. We have

I [ 0 x ] -2An-l * f(2n)(t) = I I An_ 1(x+t-r)f(2n)(r)dr+ J AIl_ 1(x+t-r)J<211)(r)dr dx

o t 0

I [ 0 x ] + J J An-I (x-t-r)f(2n) (-r) dr-f An_ 1(x-t-r)J<211)(r) dr dx,

o -t 0

where the fact that the Lidstone polynomials are odd is used. Interchanging the order of the integration in the last formula, we get

-2RnU; t)= - j J<2n)(r) [J An_ 1(x+t-r) dX]dr o 0

+ J f(2n)(r)[ J An_l(x+t-r) dx ]dr o T

+ j J<2n)(r)[J An_l(x+r-t)dx ]dr- j J<2f!)(r)[J A Il_ 1(X-t-r)dX]dr. o 0 0 T

Now, using the recurrency, we get t

-2RnCj; t)= f J<2n)(r)[A;ll-t+r)-A~(1 +t-r)]dr o

I

+ f j<2n) (r)[A~(1 +t-r)-A~(1-t-r)] dr. o

With elementary integration by parts and by means of routine transforma­tions, we get

I

-2RnCf; t)= f J<2n-l)(r)[An_ 1(1 +t-r)-An_l(l-t-r)]dr o

t

-- Ij<2fl-I)(r) [A Il- 1(1 +r-t)+An- 1(1 +t-r)Jdr

o and the representation (38) follows immediately. 0

Theorem 3.1.6. If '1': 0'([0, l])-C is a continuous linear func­tional, then the operation

t

(39) r { ok(x t) } U * g)(t) = . f(t--r)g(r)dr+ 'I'x o~ , o

with x x

1 r 1 I k(x, t)= 2"". f(x+t-r)g(r)dr- 2 f(lx-t-rl)g(lrl) sgn «x-t-r)r)dr, t -t

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128 CHAPTER 3

is a convolution oj the right inverse operator oj cf2/dt2

(40) LJ(t) = l2J(t) + tlJf(f)

in the space ~~([o. 1]) of the smooth Junctions J(t) in [0, 1] with 1(0) = 0. Proof. It is not difficult to see that (39) is an inner operation for

~&([O, 1]). Operator (26) is a special case of (40). The bilinearity and the commutativity of (39) are evident. The associativity of (39) can be proved in almost the same way as the associativity of (1). Here we can use at once the possibility of uniform approximation of the functions of ~~([O, 1 D by odd polynomials with respect to the norm

IIJI/ct = max {IJ(t) 1+ I/,(t)l}. 0 0;;;;~1

Now we shall write down explicitly a convolution for an operator, which often finds use in problems of mathematical physics.

E x amp 1 e 3. Let L be the right inverse of d2/dt2, determined by the boundary value problem

ylJ=J, y(O)=O, hy(l)+y'(I)=O, l+h=l=0'

i. e.

LJ(t) = l2J(t) + (h:1 j r:j(-r)dr:- j J(r:)dr:)t. o 0

This is an operator of the form (26) with N(f) = (hJ(I) + /,(1»/(1 +h). Then (28) gives for a convolution of L:

(41) (f* g)(t) = ;;11 [k(l, t)+h j k(x, t)dX] o

with x x

1 f 1 J" k(x, t)="2 j(x+t-r:)g(r:)dr:-"2 J(lx-t-r:l)g(lr:l)sgn«x-t-r:)r:)dr:. t -I

Letting h->=, we get operation (35), as it should be expected. 3.1.3. Convolutions of the second kind right inverses of d2/dt2• Here

we consider the class of the right inverses of d2/dt2 in ~([O, 1 D, which are defined by a boundary value condition of the second kind and with an arbitrary linear boundary value condition, determined by a linear functional N in the space ~1([0, 1 D. Such an operator L: ~([O, 1 D->~2([0, 1]) is de­fined for each J E ~ in such a way that y = LJ to be the solution of the elementary boundary value problem

(42) ylJ=J, y'(O) = 0, N(y)=O.

In order (42) to be solvable for each J E ~, it is necessary and suffi­cient to assume that N({1 })=I=O. If this condition is fulfilled, then without any loss of generality we may assume that N({1}) = 1. Then (43) LJ(t)=l2f(t)-N(l2j).

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CONVOLUTIONS CONNECfED WITH SECONJ)..()RDER UNBAR DIFFERENTIAL OPERATORS 129

The defining projector P of L has the form

(44) PI(t)=/,(O)[t-N({x})]+N(f).1.

Our main task is to find an explicit convolution of L if not in the whole space ~([O. 1 D. then at least in some subspace of it. To this end we shall use the similarity method, reducing the operator (43) to operator of the form (40). Obviously. the transformation T: fd --> CCA([O. 1 D. defined by the Volterra integration operator

t

(45) T: I(t) -ll(t) = f /(r:)d-r, o

is a similarity of operator (43), defined in ~([O, 1 D. to the operator

(46) 11<t)=l2/Ct)-tN(lj)

in ~~([O. 1]). i. e. TL=IT. Then (46) is an operator of the form (40) with 'P(f)= -N(l}). Then the operation

(] *gXt)= j /(t--r)g(7:)d7:+ 'Px{ o~ (+ J j(x+t-7:)"'g(-r)d7: o t

-+ l/(lx-t-or:!)g(,-rl) sgn «x~t--r) -r) d7:)}'

by Theorem 3.1.6, shall be a convolution of l in ~Ac[O, I D. By Theorem 1.3.6 the operation 1* g= T-l [CTf) ;. (T g)] is a convolution of L in~. Since T-l/= dl!dt, then we get the following representation of this convolution:

1 T _

(f*gXt)= I d-r f 1(7:-a) g(a) da+ 'Px{ 02k~;a:) } o 0

with x - if k(x. t) = 2" (lj)(x+ t-rXlgXr:)d7:

t

x

1 j" - 2" (lIXlx-t-rIXlg)(I-rI) sgn «x-t--r)-r)d-r. -I

Differentiating. we get

o2"k 1 fX 1 fX oxot =""2 l(x+t--r)g(-r)d-r+"2 (lx-t--rI)g(I-rl)d-r. t -I

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130

(47)

Thus we have proved the following Theorem 3.1.7. The operation

t •

(f * g)(t) = l dr! f(r-a)g(a)da

x x

-Nxolx{ + J feX+t-r)g(r)dr+ + J fClx-t-rl)gClrl)dr} t -t

is a convolution of the right inverse of d2/d(;2, defined by

Lf(t) = l2f(t)-N(l2!),

CHAPTER 3

where N is a linear functional on ({1([ 0, 1]) with N({l}) = 1. Moreover, the representation

(48) Lf={l}*f

holds. The representation (48) should be verified by a direct check. The analogon of Theorem 3.1.6 is The 0 rem 3.1.8. If 1Jf: ~([O, 1]) -+ C is a continuous linear functional,

then the operation t •

(49) (f*g)(t) = J dr J f(r-a)g(a)da+lJfx{ iJr~;,) },

o 0

with x x

1 J 1 J' rex. t)="2 f(x+t-r)g(r)dr+ 2 f(lx-t-rl)g(lrl)dr. t -t

is a convolution of the right inverse operator of fi2/dt2, defined by

(50) Lf(t) = l2f(t) + 1Jf(f)

in the space ~1«(0, 1]) of the smooth fanctions in [0, I}, sllch that the representation Lf = {I} * f holds.

Pro a f. By a direct calculation we find

{ At} { t} _ cos pi-cos U + f-t2E(f-t) cos J.t-J.2E(J.) cos fA cos * cos I'" - J.2_f-t2 1.L-f-t2'

where E(/k) = IJfA cos fl,X}. and Eel) = IJfA cos Ix} for arbitrary I, I'" with l::j::.u. We verify the associativity relation [(cos It) * (cos .ut)} * (cos vi) = (coslt)* [(cos .ui) * (cos vi)} directly. Further. we differentiate this identity 2l times on 2, 2tn times on ,u, and 2n times on v. Letting 2, I-!, v -> 0, we get ({ t21} * {t2m}) * {(;2n} ={t21}*({t211l}*{t2nD for l, tn, n=O, 1, 2, .... Hence the associativity relation (f * g) *h = f * (g* h) is satisfied for even polynomials. As in the proof of Theorem 3.1.1, it is easy to show that an estimate of the form

(51) IIf*gII1~Allflll.lIgI11

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CONVOLUTIONS CONNECTED wrm SECOND-ORDER UNBAR DIFFERENTIAL OPERATORS 131

for all I, g E ~1([0, I]) with A = const, holds. Here by II lit the integral norm 1

11/111 = /1!(t)ldt is· denoted. Since the set of the even polynomials is dense

in !l'1([0, 1 D, then from the validity of the associativity relation in the class of the even polynomials and from the estimate (51) its validity in the whole space ~([O, 1]) follows. Thus the relation (f*.g)*h=I*(g*h) is proved as holding almost everywhere. in [0, 1]. But since here both parts of the associativity relation are continuous functions, then it holds everywhere. Hence operation (49) is an associative operation in ~l([O, 1]). Since LI = {I} *1, it is a convolution of L in ~l([O, 1]). D .

E x amp 1 e. Let L be the right inverse of d2/di2 in ~([O, 1 D, defined by the solution of the boundary value problem .

(52) yH=I, y'(O)=O. hy(I)+y'(I)=O (h=t=O).

i. e. 1

(53) Lf(t)=l'J/(t) - f (I + -} -l.)/(l)dT. o

This is an operator of the form (43) with NI=(hli(l)tf(1)1/h. By (47) we find the convolution

t ~ 1

(54) U*g)(t)==I dl f 1(1-a)g(a)da-2~ r(I. t)-i- j'r(x. t)dx 000

with x x

rex. t)= II(x+t-l)g(l) dl+ f j(IX-t-ll)g(lll)dl. 1 -I

The boundary value problem (52) is often use-d in problems of mathe­matical physics.

3.1.4. Convolutions of the third kind right inverses of d'J/dt'J. Now let L be a right inverse operator of d2/di2 in ~([O. 1]). defined by a boun­dary value condition of the third kind and by a general boundary value ~ondition. determined by an arbitrary linear functional N in ~1([0, I D. If IE~l([O, 1]), we define y=Lj as the solution of the boundary value problem

(55) yt' =1. y'(O)-hy(O)=O. N(y)=O.

where h is a given constant. and N is an arbitrary non-zero linear function­al on ~1{[0. 1 D. For the solvability of (55) for each I E ~([O. 1 D it is necessary and sufficient the condition N{l +kx}=t=O to be satisfied. It is clear that without any loss of generality we may assume that

(56) N{l +hx}= 1.

Then the solution y = LI of (56) is

(57) LI(t)=l'J/(t)-N(l2/XI +ht).

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132 CHAPTER 3

The defining projector F of L has the form

(58) Ff(t) = [t-N{ x }](f'(O)-hf(O» + (1 +ht)N(f).

Since the boundary value problem (42) is a special case of (57), we shall try to proceed in a similar way. Here the approach of 0 i m 0 v ski and Pet r 0 v a [50] is presented. First, we find a similarity from (57) to an operator of the form (43).

L em m a 3. The linear transformation t

(59) Tf(t) = f(t)-h f e-h(t-')j(r)df o

is an isomorphism of the linear space ~([O, 1]) on itself, and its inverse is the transformation

t

(60) T-l f(t) = /(t)+h If(f)df.

Proof. By a direct check. 0 L e m m a 4. Transformation (59) is a similarity of the right inverse

(57) of d2/dt2 to the right inverse

(61) lfCt)=l2j(t)-N(l2j)

of d2/dt2 with N=NoT-l, i. e. TL=LT. Proof. It is obvious that T commutes with I: Tl = IT and that

T{1 +ht}= 1. Then

TLf= Tl2j-N(12f)T(1 +ht)=l2Tf-(NoT-l)(l2Tf)

and the assertion is proved. 0

Theorem 3.1.9. The operation h h

(62) (f * g)(t) = N{ 1 }(f * g)(t) - (Nol)A -h(f * g)(lx-tl) x x

1 j' 4 f + 2" f(X+t-f)g(f)df+ 2" f(lx-t-fl)g(lfl)df}. 1 -I

with

(63) h t ( • ) (f*g)(t)= f e-h(t-.) f f(f-a)g(a)da df, o 0

is a convolution of the right inverse (57) of d2/dt2 in ~([O, 1]), such that the representation

(64)

holds.

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CONVOLUTIONS CONNECI'ED WITH SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS 133

Proof. By Lemmas 3 and 4 and by Theorem 3.1.6, the operation

(65) (f-; g)(t) = T-l[(Tf) -;;; (T g)],

where; is operation (47), but with N instead of N, is a convolution of (57) in c;&'([O, 1]). This is an immediate implication of the general Theorem 1.5.6. Moreover, Lf={1+kt}*f, since T{l+kt}=l. We shall show that convolu­tion (65) coincides with (62). The identity

(66)

can easily be verified for f(t) = h(t) = cos It + (kjA) sin At and get) = fl'(t) = c~s flt +(kl fl) sin flt with arbitrary real A, fl:f0. Using the obvious identitie~ f.(t) = T-l{ cos M} and fit) = T-l{ cos flt}, the desired identity (66) can be written in the form

(67) (T-l{ cos ltD * (T-l{ cos fltD = (T-l {cos M D:; {T-l( cos flt)}. The last identity is satisfied also in the case when A or fl is 0. Since T-l{COSO.t}=l+kt, then by foCt) we can understand the function l+kt. We need the identity (67) for integer multiples of n only, i. e. for A, fl = 0, n, 2n, .... The system {cos nnt};;,,=o satisfies the hypothesis of the Weier­strass-Stone approximation theorem. Taking into account the continuity of the inverse transformation T-l, along with the continuity of the operations * and -; with respect to the uniform convergence in c;&'([O, 1]), we conclude at once that (66) is fulfilled for arbitrary f, g E c;&'([0, 1 D. The theorem is proved. 0

Example 1. Let k=O. Then N{l}=l, due to the assumption N{ 1 + kx} = 1. Then convolution (62) takes the form

t •

(f*g)(t)= f d-r f f(-r-a)g(a)do u 0

x x

-(No[)x{; ff(X + t--r)g(-r) d-r+ ~ f f(lx-t--rI)g(I-rl)d+ t -t

i. e. it coincides with (47), as it should be expected. E x amp I e 2. Let x be a linearfunctiona1 in c;&'l([O, 1 D, with x({1}) = °

and x({ x D = 1. Then, let us take the functional N = xl k. It satisfies the con­dition N({l+kxD=l and hence operation (62), which we shall denote by (f * g)h' is a convolution of the operator

Lhf(t) = [2j(t)- x([2j)(1 + kt)lk

such that Lhf=[(l+kt)*jh.Lettingk--+=, we see that lim(f*g)h=O,but h->+oo

the expression k(f * g)h has a finite limit, which we shall denote by j * g. Then

Ix-tl

(f*g)(t)=-Xx{- f f(lx-tl--r)g(-r)d-r u

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134 CHAPTER 3

x x

+-} J f(x + t-l)g(l)dr + ~ J f(lx-t-rl)g(lrl)dr}. U -I

This operation is a convolution of the operator Lf(t) = [2j(t)-xClJ). t with Lf = {t} * f. This is exactly a convolution of the type (28), provided we take into account (18) and Lemma 1.

E x amp Ie 3. Let L be the right inverse operator of d2fdt 2 defined by the boundary value problem

(68) y" = j, y'(O)-hy(O)=O, y'(1)+Hy(l) =0.

In order this boundary value problem to be solvable for each fE ~([O, 1 D, we should assume h+H+hHt-O. Since N(f)= [f'(l)+ Hf(l)]f(h +H+hH), then

(Nol)f = h+~+hH [f(l) + H / j(r) dr l o

Convolution (62) in this case takes the form

(69) U*g)(t)= h+~+hH[H(f!g)(t)-AhCl' t)- / Ah(x, t)dX], o

where h x

Ah(x, t)=-hU*g)(lx-tl)+-} f j(x+t-r)g(r)dl t

x

+ -} J f(lx-t-ll)g(l-c1)dr -I

h

and where by f * g the operation

U:g)(t) = j e-h(t-T) [J fCI-a)g(a)da]dl o 0

is denoted. 3.1.5. Operational calculi for right inverses of the square of dif­

ferentiation. The convolutions found in the previous sections can be used for developing operational calculi, intended for some classes of boundary value problems. In this we can follow the general scheme of Chapter 1. Here two approaches could be used: the convolution quotients approach and that of the multiplier quotients. Though they are equivalent, the latter ap­proach has some advantages in applications to boundary value problems of mathematical physics. Here we confine ourselves to the multiplier quotients approach, though the interpretation of the results obtained as convolution quotients is immediate. Moreover, we consider the convolutions introduced III the previous sections only.

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CONVOLUTIONS CONNECfED WITH SECOND-ORDER UNBAR DIFFERENTIAL OPERATORS 135

1. Multiplier quotients rings for first kind right inverses of the square of differentiation.

As first example, we shall consider the right inverse operator

1

(70) Lf(t)=l~,/(t)-t I (l-T)f(T)dT o

of d2/dt2, defined as the solution of the two-point boundary value problem (32). As we had seen in 3.1.2, the operation

1

(71 ) U * g)(t) = - I k(x, t)dx o

with

x x

1 J 1 j. k(x, t)=2 f(X+t-T)g(T)dT- 2 f(lx-t-Tl)g(!T!)sgn«x-t-T)T)dT, t -t

is a convolution of L in CC([O, 1]), such that Lf = {t} * j. The expression

(72) Ff(t)=f(O)(l-t)+f(l)t

represents the defining projector of (70) in the whole space CC([O, 1)). As first task, let us characterize the divisors of zero of (71). To this

end we shall find the eigenvalues and eigenfunctions of L. It is easy to see that all the eigenvalues of L are An= -ljn2n2, n= 1,2, .... The corres­ponding eigenfunctions are QJn(t) = sin nnt, n = 1, 2, .... If f(t) is an arbitrary function of CC([O, 1]), then it can easily be found that

(73) U * QJn)(t) = XnU)QJn(t), n = 1, 2, ... ,

with

1

XnCf) = (-l)n If(t) sin nntdt. nn

o

Theorem 3.1.10. A function fECC([O, 1]) ls a non-trivial non-divi­sor of zero of (71) iff

1

(74) I f(t) sin nntdt =F ° o

f orn=I,2, .... Proof. The proof follows immediately from the multiplicativity pro­

perty

(75) XnU * g) = Xn(f)Xn(g)

of the functionals Xn and from the well-known fact that the system {sin nnt}' n = 1, 2, ... , is total in CC([O, 1]) in the sense that xif) = ° for n = 1, 2, ... ' imply f= 0. Let f be a divisor of zero of (71), i. e. let there exist a functi f)n

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136 CHAPTER 3

gE~([O, 1]), g$O, such that l*g==O. From (75) it follows that Xn(f) .Xn(g)= 0 for n = 1, 2, .... Since g$O, there exists at least one k, such that XA({;)=fO. Then Xk(f)=O and thus (74) is violated. Conversely, if for some

1

m>O we have fl(t) sin mntdt=O, then from (73) it follows that/;!<{sinmnx} ()

"-= 0 and he'nce I is a divisor of zero, provided it is not iqentically zero. 0 Cor 0 11 a r y 1. A non-zero function I E ~([O, 1]) is a divisor of zero

1

of (71) iff a t least one of the Fourier sine-coefficients f I(t) sin nntdt of I is o

equal to zero. Cor 011 a r y 2. A convolutional operator

(76) MI=m*l,

with m E ~([O, 1 D, and with (71) as the convolution *, is a divisor of zero in the mUltipliers ring IDt of (71), iff m is a divisor of zero of convolution (71).

Let M be the multiplier quotients ring of (71). We shall denote the convolutional multipliers MI = m * I by M = m *. In particular, the representa­tion LI = {t} * I means that L = {t} *. In other words, the convolutional opera­tors ring is isomorphic with the ring of the continuous functions ~([O, 1]) with the multiplication (71). From now on the inverse element of L in M shall be denoted by S. i. e. S= IlL. In applications of the corresponding operational calculus, explicit representations of the basic partial fraction with S are needed.

Theorem 3.1.11. II A is an arbitrary complex number, such that A=f -k2,n2, k = 1, 2, ... , then lor each integer n=f 0 the representation

(77) I {< _1)n-1 on-1 ( sin J"f t )} (S+).)n= (n-l)! 0;,n-1 sinJ;. *

holds. In particular,

1 { sin J"ft } S+;' = sin J;. *,

where lor JI one and the same value is taken. Proof. Let us prove (77) in the case n=1. Denote cp={sinJltlsinJI}.

Since I/(S+A) =LI(1 +AL), then the desired formula takes the form L = (1 + AL)cp *. It is equivalent to L = [(1 + AL)cp] *, i. e. to {t} * = [(1 + AL)cp] *. Therefore, we should prove that the equality {t}=(1 +AL)cp is an identity. But tris is a matter of an elementary check.

The proof of (77) for arbitrary n>O proceeds as the proof of a similar representation in 2.3.1.

The example considered is a pattern for construction of multiplier quotients rings for all first kind right inverses of the square of differen­tiation. Indeed, if we take the general right inverse operator

(78) LI(t) = t2j(t)-N(l2j)t

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CONVOLUTIONS CONNECTED wrrn SECOND-ORDER UNBAR DIFFERENTIAL OPERATORS 137

of tPldt2, considered in 3.1.2, with an arbitrary linear functional N in ~1([0, 1]) with N{x}=I, then, by Theorem 3.1.3, the operation

(79) x

(f* g)(t) = - -}-tPx{J f(x+ t-T)g(T) dr t

x - J f(/x-t-rl)g(/rl) sgn «x-t-r)r)dr} , -/

with tP = Nol, is a convolution of L in ~([O, 1 D. Here the representation Lf ={ t} * f holds too. In contrast to the example considered, in the .general case we are not able to gi:ve an exhaustive characterization of the divisors of zero of the convolution (79). However, for the purposes of operational calculus this is not obligatory. In finding representations of partial fractions of the form Ij(S+A)n, it is necessary only to know whether S+A is non­divisor of zero. But from the general theory of 1.4.4 we know that S + A is a divisor of zero in Miff-II A is an eigen value of L. Hence S + A is not a divisor of zero in M iff A is not an eigenvalue of the. elementary boundary value problem

y"+Ay=O, y(O) =0, N(y)=O.

It is obvious that the eigenvalues of this problem are the zeros of the

entire function E(A) = ./~ NAsinx~I}. Then, as in the proof of Theorem 3.1.11, vJ.

it is easily seen that if A is not a zero of E(A), i. e. if E(A)=tO, then the representa tion

1 (-I)n-J iJn-1 { sin.jit } (80) (SH)n= (n-l)1 O}.n-1 .jiB(J.) * holds.

2. Multiplier quotients rings jor second kind right inverses of the square of differentiation.

First we shall consider the right inverse operator L of d2ldt2 in ~([O, 1]) defined by means of the boundary value problem

Y"=f, y'(O)=O, hy(I)+y'(I)=O, h=l=O, i. e. the operator

1

Lf(t) = l2f(t) + f(l+i-r)f(r)dr. (81)

o

In 3.1.3 we had found as a convolution of L in ~([O, 1]) the operation / ... 1

(82) (!*g)(t) = f dT ff(r-CJ)g(CJ)dCJ-2~ r(l, t) - + f rex, t)dx 000

with x x

rex, t)= f!(x+t-r) g(T)dr + !fClx-t-rj)gC\r!)dr. / -/

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138 CHAPTER 3

Then the representation Lf = {I} * f holds. The eigenvalues of the problem

(83) yl/+2y=0, y'(O)=O, hy(l)+y'(I)=O

are the zeros of the entire function £(2) =~ cos VI -VI sin vij /h If 2 E C is such tha,t £(,t):f0, then we get

(84) I {{-I)1~-J on~l ( cos lJ:t)} {S+).)n= (n-I)I o).n-l E{l) *.

We shall not consider the general case, since everything proceeds in the same way as in the previous subsection.

3. Multiplier quotients rings for third kind right inver~es of the square of differentiation.

In 3.1.4 we have found a convolution for the general right inverse of fi2jdt2 of the third kind. For a corresponding operational calculus we shall confine ourselves to the right inverse considered in Example 3 of this sec­tion. This right inverse L of d2jdt2 is defined by the elementary boundary value problem

yl/=f, y'(O)-hy(O) = 0, y'(I)+Hy(l)=O,

under the assumption that h+H+hH:fO. It has the form

(85) Lf(t)=L2f(t)- h~~~2H [/f(T)dT + H f~I-T)f(T)dT]. o 0

The convolution of L in ~([o, I]), found in 3.1.4, had the form (69). The exact form of this convolution is not necessary for general considerations and for finding representations of the partial fractions Ij(S+,t)n as convo­lutional operators in this convolution. It is enough to know the representa­tion Lf={I+htl*fonly. The element S+2 of.M is invertible iff 2 is not a zero of the entire function £(2)=(h+H)cosVf+(hH-2) sin vf!VI. In such a case the representation

(86) I {_l)n+J i}n-l (COS.JJ:t+h sin JJ:tllJ: ")' }

(S+J.)n = {(n-I)I 'i}).n-l (h+H+hH)E().) *

holds. In particular

_1 _ = {cosJ""i t+h sin JJ:t Iii} *. S+J. (h+H+hH)E(}.)

3.2. CONVOLUTIONS OF INITIAL VALUE RIGHT INVERSFS OF LINEAR SECOND-ORDER DIFFERENTIAL OPERATORS

In this section we consider the problem of determining convolutions of right inverse operators of linear differential operators of the second order with variable coefficients, defined by initial value problems for a point of

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CONVOLUTIONS CONNECI'ED WITH SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS 139

the interval considered. First we treat the case of non-singular linear dif­ferential operators of the second order, and then as an example for a sin­gular second-order linear differential operator we take the Bessel differential operator. In both cases the basic method for finding convolutions of the corresponding right inverses is the similarity method, developed in general terms in 1.3.3. The Oelsarte-Povzner transmutation operators and the Sonine transform are used as similarities.

3.2.1. Convolutions of the initial value right inverse of non-singular second-order linear differential operator. In this subsection we consider the general linear differential operator of the second order

d2 d (1) D= a(x)dx2 + b(X) dx +c(x)

in an interval L/, assuming that the coefficients are complex-valued and satisfy the smoothness conditions a(x) E C6'2(L/), b(x) E C6'l(L/) and c(x)E C6'(L/). More­over, we assume that a(x) is real-valued, and a(x»O. Let Xo be an arbitrary but fixed point of L/.

De fin i t ion 1. A linear operator Lo: C6'(L/)-+C6'2(L/) is said to be the initial value right inverse of D for the point x = xo, if for each f E ct'(L.J) the function y = Lf is the solution of the initial value problem

(2)

In order to express explicitly the defining projector F of L, we intro­duce the fundamental system Yl(X), Y2(X) of the operator D for the point xo. These are the solutions of Dy = 0, which satisfy the initial value condi­tions Yl(XO) = 1, y~(xo)=O, Y2(XO) = 0, y;(xo)=1. Then for Ff=(I-LD)f we get

(3)

and it can be considered on ct'1(L.J) (B 0 z hi nov, 0 i m 0 v ski [17]). We shall show that Lo has a convolution in C6'(L/). Without an essential loss 0 f generality, we may consider the operator

d2 (4) D= dt2 -q(t)

instead of (l). L e m mat. The transformation T: (6'(.d)-+C6'(Ao) given by

(5) T : f(x)~g(t)f[<p-l(t)l, with

X '1'_1(1)

f dE; {f 2b(i;)-a'(i;) } q>(X) = ~a(i;) and get) = exp 4a(i;) d~, Xo Xo

where Llo=q:{LI), transforms a(x)(d2jdx2) + b(x)(djdx) + c(x) into (d2jdt2)_q(t), [ 4b2(x)-a'(x)2 d (2b(X)-a'(x) )1 .

where q(t) = -c(x)+ 16a(x) +a(x) dx 4a(x) X='1'-l(t) mthesense of tke similarity

(6) [ d2 d ] [d2 .] T a(x)dx2 + b(x) dx +c(x) = dt2 -q(t) T.

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140 CHAYl'ER3

The proof consists in an elementary verification and in essence it is contained in every textbook on ordinary differential equations.

Further we consider operator (4) only, denoting it by D. and let the inter­val in which t varies be LI. But the fundamental system Ul(t). ua(t) shall be con­sidered for the point t=O. Under these assumptions the defining projector P of Lo has' the form

(7) Pf = f(0)u1(t)+ /,(O)~(t). Now we shall show that there exists a similarity T of Lo to la, which

is a Volterra transformation of the second kind t

(8) Tf(t)=f(t)+ fK(t. -r)/(r:)d-r. o

L e m m a 2. Let q(t) E fCl(LI), and let K(t, -r) be a solution of the par­tial differential equation

(9)

in the domain G={(t. -r): tELI.min(t. O)<-r<max(t. O)}. which satisfies the condition

t

(10) K(t. t)=-+ fq(-r)d-r. o

If T is transformation (8) with this K(t, -r) as kernel, then for every f (t) E fC2( LI). the identity

(11) (TD-::! T )f(t)=f(O)K~(t. O)-/,(O)K(t. 0)

holds. Pro a f. Obviously. T is a linear isomorphism of fC(LI) onto itself. i. e.

an automorphism. and fCl(LI) and fC2(A) are invariant subspaces of T. It is clear that T preserves the values of the functions and of their derivatives in t=O. i. e. Tf(O)=f(O) and (Tf),(O)=f'(O). The assertion of the lemma follows from the easily verified identity

t

(12) (TD-:t: T)f(t) = f[Kn(t. -r)-q(-r)K(t. -r)-Ktt(t. -r)Y(-r)d-r o

-[ 2! K(t. t) + q(t) ]f(t)+f(O)Kit. O)-/,(O)K(t. 0). 0

It is not difficult to show the existence of kernels which satisfy (9) and the boundary value condition (10). To see this. let us make the characteristic change of the variables

(13) u=(t+-r)/2. v=(t--r)/2.

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CONVOLUTIONS CONNECTED wrrn SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS 141

Then for k(u, v)=K(a+v, a-v) we get the equation

(14) kU11= -q(a-v)k

in the corresponding domain 0'. If along with (10) we impose the con­dition

(15) K(t, O)=a(t)

with an arbitrarily given function aCt) E 'C(LI), then for k(u, v) we get the conditions

u

(16) 1 j' k(a, 0) = - 2" q(~)d~, k(a, a) = a(u). o

It is easy to see that the boundary value problem (14), (16) can be reduced to the second kind Volterra integral equation

u U 11

(17) k(u, v)=a(v)-+ fq(~)d~- fd~ fq(~-11)k(~, 'Y)d'Y). 11 11 0

From the well-known theory of the Volterra integral equations (see, c. g Mar c hen k 0 [74J, p. 19) follow the existence and the uniqueness of a solution of (17) in the whole domain a. This solution could be found e. g. by the successive approximations method. It is essential to note that (17) has a solution for each q E 'C(Ll), but it may happen k(a, v) to be only once differentiable, if q(t) is not differentiable. Nevertheless, if T is (8) with the kernel

(18) (t+-e t--e) K(t, -,;)=k -2-' -2-'

then it can be shown that T: 'C2( LI)-+'C2(LI) and (11) is also valid. Since we do not use this property, we shall confine ourselves to a weaker propo­sition.

Lemma 3. If q(t)E'C(LI), and K(t, -,;) is (18), then Tf(t)=f(t) t

+- f K(t, -,;)f(r)d-,; is a similarity of Lo to 12 in 'C(LI), i. e. the relation o

(19) TLo=12T

holds. Proof. If q E 'C1(LI), then (19) follows immeqiately from identity (11).

Indeed if we substitute Lof(t) instead of f(t) , for f(t) E 'C(LI) we get

d2 TDLof(t)-dt2 TLof(t)=(Lof)(O)K.(t, O)-(Lof)'(O)K(t, 0),

i. e. (d2/dt2)[TLof(t)]=Tf(t). Since Lof(O) = (Lof)' (0) = 0 then TLof(t) = 12 Tf(t). Now, let us consider the general case q(t) E 'C(LI). In order to prove (19)

in the class 'C(LI), we approximate q(t) by a sequence qn(t) E 'C1(LI), n = 1,2, ... , in the topology of the almost uniform convergence in 'C(LI). Let 2i with

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142 CHAPTER 3

jeLl be a compact subinterval of LI, containing the point t = O. Then the sEquence {qit)};;'=l is iniformly convergent to q(t) in ii. Since ~(J) is a Banach space, then using the usual methods of the theory of Banach spaces, we can easily see that the operator sequences {Tn}:=l and {Lo.n }~=l' which correspond to the functions of the sequence {qn}:=l' are convergentto the

restrictions of T and Lo to ~(J). Then, in order to establish (19) in ~(LI), we let n-= in TnLo,nf=12Tnf. 0

Later on, for definiteness, we suppose that the arbitrary function a(t) in (17) is chosen to be O.

The 0 rem 3.2.1. The operation o

(20) f * g= T-l[(Tf) * (rg)],

where * denotes the Duhamel convolution in ~(LI), is a continuous convo­lution of the initial value right inverse operator Lo of D=(d2jdt2)-q(t) in

o ~(LI), and the representation Lf = u2 * f holds.

The proof follows immediately from Theorem 1.3.6. The continuity of (20) is ensured by the continuity of the Duhamel convolution and by the

o continuity of the transformations T and T-l. The representation Lf = U2 * f follows directly from the representation l2f = {t} * j and from u2 = T-l{ t}. 0

The 0 rem 3.2.2. The convolution quotients ring of ~(LI) with respect to convolution (20) is isomorphic to the Mikusitiski ring for ~(LI).

Proof. It is easy to see that the similarity T is an algebra isomor-o

phism of the convolutional algebras (~(LI), *) and (~(LI), *). Therefore, from the results of 1.4.3, their convolution quotients rings are isomorphic. 0

Hence, there are two ways for developing operational calculus for the operator Lo· One of them is to proceed directly, using convolution (20) and finding directly the corresponding partial fractions. The other approach con­sists in transferring all the facts of the classical operational calculus to facts of the operational calculus we are to develop. By means of convolu­tion (20) a representation of all continuous linear operators M: ~(LI)-~(LI), with an invariant subspace

(21 ) ~g(LI)={fE ~2(LI), f(O)=f'(O)=O},

which commute with D = (d2jdt2)-q(t) in ~g(LI), can be found (for details see B 0 Z hi nov, Dim 0 v ski [201). Here for the sake of simplicity we confine ourselves to the simpler case of the one-sided interval LI = [0, =).

The 0 rem 3.2.3. A continuous linear operator M: ~([O, = »-~([O, = », which maps ~WO, =» into itself, commutes with D = (d2jdf2)-q(t) in ~6([0, = », iff it admits a representation of the form

(22)

where m E ~1([0, co», and m' is a function with locally bounded 'variation in [0, =).

Proof. It is easy to see that the problem for finding the operators M which satiSfy the hypothesis of the theorem is equivalent with the problem

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CONVOLUTIONS CONNECTED wrrn SECONJ).()RDER LINEAR DIFFERENTIAL OPERATORS 143

for finding the commutant of Lo in tt!f([O, = ». By means of the similarity T, the last problem is reduced to the problem of finding the commutant of l2 in tt!f([0, 00». In 1.1.3 it was shown that a linear continuous operator M: tt!f(LJ)-tt!f(LJ) commutes with the Volterra integration operator l in tt!f(LJ) iff it admits a representation of the form

(23)

where m=M{I} is a function with a locally bounded variation in [0, 00). Since in the case of the one-sided interval the commutants of land l2. co­incide (see B 0 Z hi nov. Dim 0 v ski [20]), it is clear that (23) represents the commutant of l2 in tt!f([0, 00». Let M: tt!f([0, 00 »-tt!f([O, 00» be an ,arbi­trary continuous operator, which commutes with Lo in tt!f([0, 00». Let us consider the continuous linear operator M = T MT-l, which obviously com­mutes with l2 in tt!f([0, 00». Therefore a representation of the form (23) holds for M. Since:(lm* Tf)(O)=(lm* Tf)'(O) =0, then from formula (11) we get

_ _ 0

Mf = T-IMT f= T-l (d2jdt2)[l m* Tfj =DT-l[(Tm) * (Tf)] =D[ m *f],

where we have denoted m= T-l(lm). Obviously, m E ~l([O, 00». It is not so obvious that m' is a function with locally bounded variation, when q E tt!f([0,00». Then the kernel K(t, r) is not from the tt!f2-class. But this assertion follows easily from t1;te identity

t

(24) (Tf),=/,(t)+K(t, t)f(t) + fI<.tCt, r)f(r)dr. o

Since aCt) is taken to be 0, from integral equation (17) it can be seen that

(25) Kt(t, r)=-+q(-!~~)+i-q(t-;$)+F(t, r),

where F(t, r) is a smooth function from tt!f1(O). But it can be shown directly by a linear substitution that the functions

t t

Iqe~$ )f(r)dr, f q e~$) f(r)dr o 0

are smooth ones, provided f E tt!fl([O, = ». If /' is a function with locally bounded variation, then the function Tf is a smooth one and (Tf), isa function with locally bounded variation, and vice versa. 0

In the case considered, when LJ = [0, 00). an integral transformation can be defined, which is to play the same role as the role played by the Laplace transform in the classical operational calculus. Let tt!fexp denote the subspace of tt!f([O, 00», consisting of the functions which are O(eat) for t--++ 00,

and let tt!fT = T-l(~exp). Then we define the integral transform 00

(26) ~{f; p}= fe-Pt(TfXt)dt o

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144 CHAPTER 3

00

as the composition of the Laplace transform ~{f; p} = f e-pt f (t)dt and the

o similarity T.

The 0 rem 3.2.4. For arbitrary f, g E rri T the identities

1 (27) St{Lof; p} =~ p2 StU; p}

and o

(28) StU * g; p} =StU; p} St {g; p}

hold. Proof. Since f, gErriT, then Tf, TgErriexp and hence Tf'!'TgECCexp •

o Therefore f * g E rri T· Then

o StU * g; p} = ~o T[T-l«T f) * (T g))],

=~«Tf)*(Tg))=(~oTf).(~oTg)=St{f; p}.St{g; pl.

Thus (28) is proved. Then (27) can be obtained as a special case of (28). o

Indeed, since by (21) Lf = u2 * f E rri T' due to u2 = T-l{ t} E rri T' then, using the properties of the Laplace transform, we get

1 1 ~{Lf; p}=~{TLf; p}=~WTf; p}= p2 ~{Tf; p}= p2 ~{f; p}. 0

Cor 0 II a r y. Let ker Dc rri T and let f be such a function from rri2([0, 0':»)) that Df E rri T. Then f E rri T' and the identity

(29) St{Df; p}=p2St{f; p}-f(0)St{u1 ; p}-f'(0)~{U2; p}

holds. The proof follows from the identity LDf = f-f(0)u 1-f(0)u2 and

from (27). In order to be able to verify effectively the hypothesis of the above

theorem, the following remark can be useful. If

(30)

t

o~~)f q(r)dr/< ec,

then the similarity T maps rriexp into itself, and hence rriT = rriexp• In order to show this, we may use an estimate for the kernel K(t, r) of T, given in Marchenko [74], p. 26.

3.2.2. Convolutions of the initial value right inverses of the Bessel diffeFentic't1 operators. Next we shall consider a classical example of a linear differential operator of the second order with a singularity. We shall find convolutions of the initial value right inverse of the Bessel differential ope­rator

(31) d2 Id ,,2 d ,d

B. = dt2 + T dt - t2 = t-v- 1 dt t2' T 1 dt t-v

with v:;;; 0 for the singular point t = O.

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CONVOLUTIONS CONNECI'ED WITH SECOND-ORDER UNBAR DIFFERENTIAL OPERATORS 145

The basic fundional space in which we shall define the initial value right inverse Lv of Bv consists of the functions t(i) , defined for O<i< = and having a representation of the form

(32) f(t) = tP lct) with a function ~f(i), defined and continuous for O~i< 00 and with an arbi­trary p>v-2- We denote this space by 'i&'v-2. If f(i) E'i&'k, then the space of such fj1nctions (32) is denoted by 'i&':_2.

D e fin i t ion 1. The initial value righi inverse of Bv is said to be the linear operator Lv: 'i&' v-2---->'i&'; such that for f E 'i&' v-2' the function L;j(t) is the solution of the initial value problem

(33) BvLv/et) = f(t), lim t-v Lv/et) = 0. t->+O

It is easy to see that Lv has a representation of the form

1 1

(34) Lv/ (t) = ~ f J t1v/2 t2/2 f(t~ilt2)dtldi2. o 0

Let us find the defining projector Fv of Lv. As we have seen in 1.3.4, the restriction of each defining projector of Lv onto the subdomain 'i&'; of the domain of Bv is

(35) Ff(t) = (I - LvBv)f(t) = iv lim -,;-vf( r) = 0, .->+0

i. e. Lv and Bv are mutually inverse on 'i&'~- It is better to consider Fv -2

on the space qj v of the functions of the form

(36) f(i) = f Jv + J(t)

with a constant fo and with a function Jet) E 'i&';. If fE 'i&';, then

(37') Fv/et) = fotv with fo= lim -,;-vf(r). .->+0

The 0 rem 3.2.5 (D i m 0 v ski [43]) The linear iransform Tv: 'i&'v-2 [v+I/2]

--> 'i&'2v_1 C'i&' __ I, defined by 1

tv+1 f -(38) Tv/(t) = 1'(,,+1/2) (1-r)v-I/2r-v/2f(t ~r )dr

is a similarity of Lv io l2, i. e.

(39)

o

t

where I is ihe Volterra integration operator If(t) = ff(r)dr. o

Proof. Let us note that here the operator [2 is considered as defined not on its "natural" domain'i&'_l([O, 00)) but on the subspace 'i&'~v"!=."2] of

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146 CHAPTER 3

f(f _1([0, 00). We verify the desired identity T vL./(t) = [2T,f(t) at first for powers of t, i. e. for functions of the fbrmf(t)=tP with p>y-2. Then let

n

f(9 = tPf(t) with p>y-2 and withJE ~([O, oo». Let f:(t) = ~ anktk, n = 1 k=O

2, .. ', be an almost uniformly convergent sequence to J (t) in ~([O, oo]). If fn(t)=tplnCt),'then TvL./nCt)=l2T./nCt) for n= 1, 2, .... For n--=, we get relation (39). D

In non-explicit form transformation (38) with I Y I < 1/2 can be seen in

the so-called "Sonine integrals", which transform Bessel functions into sine and cosine. That is why, following Del sa r t e [28J, p. 439, we name (38) the Sonine transform. .

L e m m a 4. The Sonine transform Tp is invertible, and its inverse transform T;I has a representation of the form

(40) r (1 d )V+I/2

T;lf(t) = 2v+ 1/2 7 dt {J(t)}

when Y+ 1/2 is an integer, i. e. when y is half of an odd integer, and in the form

1 -I tV (~ ~)[v+l/2J+I r -{ v+l/2} ,-

(41) Tvf(t)=2[y+I/2]t-lr(1_{v+l(2}) t dt . (l---r) f(tyr)dr o

whrn y+ 1/2 is not an integer, where by [')1+ 1/2J and {')1+ 1/2} are denoted the integer a-td the fr.lciional paris of ')1+ 1/2 correspondingly. In the first case T-I is deFined on ~v+l/2, and in tke second in ~[v+l/21+1 v J' 2v-1 2v-1 .

Proof. The invertibility of Tv is a simple consequence from the Titch­marsh theorem on convolution (see 1.1). Indeed the equation Tv{g(t)} = 0 can be written in the form {tv-l/2} * {t- vI2 g(JF)} =0, where * denotes the Duhamel convolution. Therefore, g( J{) = 0, even when we consider * in a finite interval [0, Aj, since t,,-112 does not vanish in any right neighbourhood of t = O. The function T;lf(t) = get) we search as the solution of the first kind integral eq.uation r,1J.=f(t), whic'l can easily be written in the form [v+112{t-v/2 y (Jt )} =J(.jt ). The formal solution of this equation is

y(JF) = t Vi2(d: y+112{J(.jI)},

where (d/dt)v+I/2 is the operator of fractional differentiation of order ')1+ 1/2 when y+ 1/2 is not an integer, and of ordinary differentiation of order ')1+ 1/2, when it is an integer. Formula (41) is obtained using the properties of the operator (d/dt)v+112. D

Having at our disposal the similarity T v of Lv to [2 and using the re­sults of 1.3.3, we can at once write down a convolution of Lv in f(fv-2. This is the operation

v

(42) (f * g)(t) = T;I[(T./) * (Tvg)J·

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CONVOLUTIONS CONNECTED wrrn SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS 147

In the following theorem an explicit representation of (42) is given. The 0 rem 3.2.5. Convolution (42) admits a representation of the

form

(43)

with 1 1

(44) Uxg)(t)=t2 J n~:g=~~r2 f(tM)g(tV(l-t1)(1-t2))dt1dt2

o 0

and with the operator A.: ~2.-2->~3"-2' defined by 1

(I J;; t. 2·+ 1r(v)

A.h(t)= {

J(1-T).-lT-'hCtV-;)dT for 1'>0,

(45) o

l J; h(t) for 1'=0.

Proof. We verify identity (43) at first for powers of t, i. e. for func­tions of the form f(t)=tP and g(t)=tq with p, q>1'-2. If now f, gE~'--2 are arbitrary functions of the form f(t) = tPICt) and g(t) = tqgct) with} (t) g(t)E ~ ([0, 00)), then we take polynomial sequences (fn(t)}~l and {gn(t)}.~'=l

converging almost uniformly to let) and get). correspondingly. If n -> 00 in

the already proved identity fn; gn =A.(f~Xgn) with fnCt)=tP [,,(t) and gnCt) = tq gn(t) , then (43) follows using the continuity of A. and X.

Representation (43) is an interesting one, Sll1Ce it contains fewer inte­grations than (42).

Now, having at our disposal the similarity T" of L. to l2 along with the explicit representation (43) of the convolution (42) of L. in ~.-2' with­out any difficulties the elements of the corresponding operational calculus can be developed. Let S. stand for the inverse element of L. in the mul­tiplier quotients ring SJJl:. of convolution (42), i. e. S.= l/L •. Let us find the connection between B. and S •. From expression (35) of the defining projec-

-2 tor F. of L. in ~'" we get . (46) BJ=SJ-foS.{t·}

with fo=lim C> f(T). For an effective use of the corresponding operational .--.+0

calculus, we need explicit expressions for the partial fractions of the form S.{tv}/(S,,-,lt. It is not difficult to show that

(47) ~:~J. = {J.cRt} * and

(48) S.{tV}

(S. -J.)n ----J( '-It) * for n~1. { 1 an- 1 -}

(n-l)! aJ.n- 1 • V

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148 CHAPTER 3

E x amp I e. In axially symmetric problems of mathematical physics an im­portant role is played by the Bessel differential operator

(49) d2 1 d Bo= dt2 +t di

with y = O. We are interested in the initial value right inverse operator Lo of Bo, given by

(50)

Then the corresponding Sonine transformation, serving as a similarity of Lo to 12, has the form

t

(51) Tol(t)=:' r 'tf(-.:)d't • Jot. J t2_'t2

o

Since in this case ..10 from (45) is a multiplication with the constant /;;;/2, i. e. AJz(t) = (J-;;/2)h(t), then convolution (43) of Lo in '1&'_2 takes the form

1 1

(52) (f; g)(t) = ~ t2 f f t(tJ t1t2 ) get J(1-t1) (1-t2» dt1dt2•

o 0

Here the defining projector Fo of Lo is more appropriate to be consi­dered in the space ~6 of the functions of the form

I(t) = 10 In t+ let) with constants f 0 and with I (t) E '1&'6. Then

(53) Fo/(t) = 10 In t with 10 = lim f(.:). <-->+0 In r

Two types of partial fractions with So are of interest:

(54)

and

(55) s~~~}~} ={~ Yo(J-At)+ (In J~-;.-Y ) Jo(J-2t)} * ,

where y=0.577215 ... is the Euler constant. The corresponding partial fractions 1/(So-lt and So{ln t}/(So-lt with

integer n> 1, can be obtained from (54) and (55) by a formal differentia­tion with respect to l.

In the same way as in the previous subsection, an integral transforma­tion connected with the initial value right inverse operator of the general non-singular linear differential operator of the second order had been in-

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CONVOLUTIONS CONNECTED WITH SECOND·ORDER LINEAR DIFFERENTIAL OPERATORS 149

troduced, as for the Bessel differential operator there exists analogous in­tegral transformation. This is the Meijer-Obrechkoff integral transformation (0 b r e c h k 0 ff [83]).

De fin i t ion. The Meijer-Obrechkoff transformation is said to be the composition of the Laplace transform and the Sonine operator (38), i. e. the integral transformation

00

(.56) ~,,{f(t); p}= f e-Pt(T,,J)(t)dt. o

We consider the Meijer-Obrechkoff transformation in the space t6'!~2 of the functions of t6','_2' which are O(eat ) for t -+ 00 with real a.

Taking into account some elementary properties of the Laplace 'trans­form, and identities (39) and (42), it is easy to show t see Dim 0 v ski [33]) the following operational properties of the Meijer-Obrechkoff integral trans­formation:

(57)

and

(58)

1 ~.{L.t(t); p}= p2 ~,.{j(t); p}

~v{(f;g)(t); p}=~v{f(t); p} ~v{g(t); pl·

Usually, Meijer transformation is said to be the integral transfor­mation

00

~,. { tU); p} = f tK.( pt) f(t) dt. o

In 1954 Obrechkoff [83], p. 85, studied its modification 00

~,. {f(t); p} = p-v! tK.(pt) f(t)dt.

We shall show that this modification up to a numerical multiplier coincides with (56).

The 0 rem 3.2.6. The representation 00

(59) 2v+1 r ~v {j(t); p} = .j; p-v. tK.(pt)f(t) dt, o

where K.(x) is the MacDonald function, holds. Proof. First, we shall show the validity of (59) for real p>O, and

then its validity for complex p shall follow from the principle of analytic conti­nuation. Writing the Sonine transformationT. in the form

t

(60) T f(t) = 2 [Ct2-r2)V-1/2rl-"fi(r)dr. • ,r(,,+1/2) . .

o and changing the order of the integration in (56), we get

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150 CHAPTER 3

OC 00

~p{!(t); p}= r(,,11{2)f r1-"!(r)dtf (i2-r2)v-l/2e-Pl dt U T

It remains to use the well-known integral representation 00

T(y+ 1/2)K,(z) = V;:; (~r f e-zt(t2_1)v-l/2 dt, Re 1'> - ~, Re z>O 1

of the function Kv(z) (see E r del y i et al. [56], p. 82) in order to get (59). The Meijer-Obrechkoff transformation can be used for a functional de­

velopment of the operational calculus for the initial value right inverse ope­rator Lv of Bv.

A convolution of the integral transformation 00

i.{f(t); p}= 2 f (pt),·/2K,.(2Vpt)!(t)dt, o

related to the Meijer-Obrechkoff transformation, had been found by l( r it t z e I [71] in 1965.

3.3. CONVOLUTIONS OF BOUNDARY VALUE RIGHT INVERSES OF LINEAR SECOND-ORDER DIFFERENTIAL OPERATORS

In this section convolutions for some general classes of right inverse operators of an arbitrary non-singular linear differential operator are found. The boundary value problems connected with these right inverse operators include as a special case the classical Sturm-Liouville eigenvalue problem and boundary value problems with non-local boundary condition. That is why specializations of these convolutions can be used as convolutions for the finite Sturm-Liouville integral transformations. As an example of a sin­gular second-order linear differential operator the Bessel differential opera­tor is considered. Thus convolutions for the classical finite Bessel transfor­mations are found.

3.3.1. Convolutions of right inverses of non-singular second-order linear differential operator, determined by a Sturm-Liouville and a ge­neral boundary value conditions. As in the first part of the previous sec­tion we consider again an arbitrary linear differential operator of the se­condorder

(1)

in a segment J=[a, b] under the assumption that

a(x)Ect2(J), b(x)Ect1(J) and c(x)Ect(J).

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CONVOLUTIONS CONNECTED WITH SECOND·ORDER UNBAR DIFFERENTIAL OPERATORS 151

Let cIJ: ~le ,1) -> C be an arbitrary non-zero linear functional in fi&'l( ,1) continuous in the norm of ~1(L1): IIfllcl=max {lfex)I+lf'(x)I}. The right

xfA inverse operator L: ~(L1) -> ~(L1) of D in ~(L1) we are to consider, is defined in such a way that y=L f is the solution of the elementary boundary value problem

(2) Dy=f, ay(a)+.By'(a)=O, cIJ(y)=O,

where a and .B are numbers which are not both zero. In order problem (2) to have a solution for each fE~([a, bj), it is necessary and sufficient cIJ(aY2-.BY1)=j=0, where Yl' Y2 is the fundamental system of D for the point x=a, i. e. DYi=O, i= 1,2; Yia)= l,y;(a)=O,Yia)=O, y;(a)= 1. Then the right inverse L of D we are interested in can be represented in the form

(3) Lf=Lof-y cIJ(Lof),

where Lo is the initial value right in verse of D for the point x = a, i. e. Lo is defined by DLof=f and (LofHa)=(Loj)'(a)=O. Here Y stands for (aY2-{JYl)/cIJ{aY2- .BYl}·

Here we shall show that L has a non-trivial convolution in ~([a, bD and we shall find an explicit representation of such a convolution. In this we follow B 0 z hi nov and Dim 0 v ski [18, 19]. Using the similarity method, we reduce this problem to an analogous problem, but for the square of the differen­tiation, already solved in sect. 3.1. As in the previous section, it can easily be seen that instead of the general operator (1) we can consider, without an essential loss of generality, the operator

( 4) d2

D = dt2 -q(t), q E ~([O, 1 D,

in the standard segment [0, 11. Indeed, after the analogous change of the x b

independent variable by t=lP(X)=! ,d; I r ,d; ,the segment [a, b] goes a vat;) a vam

into [0,1]. Further, by a well-known substitution for the function, the coef­ficient of d/dx in the operator D can vanish. Thus we get an operator of the form (4), similar to the given operator (1) by a similarity of the change of the v ar iables type.

Henceforth we consider operator (4) and boundary value problem

(5) DLf(t)= f(t), a(Lf) (O)+.B(Lf)'(O)=O, cIJ(Lf) =0,

for the right inverse L of D in ~([O, 1 D. Here cIJ is again a non-zero linear functional in fi&'1([0, 1 D, continuous in the norm

(6) IIfllct= max {If(x)I+If'(x) I}. O;'i;x;'i;!

We suppose that if ul(t), u2(t) form the fundamental system for D in the point t = 0, then the condition w = cIJ(.Bu l -au2) =j= ° for solvability of (5) is fulfilled. If we denote u(t) = (.Bu i -au2)/w, then L has the representation

(7) Lf(t)=Lof(t)-u(t) cIJ(Lof),

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152 CHAPTER 3

where Lo is the initial value right inverse of D for the point t=O. We shall show that L is similar to a right inverse operator of d2jdt2 of the type considered in sect. 3.1. To this end, we should distinguish the two cases: (a) P=FO; (b) p=O.

(a) tJ=FO. Lemma 1.IfP=FO, then there exists a functionK(t,T),KEC{f'l(O}

'where 0= {(t, T): 0~·-.t::;-1, Occ;c:-t}, such that the second kind Volterra trans­formation

t

(8) T f(t) = f(t)+ J K(t, T)f(T)dr ()

is a similarity of L, given by (7) to the right in'verse operator'

(9)

of d2!di2 with (p = cj) 0 T-l, i. e. TL = I T. Proof. As in 3.2.l, at first we consider the case of smooth q(t), i. e.

q E C{f'l([O, 1 D. Then let K(t, r) be the solution of the partial differential equation

(10) Ktt - KH + q( r)K = 0

in the domain 0,-,= {(t, T):O~.::::t:;c;l, O;:;:r;:~t} and satisfying the boundary va­lue conditions

t

(11 ) K(t, t)= -h -~ j q(r)dr ()

and

(12) K.(t, O)-hK(t, 0) = 0

with h= -a/p. In the same way as in 3.2.1 the characteristic substitution U=(t+r)j2, v=(t-r)j2, reduces the problem (10), (11), (12) to the problem

kuv = -q(u-v)k,

u

(13) k(ll, 0)= -h-~ J q(~)d;, o

ku(u, u)-kv(u, u)-hk(u, u)=O

for the function k(u, v)=K(u+v, u-v). By means of successive integration on u and v of the partial differential

equation (13), and using the boundary conditions, we get the following second kind Volterra integral equation

v u

(14) k(ll, v) = -he-2hll-e2hVI e2h~ q(!;)d$ - ~ jq($)d; () v

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CONVOLUTIONS CONNECTED WITH SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS 153

'V ~ U 'V

_e2h'O f e2h; d; J q(; - 'Y})k(;, 'Y}) d'Y} - J d; J q(;-'Y})k(;, 'Y})d'Y}. o 0 '0 0

Since this integral equation always has a solution k(u, v) in the whole do­main, then in reverse order it can be seen that the function K(t, 1") =k(t+1")/2, (t-r)/2) is a solution of boundary value problem (10), (11), (12). In the same way as in 3.2.1, it is seen that for each I E ~2([0, 1]) the identity

(15) TD I(t) = !: T l(t)-(f(O)-hl(O»K(t, 0)

holds. Substituting LI(t) instead of I(t), and using the defining condition (Lf)'(O)-h(LI){O)=O for L, we get

(16) d2 dt! T L I(t) = T /(t).

From (16) the similarity relation TL=LT follows immediately. Thus the assertion is proved in the case P=I=O. In the same way as in the proof of Lemma 3 in sect. 3.2 it is seen that in this case the integral equation (15) has a smooth solution k(u, v) in the corresponding domain and that trans­formation (8) with the kernel K(t, r)=k«t+r)/2, (t-r)/2) is the desired si­milarity. This can be shown in a standard manner by approximation of q(t) by functions qn(t)E CCI ([O, 1]) in the topology of CC ([0, 1 D. 0

Theorem 3.3.1. The operation t

(17) U*g)(t)=jl ol (J(T!)(t - rXTg)(r)dr ) o

x

-2~ Wx{ TiloT-;lolx(J(Tf)(x+t-r)(Tg)(r)dr 1

x

+ J(TI)(lx-t-rl)(Tg)(lrl)dr)} -I

is a convolution 01 L in CC ([0, 1 D. Proof. By Theorem 1.3.6, the operation 1* g= jI[(T!) :; (Tg)], where

t T x E

(f-; g)(t) = I d1" II(r-a)g(a)da- 2~ Wx {T-;l I d;(jIU+ t - r)g(1")dr o 0 0 t

x

+ J 1(1 x-t-1" I)g(l1" I) dr)} is a continuous convolution of (9) in ~ ([0, 1 D -t

Obviously, it coincides with operation (17). 0 Cor 0 II a r y 1. The operator L has the representation

(18) L} ={-; u(t) }*I

as a convolutional operator.

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154 CHAPTER 3

Indeed, since Tu(t) = -/J/w=const, and Ij={l}" f, then L f=(T-l {l})

*f={- (; u }*f. (b) /J=O. Then the right inverse L of D is defined as the solution of the boun­

dary value problem

(19) DLf(t)=f(t), (Lf)(O)=O, r[J(LJ)=O

and it has the representation

(20) L f(t) = Lof(t)-U2(t) cP (Lof),

where without a loss of generality it is assumed that cP (u2) = 1. Let us re­mind that u2(t) is the solution of Du=O with u(O)=O, u'(O)=1.

L e m m a 2. If q(t) E ~ ([0, 1 j), then there exists a function K(t, r:) from ~l(O), where O={(t, r:): O;;t;::; 1, OSr:;£t}, such that the second kind Volterra transformation

t

(21 ) Tf(t) =f(t) + I K(t, r:)f(r:)dr: o

is a sinilarity of the operator L to the right inverse operator

(22)

oj d2/dt2. Proof. As in Lemma 1, first we consider the case q(t) E ~1([0, 1]) and

define K(t, r:) as the solution of the boundary value problem

Ktt-Kn + q(r:)K= 0 (23)

t

K(t, t)= - { J q(l;)dl;, Klt,O)=O o

in the domain O. In the same way problem (23) is reducible to the second kind V olterra integral equation

II 'lI

(24) k(u, v)= --} I q(l;)dl;- { J q(l;)dl; o 0

'lI I: II /: -- I dl; J q(I;-1])k(l;, 1])d1]- J dl; I Qll;-1])k(l;, 1])d1] () 0 'lI 0

for the function k(u, v)=K(u+v, u-v). But (24) has a smooth solution for q(t) E ~ ([0, 1 ])too. Then the function KCt, r:) = k«t + r:)/2, (t -r:)/2) is the kernel of a similarity (21) of the desired kind. 0

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CONVOLUTIONS CONNECI'ED WITH SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS ISS

The 0 rem 3.3.2. The operation x

(25) (f*g)(t)= - 4 4>x{Tt l 0 T;l olx (J (Tf)(x+t-r) (Tg)(r)dr t

x

---J (Tf)(1 x - t - r I)(Tg) (I tI) sgn «x - t-r)r)dr)} --t

is a continuous convolution of the operator (20) in ~ ([ 0, 1 D. Proof. By Theorem 1.3.6, the operation f*g=T-l[(TfF(Tg)],'with

x x

(f *' g)(t)=-44>xoT;lolx{J j(x+t-r)g(r)dr- J fClx-t-rl)g(lrI)X t -t

X sgn«x-t-r)r)dr} is a convolution of L in ~ ([0, 1 D. This is exactly ope­

ration (25). 0 Cor 0 11 a r y. The operator L has the convolutional representation

(26)

The convolutions (17) and (25) give a complete solution of the problem stated in the beginning of this section. They can be used for developing operational calculi for corresponding boundary value problems. If S= I/L is the inverse element of L in the multiplier quotients ring of a convolution of the kind considered, then the first problem in the operational calculus for L is the establishing of the relation between Df and Sf for arbitrary fE~2([0, 1 D.

Theorem 3.3.3. Let fE~2([0, 1]). Then in the multiplier quotients rings of the convolutions (17) and (25) the relation

(27) Df=Sf- ! 4>(f)-[af(O)+p/,(O)]S{v(t)},

(28) Df=Sf-4> (j)-f(O)S {v(t)},

with v(t)=u1(t)-4>(U1)U2(t) when p=O, holds. Proof. It is easy to see that in both cases the defining projector F­

of the right inverse operator L of D has the form

(29) Pf= 4> (f)u+[af(O)+ p/,(O)]v,

where in the first case u=(pu1-au2)/w, v=(4)(U1)U2-4>(U2)U1)/W, and u=u2,

V=Ul -U24>(Ul ) in the second case. Taking into account representations (18) and (26), the desired relations (27) and (28) are obtained at once from (29). 0

Example. Let 4>(f)="f(I)+~/,(I). Then the right inverse operator L of D is closely connected with the classical Sturm-Liouville problem. In the next subsection we shall consider in more detail some applications of the convolutions obtained for this problem.

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156 CHAPTER 3

Now we shall show briefly the possibility of extending the convolu­tions (17) and (25) as continuous convolutions of the operators (9) and (20) in the whole space !l! ([0, 1 D.

L e m m a 3. Let tP be a bounded operator in t;6' ([0, 1 D. Then the ope­ration

t

(30) (1* gXt) = tPx{ J f(t+x-r)g(r)dr} x

admits an estimate o! the form

(31) 11!*glll:::;A 1I!111 .lIgII1 1

for f, gEt;6'([O, 1]), with A=lItPllc([O,ll), where 1I!1I1= JI!(r)ldr denotes the o

integral norm. I

Proof. If tP(j)= If (r)dfJ{ r) is the Riesz representation of tP with a o

I

complex measure ,u(r)=,uI(r)+i,u2(r), then let us denote ItPl(f)=I !(r)dl,ul(r) o

with 1,uI(r)=I,ull(r)+i/,u2!(r), where 1,u1 1(r) and 1ft2I(r) stand for the va­riation of /hI and ,u2 on [0, r]. Then we get

1 I x

1I!*glll = II(f*g)l(t)dtsltPlx{I sgn(t -x)dt r If(t+x-r)I.lg(r)ldr} o 0 t

x x 1 t

=ltPlx{J dt IIf(x+t-r)I.lg(r)ldr + I dt II!(x+t-r)I.lg(r)ldr} o t x x

x ~ 1 1

~ltPlx{Ilg(r)ldr II!cx+t-r)ldt+ Jlg(r)ldr JI!(x+t-r)ldt}. o 0 x ~

Since • x 1

I lJ(x+t-r) I dt = J lfCu) I du s II!(U) I dU=II!lh o x-. 0

and I t+x-. 1

II!(x+t-r) I dt= I lJ(u) IdUSIlf(U)1 du=II!lI t ,

• x 0

then x 1

!1!*glll~ltPlx{(Jlg(r)ldr+ jlg(r)ldr ).II!lll}=1I!1I1.lIgI11.ltPlxC{1}). o x

But 1 tPx 1(1)=11 tP Ib[o,l)), and hence (31) is proved. 0

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CONVOLUTIONS CONNECfED WlTII SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS 157

The 0 rem 3.3.4. If * is one of operations (17) or (25), then for f, gE '6' ([0, I D the estimate

(32) Ilf * g IIl;SA IIf 11 1 .11 gill'

with a sllitable constant A, holds. The operations (17) and (25) are ex­tendable on the space 2([0, I]) as continllolls convolutions of the corres­ponding operator L.

Proof. By the expressions of (17) and (25), using the boundedness of the similarities (8) and (21), considered as opera tors on 2 ([0, I D, the above estimate (31) can be obtained directly from (31). The last assertion of the theorem is an immediate consequence of the estimate (32). 0

3.3.2. Convolutions of the finite Sturm-Liouville integral transfor­mations. Now let D=(d2jdt2)-q(t) with a real-valued q E '6' ([0.1 D. A Sturm-Liouville right inverse of D is named the operator L:~' ([0, 1])---+ '6'2([0, 1 D, defined by the solution y=Lf of the elementary boundary value problem

(33)

Dy=f,

Nu.(Y) = cos ay(O)+sin ay'(O) = 0,

N p(y)= cos fJ y(l )+sin fJ y'(I)= 0.

We assume that 1= ° is not an eigenvalue of the Sturm-Liouville eigenvalue problem

Dy-ly=O (34)

N,,(Y) = 0, NP(y) = 0,

which is equivalent to the requirement L to exist on '6' ([0, 1 D. It is well known that the Sturm-Liouville eigenvalue problem (34) has

an infinite system of eigenvalues 11, 12, ••• , l,Z' ... , and to it corresponds a total system of eigenfunctions ({JI' ({J2' . .. , ({J1l' ••• It is total in the sense

1

that if for a function jE21 ([0, 1]) we have I j(t)({JIl(t)dt=O, for n=l, 2, 3, o

... , then f= 0 almost everywhere in [0, 1]. In particular, if jE '6' ([0, 1 j), then I~ ° identically. By means of the total system of the eigenfunctions of the Sturm-Liouville problem (34) the finite Sturm-Liouville integral transformation is introduced (see C h u r chi 11 [25], p. 325).

o e fin i t ion. A finite Sturm-Liouville integral transformation is said to be the correspondence

I

(35) Til {f}= I j(t) ({In(t)dt, ll= 1,2,3, ... , o

which associates the Fourier coefficients sequence of each IE '6' ([0, 1 J) with respect to the eigenfunctions ({In' II = 1, 2, ... , of the Sturm-Liouville eigen­value problem (34).

If ({In is the eigenfunction of (34), corresponding to the eigenvalue 11l ,

then ({In is an eigenfunction of L, but with the eigenvalue Ijln' L e.

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158 CHAPTER 3

(36)

From Theorem 1.3.3 it follows that ({In is a divisor of zero of each convo­lution of L, and a divisor of zero in particular of convolutions (17) and (25). From Theorem 1.3.4 we have

(37) ({Jm * ({In= 0, for m=!=n,

for each convolution * of L. The basic operational property of the finite Sturm-Liouville integral

transformation, connected with the eigenvalue problem (34), is the relation

(38) Tn {Df}=lnTn {f}+N~ {({In}N, {f}-N;{({Jn}Np{f}

with

(39) N~(f)= -sin a/(O)+ cos a/,(O),

NI. f)= -sin PI(l)+cos P/,(l)

(see C h u r chi 11 [25], p. 327). Substituting LI(t) instead of I(t) in (38) we get

(40)

Hence the finite Sturm-Liouville integral transformation "algebraizes" the cor­responding Sturm-Liouville right inverse of D.

For the sake of definiteness of the following considerations, we shall confine ourselves to the case cos a=!=O to which the convolution (17) cor­responds.

L e m rna 4. Por each lEI/&' ([0, 1]) the identities

(41) j * ({In= Xn (f)({Jn' n= 1, 2,3, ... ,

hold. Here {xn} is a system oj linear junctionals which are multiplicative with respect to convolution (17) oj the operator L, i. e.

(42) Xn(f * g)=Xn(f)xn(g), n=1,2, ....

The system {Xn} is biorthogonal to the eigenjunctions system {({Jm}, i. e. Xn«({Jm)=O lor n=!=m, and Xn«({Jn)=!=O lor each n.

Proof. From L({Jn=+-({Jn it follows L(f * ({In)=j*(L({Jn)=+ (f*({Jn),L e. An An

that either 1* ({In = 0, or j * ({In is an eigenfunction of L with the same eigen­value 1/1n. Since the eigensubspace corresponding to each eigenvalue of the problem (34) is one-dimensional, then there exist constants Xn=Xn{f), such that (41) holds. Obviously, Xntf) are linear continuous fundionals in 1/&'([0,1]). Using (41) several times, we get (j * g) * ({In = Xn( j * g)C:Pn = j * (g * ((In) = I * (Xn(g)C:Pn) = Xn(g) (I * ((In) = Xn(g)Xn( f) ({In' whence (42) follows immediately. Now we shall show that ({In*({Jn=!=O for each n=1,2, ... , i. e. that xnC({Jn)=fO for n=l, 2, .... Indeed, let us assume the contrary, i. e. that there exists an no with ({In. * ({In, = O. Since from (37) it follows that ({Jm * ({In. = 0 for each m=!= no, then j*({Jno=O, for each n= 1,2, .... If I is an arbitrary finite linear combination

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CONVOLUTIONS CONNECTED WTI1l SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS 159

of the eigenfunctions lPn' then j * IP"" = O. Using the well-known denseness in ~ ([ 0, 1]) .of these linear combinations with respect to the integral norm II 111' we get I*IPn,=O for each jE~([O, I)). Since the function u(t), which represents the right inverse operator L of D in (17) is not a divisor of zero of this convolution, then it follows that lPn, == O. But this contra­dicts the assumption that 'PII. is an eigenfunction. From (42) follows Xn(IPm) =0 for m=f=n. 0

L e m maS. For each IE ~ ([0, 1]) the identities

(43) I · T {j} 'l.n(rrn)r:r;n 1 2 liIqJn= n . T{ } , n= , , ... , n ern

hold, with Xn defined as in the previous lemma. Proof. Identities (43) can easily be verified for the eigenfunction 1= qJm

m= 1,2, .... Indeed, if m=f=n, then from (41) it follows that if l=qJw then T n(j) = 0 and 1* qJn = 0, and (43) holds. If m = n, then from (41) it follows that (43) is fulfilled. The identity (43) can be proved in the general case by approximation with respect to the integral norm, in the same way as the approximation was used in the previous lemma. 0

The 0 rem 3.3.4. /j 1* g is (17), then

(44)

jor every I, g E ~ ([0, 1)), i. e. the operation 1* g is a convolution 01 the linite Sturm-Liouville integral transjormation.

Proof. By Lemma 5 it is clear that

Xn(f) =Tn{f}Xn{ern} . Tn {ern}

Now, using the multiplicativity 'of Xn with respect to the convolution *, we get at once

i. e.

Tn(f )'l.n(ern) Tn(ern)

Tn(g)'l.n(rpn) Tn(f * g)'l.n(rpn) Tn(rpn) Tn(rpn)

Tn(f*g)= Tn(f)Tn(g)Xn(rpn). Tn(rpn)

The still unknown constants can be determined from the representation Lj 1

=u*l. Indeed, u*qJn=LqJn = ;:-IPn and n

Hence Xn( IPn) = l:T~~~~ , and (44) is proved. 0 Quite similar is the result for the corresponding Sturm-Liouville finite

integral transformation in the case /J=O. Thus the problem to find explicit convolutions for the finite Sturm-Liouville integral transformations is solved.

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160 CHAPTER 3

3.3.3. Convolutions of the finite Bessel integraJ transfonnations. The considerations of the previous subsection in general are not applicable to a singular operator D. A classical example is the Bessel differential operator

B - d2 + 1 d ,,2 _ t- -1 d t2 +1 d t-v --'0 v - dt~ t dt --t2 - v dt v dt ,1'~,

considered in the finite interval [0, 1]. For it one cannot consider arbitra­ry eigenvalue problems, as those of Sturm-Liouville. In 3.2.2 we have considered a right inverse of B., defined with an initial condition in the singular point t = 0. The definition of this right inverse operator, which in this subsection will be denoted by Lv,o, is the following. We consider the space '6'.-2 of the functions, defined for O<t< 1 and admitting a representa­tion of the form j(t)=tPJ(t) with p>1'-2 and a continuous function l(t) in O~t:s1. If !E'6'v-2. then we define the function y=L, .. o! as the solution of the initial value problem

BvY = j, lim t-vy(t) = 0, 1 ..... +0

Then L,.,o, as in 3.2.2, has a representation of the form 1 1 v v

(45) L, .. oj(t)= ~ f f t-; 2 t} !(t~tlt2) dt1 dt2· o 0

We have L v •o: '6',.-2 -----+ '6'~, where by '6'; we have denoted the space of the

functions of the form j(t)=tPI(t) with p>v and with Jet) E '6'2 ([0, 1]). Further we consider the most important case of an integer v=n~O only. The general case can be treated in the same way, but using operators of fractional differentiation.

At first, we consider an arbitrary right inverse operator Ln of Bn in Cn- 2 '

defined by a linear functional q; in the space '6' n-2 of the functions of the form !(t) = tP Jet) with p>n-2 and with fct) E '6'([0, 1 D, in such a way that y=Ln! for! E '6'n-2 to be the solution of the boundary value problem

(46) B"y=!, cJi(y)=O. In order problem (46) to be solvable in the whole space '6' n-2' we should assume cJi({xn})=f0' Then, without any loss of generality, we may assume

q;( {xn}) = 1.

Then, the right inverse Ln of Bn, we are interested in, is well defined and it has a representation of the form

(47) Ln! = Ln,o! - cJi(Ln,oJ)W}· Our aim is to show that (47) has a non-trivial convolution in '(;'n-2' and

to find an explicit representation of such a convolution. Here we again rely on the similarity method. In 3.2.2 we have shown that the transformation

1

(48) T f,(t)- tn+! f(l- )"-1/2 -n/2 F,Ct '-) d n1' - r(n+l/2) r r J' yr r,

o

referred to as the Sonine transform, is a similarity of L",o to [2, i. e.

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CONVOLUTIONS CONNECI'ED WITH SECOND-ORDER UNBAR DIFFERENTIAL OPERATORS 161

(49) TnLn,o=l2Tn-

But, unfortunately, T" is not a similarity of L" to a right inverse operator of cf2jdt2, except in the case n = 0, since

The Sonine transform T" can be so modified as to become such a simi­larity.

o e fi nit ion 1. The modified Sonine transform Tn is said to be the transformation

1

(50) Tnf(t) = (Tnf(t))<2n) = ( :t rn{r(~;~/2) f(l-r)n-1 /2r-nI2 f(t~~) dr}. o

It is easily seen that the image T n(CC n-2) of CC n-2 under Tn is a sub­space of CC_1, and

L e m rna 6. The modijied Sonine transform Tn is a similarity of the right inverse Ln of Bm defined by (47) to the right inverse operator of d2jdt2, dejined by

(51 )

22n+l n I where rp= J rp oT-1 is a linear functional in CC-1• ct n

Proof. The only thing we should show is to establish that the functional rp oT;1 is in fact defined on the whole space CC-1• To this end we shall use the inversion formula (41). We get the following inversion formula for Tn:

1

(52) 1';lf(t)= 2n::Jct (+ :tr+ltf(l-r)-112(l2nf)(t~~)dr. o

Since the function l2nf is at least 2n times differentiable, then the inverse transformation f;1 is defined in the whole space CC_ t , and 1'-;;1 (CC-1) C CCn- 2•

Hence, Ln is a well defined operator in CC_1, and t "Ln = ["T ", and thus all is proved.

The 0 rem 3.3.5. The operation

(53) (J*g)(t)= -rpx\(L~,o)x(L~,oM1';l)if;l)t x

[+ I(1'" f)<2n) (x + t-r)(i\g)(21l)(r) dr t

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162 CHAPTER 3

x --} fc Tn j)(2n)c Ix - t - 'dXT ~)(2n)(1 r I)sgn (x-t-r)rdr ]) -I

is a convolution oj the operator Ln in 'l&'n-2' such that

(54) L,J={ 22ni~nl tn } *j.

Proof. By Theorem 1.3.6, the operation j*g=T;;l[(Tnf);'(Tng)], x

where the operation; is given by (1* g)(t) = -(4J 0 I)x { + f j(x+ t -. r)g (r)dr 1

x

--{- f j(lx-t-rl)g(lrl)sgn(x- t- r)rdr} is a convolution of Ln in 'l&'n-2· -I

But (53) is only the explicit form of this convolution. The representation (54) is a consequence of the representation In/={t} * j, established by Theo­rem 3.1.3.

E x amp I e. Let us consider the simplest but important special case n=O. For the sake of simplicity, we confine ourselves to the case 4J(j) =/(1). For n=O the Sonine transform To, which now has the form

(55) 1

TojCt) = _~f_~fJ';>-dr, In JtL.,;2 o

is a similarity of Lo to the right inverse 1

(56) - 2t f Lo/(t) = 12 j(t)--;: arccos r/(r)dr o

of d2fdt2• Then convolution (53) takes the simpler form

1 x

(57) (f * g)(t) = -2":--; f( Tol)/{I(Tof)(x+t-7:)(ToB)(r)dr

where

o 1

x

-f(To/)(1 x - t - r I) (Tog) (I rl ) sgn (x - t-r) rdr} I dx , v1- x2

-I

t

T-l /(t)=~- ~JUf(U)dU o J:n; t dt JtLu2

o

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CONVOLUTIONS CONNECfED WITH SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS 163

The convolutions (53) are of interest both on their own right and as containing a definite solution of the problem for finding convolutions of the finite Bessel transforms (C h u r chi II [25], p. 424).

o e fin i t ion 2. The finite Hankel integral transform with a parame­ter h is said to be the correspondence

1

(.58) .p~~}{ f} = f f (t)t in(tt i)dt, j = 1, 2, .. , , u

where ttj, j= I, 2, ... , are the positive zeros of the function hin(t)+tJ~(~). In other words, this is the map of a function fE~r.-2 into its Fourier coef­

ficients sequence on the system {in(ttjt)}'!=1 using the inner product !

(j, g) = ftf(t)g(t)dt. o

The basic operational property of the finite Hankel integral transform (58) is given by the identities

(59) .p~}{Bnf} = -tt; .p~~} {f}+in(ttj)[h f(l)+ /'(1 )], j= 1, 2, ... (see C h u r chi II [25J, p. 425). This shows that it is connected with the singular eigenvalue problem

(60) Bny-ly=O, hy(l)+y'(I)=O

for the Bessel differential operator Bn in [0, 1]. It is near to guess that (58) is connected with the right inverse operator of Bn, defined by

(61) BnLnf(t) = f(t), h(LnfXl)+(Lnf)'(l)=O

for fE ~n-2' Lemma 7. The right inverse Ln of Bm dejined by (61), has a repre­

sentation of the form ! !

(62) tn {2n-l f 1 f } Lnf(t)=Ln,oj(t)+ h+n ~ t'+! j(r)dr+2i/ r1-nf(r)dr U 0

in ~n-2' A convolution of Ln in ~n-2 can be obtained from (53) by the

substitution q.>(f)= [hj(1)+ j'(I)]!(n+h). Then Lnj= L2i~ n I tn} * j.

The proof is immediate. Now we shall show that this convolution j * g of Ln in ~ n-2 is a con­

volution of the finite Hankel integral transform (58) too in the sense of the following

De fin it ion 3. An operation j * g in ~ n-2 is said to be a convolu­tion of the finite Hankel integral transform c:.(h). iff relations "c:!n,}

(63) .p~~}{f * g} = w~~} .pA~}{f} .p~~}{g},

with non-zero constants W(h)., " = 1, 2, ... , hold. n,}

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164 CHAPTER 3

Theorem 3.3.6. The operation (53) with 1:i(j)=[hj(1)+f(l)]/(h+n) is a convolution oj the finite Hankel integral transjorm (58), such that

(64)

j= 1,2, ...

~(h){ f* } - _ J; ~(h) {j} ~(h) {g} 'l:,!n,j g - 22n+1 n! Ih+n)Jn(llj) 'l:,!n,j 'l:,!n,j ,

Proof. Relation (64) can be proved just in the same way as the proof of Theorem 3.3.4. To relation (44) corresponds the relation

(65) .p~}{ j * g} = - :z.p(~). { } .p~~}{ f }.p~~}{g}, III fl.} U

22n+1n! where u(t)= J" tfl is the function representing Ln in convolution (58).

It can easily be seen that

and this together with (65) gives the exact values of the constants w~:} in (63) for the convolution considered. 0

Thus the convolution problem for the finite Hankel integral transforms, posed by C h u r chi II [25], p. 424, is solved.

3.4. APPLICATIONS OF CONVOLUTIONS TO NON-LOCAL BOUNDARY VALUE PROBLEMS

The natural domain of applicability of the convolutions studied in this chapter are some classes of boundary value problems - both local and non­local - for linear differential operators of the second order. The study of such boundary value problems is interesting both from purely mathematical point of view and for its application to linear non-local problems of mathematical physics. Along with the convolutions for boundary value problems for linear differential operators of the second order, the convolutions for boundary value problems for the differentiation operator, studied in Chapter 2, can be used too. Since the object of the classical problems of mathematical physics are boundary value problems for partial differential operators, such convolutions should be extended in spaces of functions of several variables. This can be accomplish­ed in a rather standard manner (see [46]), if we restrict our considerations to rectangular domains. Then the convolutions needed for treating such boundary value problems are in fact tensor products of one-dimensional con­volutions. Nevertheless, there are some cases of problems of mathematical physics in which one convolution suffices. In such cases we can look on the other variables as on parameters. This is a well known approach in using integral transforms for solving such boundary value problems.

In this section two kinds of problems are considered. First, in more detail the eigenexpansions of some spectral problems for the square of dif­ferentiation are studied. This is done by using the convolutional approach. The results obtained are used for explicit Duhamel representations of the

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CONVOLUTIONS CONNECfED wrm SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS 165

solutions of Bitsadze-Samarskiy and Samarskiy-Ionkin problems. This is an illu­stration of the use of the convolutional method in mathematical physics.

3.4.1. Eige:1expansions for non-local spectral problems for the square of differentiation. In sect. 3.1 the general spectral problem

(1) y"+A2y=0, y(O) = 0, qi(y)=O

with a non-zero linear functional qi in Iifl([O, 1]) was considered. There the inessential restriction qi{x} = I was imposed, i. e. that A=O is not an eigen­value of problem (1). In the same section it was shown that the operation

~

(2) (f* gXx)= -(qi 0 l)~{ + I j(~+X-1J) g(1J)d1J x

~ -+ JfCl ~ - x - 1J I) g(l1J I) sgn (~-X-1J) 1Jd1J} -x

is a convolution of the right inverse operator Lf(x)=12 j(x)-x<P(f2j) of d2jdx2, determined by (Lf)(O)=O and qi(Lf)=O. It can be represented as the convolutional operator

(3) Lf(x)={x}*f·

It is easy to see that each eigenvalue of (1) is a zero of the entire function of exponential type

(4)

and conversely. The multiplicity of a zero An of £(A) determines the dimen­sion of the corresponding eigenspace of An'If /-In> 1, then along with sin Anx, there are f-ln-l associated functions in this eigenspace XI.n; I'n' and it is the span of them. Then

(.5)

According to Theorem 1.3.5, there is an element fPn E x< 01' such that III n

the operator Pnj= fPn * f is a projector of 1if([0, 1]) onto x<n;l'n' We call write down this element explicitly.

The 0 rem 3.4.1. Let An be an eigenvalue of spectral problem (1) and let Gil be a contour containing An and no other zero oj £(2) inside. Then the operator

(6) with

(7)

is a projector of 1if([0, 1]) onto Xloll;l'llo

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166 CHAPTER 3

Proof. It is easy to verify that gln E Z€~n;l'n. It remains to show that

gln*gln=gln' To this end we take contours C~ and C: such that C: lies inside C~. Then, expressing gln<x) once as an integral on C~, and then as integral on C:. we get

1 II {sin }.x} * {sin .ax} (gln*gln)(X) = (Jrl)2 E(J.)E(.u)

e'xe" n n

dldl-l

=-I-f f EI.u).u sin J.x-E(l)J. sin .ux d)' d (ni)2 (.uLJ.2)E(J.)E(.u) I-l

c'xC" n n

= J...j'sin J..x [J...f £~Jdl- -.!..j'sin .ax [J...f~] d = (x) :d e' E(J.) ni e" ,u2_;.2 ni e" E(.a) ni e,.u2-J..2 I-l gln '

n n n n

where we use the evident formulas

J.- j' ~d.a'2 =0 and -"!"'f 2J..d~2 =--1. 0 m 1'--" nt 1'-"

en c' n n

Corollary. If In is a simple zero of E(l), then gln(x)=2sin lnxjE'(ln) and

< (8) Pnf(x)= - J.n~'(J.n)w~{I sin In(~-r])f(7J)d7J} sin lnx .

o

As in Chapter 2, sect. 2.2, we can introduce a formal eigenexpansion for problem (1).

De fin i t ion 1. Let J(x)E ~([o, 1]) and leU!,~, .•. be the eigenvalues of (1). The formal spectral expansion of f(x) for eigenvalue problem (1) is said to be the correspondence

(9) f(x)---- I(f*gln)(x). n=l

It is not supposed the series in (9) to be convergent, but nevertheless this formal expansion is of considerable interest, if it has the uniqueness property, i. e. when from (f*gln)(x}=-O for n= 1,2, 3, ... it follows f(x)"20. Here we shall consider two non-local eigenvalue problems (1) which have the uniqueness property.

(10)

1. 'The Bitsadze-Samarskiy spectral problem [5] Let us consider the eigenvalue problem

y" +12y=0, y(O)=O, y(1)--y(lj2)=0.

The functional w(f) should be taken in the form W(f)=2[f(I)-f(lj2)]. There are two series of eigenvalues: an=2n:n:, n= 1, 2, ... , and iJm=2:n:j3+4m:n:, m=O, ±1, ±2, .... In this case formal expansion (9) takes the form

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CONVOLUTIONS CONNECI'ED wrm SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS 167

00

(11) f(x)"",,~ansin2nnx+ ~ bm sin (2nj3+4mn)x n=1 m=-oo

with '1 1/2

an= 2-(~1)n If sin2nn~f(~)d~-·(-1)n f sin2mr~f(~)d~] u 0

and 1 1/2

bm= ~ [fsin (:; +4mn) (1-~)f(~)d~- fSin (;;r +4mn)( +-~)f(~)d~']. u 0

Lemma 1. Expansion (11) for the function f(x)=L{x}=x3j6-7x/24 represents it and is uniformly convergent.

Proof. Computing the coefficients in (11), we should prove the identity

(12) x 3 7 x 1 [~ sin 2rmx 6 - 24 = 4;r3 ~1 nS[2-(-1)nJ

18 I sin (23f/3+4m;r)x 1. m=-oo (1 +6m)S J

We use the known identity

(13) ~ sin (m-a)9 ~ m-a =n,

m=-oo

valid for O<6<2n and for non-integer a (see T. B rom w i c h. An Introduction to the Theory oj Infinite Series. Cambridge, 1926, p. 371). From (13) the following two identities can be obtained:

and 00

~ m,=-OO

~ cos(2;r/3+4m;r)x 1-x ~ (2;r/3+ 4m;r)2 =-4-

m=-oo

sin (23f/3+4m;r)x (23f/3+4m;r)S

x(2-x) f 0< <_1_. 8 or =x= 2

From the elementary theory of Fourier series the identities

~ sin 4rmx XS X2 x ~ (4rm)'l [2-16 + 96 n=1

and

~ sin 2(2n-1);rx _ x(1-2x) ~ [2(2n-1);rJ3 - 32 ' n=1

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168 CHAPI'ER3

valid for 0~x<1/2, can be used. By means of these four identities, it can easily be shown that (12) is valid in O~x~ 1/2. For 1/2::::; x:::;; 1 the validity of (12) can be shown in the same way, but now using the identity

cos (m+a)8 :;r ---'---'---'--=-- 0<e<2n a non-integer m+a tg;;ra' ,

nl=-oo

(see T. B rom w i c h, 1. c., p. 371, ex. 5). 0 Lemma 2. Let fE "tt([O, 1]) and f(O)=/,,(O)=O, f(1)-j(1/2)=f"(1)

-j"(1/2) =0. Then expansion (11) gives a uniformly convergent series, converging to j (x) in [0, 1].

Proof. From the generalized Taylor formula (24) of 1.3.4, we have

j(x)=L2 J<4)(X) = (L{x}) * f(4)(X) = (x3j6-7xj24) * J<4l(x).

Then, by convolutional multiplication of (12) by J<4)(X), we get expansion (11) of f(x). It is uniformly convergent and represents f(x) in [0, 1]. 0

The 0 rem 3.4.2. (Uniqueness property of (11 )). Let f be an arbitrary function of "t([0, 1 D. Ij all the coefficients an and bm in (11) are 0, then f(x)==O.

Proof. Let an=O, 1Z=1, 2, ... , bm=O, m=O, ±l, ±2, .... Then the corresponding coefficients an and bm of the function f(x) = Uf(x) are

Since Ax) satisfies the hypothesis of Lemma 3, then lex) == O. Applying d2/dx2 to the last identity, we get f(x)==O. 0

Now we return again to problem (1). There are cases in which a sim­pler convolution than (2) can be found in "t([0, 1 D. This is the case when the functional IP is a smoothing one.

The 0 rem 3.4.3. Let IP= 'I' 0 l with a linear functional 'I' in "t([0, 1 D, and P.{fJ/2} = lPe{N = 1. Then the operation

x

(14) (f; g)(x)=f(X)I['(l~)+g(x)'I'(lj)-P{1} ff(x-nl:;(~)d~ o

I; ~

-pd-} ff(~+x-1'J)g(1'J)d1'J--} f f(I$-x-1'J°I)g(I1'Jl)sgn a-x-1'J)1'Jd1'J} x -x

is a convolution of the ri~ht inverse L of d2/dx2, determined by (1). It has the function {x} as unit, i. e.

(15) {x};j=j for all jEf6'([O,l]).

Proof. In order to show that (14) is a convolution of L in "t([0, 1]), we can rely on the identity

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CONVOLUTIONS CONNECTED WITH SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS 169

Then the associativity relation (j; g) *-h = f ;(g ; h) follows from that for *, using the fact that (f * g)(O)=O and ([J(f * g)=O for allf, gE~([O, 1 )]. Identity (15) can be verified directly. Since Lf={ x}* f, then L}= (L{x})* j={x3j6- xlJ1i,~2/2)} *} and hence ;;; is a convolution of L in ~([O, 1 D. 0

L e m m a 3. Let the functional ([J in (1) be the same as in Theorem 3.4.3. Then, if 2n is an eigenvalue of (1), the operator

(16)

with

(17) - 1 f sin2x fPn(x) = - lli 22£(2) d2

en

is a projector of ~([O, 1]) onto the eigenspace XAn;Jln of 2n'

Proof. Since q;n = LfPn, then using Theorem 3.4.1, we obtain

Pnf=f*fPn=f; ;Pn' 0

Using convolution (14), the formal eigenexpansion (9) can be written in the form

00

(18) f(x),....., I (f;;Pn)(X)' n=l

We shall use convolution (14) for a study of a special non-local spec­tral problem we shall need later.

(19)

2. The Samarskiy-Ionkin spectral problem [65, 66]. This is the eigenvalue problem

1

y"+22y=0, y(O)=O, JY(~)d~=O. o

1

Here ([J(f)=2 f f(~)d~, and o

£(2) = 2(1 ~~os 2) .

The eigenvalues of (19) are 2n=2nn, n= 1, 2, ... , and each of them has multiplicity 2. Hence, the eigenspaces Xl.n ;2 are two-dimensional. Then con-volution (14) takes the form

1 1

(20) ({; g)(x) = 2f(x) f g(nd~+2g(x) ff(~)d~ o 0

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170 CHAPTER 3

x 1

-2 ff(x-~)g(~)d~ - ff(1 +x-ng(Od~ o x

1

+ f f(ll-x-~I)g(I~I)sgn(1-x-~)~d~. -x

L e m m a 4. The convolutional projectors (16) for (19) have the form

(21 ) Pnf(x)={ -2x cos2nnx} -; f(x) 1 1

= [4 I (1 -~) f (~) sin 2nn~d~1 sin 2flnX- [4f f (~)(1- cos 2fln~)d~]xcos2flnx. () a

Pro a f. It is easy to get from (17)

(22) ;fn(x) = -2x cos 2nnx.

Then Pnf(x)=(;Pn -; f)(x) can be transformed into (21) in a standard man­ner. 0

Theorem 3.4.4. Let fE~([O, 1]). If Pnf=O for n=1,2, ... , then f(x)~O.

Proof. If Pnf=O, then PnLf=O for n=1, 2, .... The function Lfsatis-1

ties both (Lf)(O)=O and f (Lf) (~)d~=O. From (21) we have a

1

f(Lf)(n cos2nnM~=0 and o

1

I(1-0(Lf)(~) sin 2nn~d~=0. o

From the first of them it follows that Lf(x) is odd with respect to the point x=1/2, i. e. (Lf)(x)=-(Lf)(1-x). The second of the above equations

1

gives fCLf)(n sin 2nn~d~=0. Hence Lf(x) should be even with respect to o

the point X= 1/2. This is possible only for Lf(x)~O. Hence f(x)~O. Hence spectral development (18) of a function f(x) E ~([O, 1]) determines

it uniquely, though it may be non-convergent. 0 Lemma 5. Expansion (18) for the function f(x)=L{x}=x3/6-x/12

gives a uniformly convergent series, which represents it in [0, 1 J. Proof. The identity

(23) x 3 _ ~ = ~ { 2x cos 2nnx 6 12 ~ (2nn)2

n=l

4 sin 2nnx } (2f1n)3

can be proved by elementary summings of the Fourier series involved. 0

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CONVOLUTIONS CONNECI'ED wrrn SECOND·ORDER UNBAR DIFFERENTIAL OPERATORS 171

1

Theorem 3.4.5. Let f(x)E~2([0, 1]), f(O) =0, and .r fmd~=O. Then o

"" 1

(24) f(x) = }: {[ 4f (1- ~)f (~) sin 2 nn~d~ ]sin 2nnx n=1 0

1

-[4ff(~X1-COS 2nn~)d~]X cos 2nnx }, o

the series being uniformly convergent on [0, 1]. Proof. Under the hypothesis

f(x)=LI" = {x3/6-x/12};;; 1". Then by convolutional multiplication of (23) by f'(x), we get

V> "" ""

f (x)= }: (Lin); I" = }:,pn ;; LI" = }:,pn ;;; f n=1 n=1 n=1

which is (24). 0 Now let us consider the multiplier problem for the expansion (24)

in ~([O, 1 D. De fin i t ion 2. An operator M: ~([O, 1]) -+ ~([O, 1 D is said to be a

multiplier of the formal eigenexpansion (18), if (/.)

Mf~ }: I1-nU; ,pn) ,,=1

with numerical multiplier sequence 11-1' 11-2"'" f1,~, •••• Using the uniqueness Theorem 3.4.4 it can easily be shown that each

multiplier of (18) is a multiplier of convolution (20) too. The converse is not always true. Therefore, each multiplier of the Samarskiy-Ionkin eigenex­pansion (18) has the form

(25) Mf=m; j, with m E ~([O, 1 D. It remains only the form of m to be specified.

The 0 rem 3.4.6. A linear operator M: ~([O, 1]) -+ ~([O, 1]) is a mUl­tiplier of the Samarskiy-Ionkin eigenexpansion (24) iff it has the form (25) with

m(x)=xh(11-2xJ)

'where h is an arbitrary function from ~([O, 1 D. Proof. It is easy to see that in order (25) to be multiplier of (24)

m should have expansion of the form 00

m(x),....., }: I1-n ,pn (x). n=1

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172

Hence 1

I(1-;) m(;)sin 2nn;d;=0, n= 1, 2, .... o

CHAPTER 3

This means that (I-x) m(x) should be an even function with respect to the point x= 1/2, i. e.

(I-x) m(x)=xm (I-x).

Hence the function m(x)/x should be continuous on [0, 1] and e~en with respect to the point x= 1/2. The general form of the even functions with respect to x= 1/2 is h(ll-2x I) with h(x)E CC([O, 1 D· 0 .

It deserves to be mentioned that the multiplier problem jor the Samar­skiy-lonkin jormal expansion allows an easier solution than for the classical Fourier expansion.

3.4.2. Duhamel-type representations of solutions of non-local boundary value problems for partial differential equations of mathematical physics. Now we shall apply the considerations of the previous section to some pro­blems for equations of mathematical physics. Since our aim here is rather to illustrate the applicability of the convolutions found than to treat elabo­rately the most general problems, we confine ourselves to only two re­presentative examples.

The simplest and well-known example of a Duhamel representation concerns the boundary value problem

(26) u(O, t)=O, u(I, t)=j(t); u(x, 0)=0

for the heat equation. The solution u(x, t) of (25) can be represented by the Duhamel integral

t

(26') u(x, t)= !t fU(x, t-r)j(r)dr, o

where U(x, t) is the special solution of (26) with j(t)== 1. We observe at once the appearance of the Duhamel convolution in (26'). It is rather na­tural to seek another boundary value problems for the heat equation, when similar representations hold, but with other convolutions. Especially, we are interested in Duhamel-type representations with respect to the space variable.

Let us consider the general boundary value problem

(27) u(O, t)=O, qJ~{u(;, t)}=O; u(x, O)=j(x)

in D={(x, t):O<x<l, t>O} with non-zero linear functional qJ in CC([O, 1]). It is not a strong restriction to assume that qJ{ x} = 1. To this problem the following theorem can be proved.

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CONVOLUI'IONS CONNECTED WITH SBCOND-ORDER LINEAR DIFFERENTIAL OPERATORS 173

Theorem 3.4.7 (Dimovski, Mineif [48]). Let there exist a solu­tion U(x, t) of (27} with f(x)==x3/6-w#s/3)x.lf fix) E CC,[O, l]),f(O)=/"(O) =0, and cJ>Cf)=lfI(f")=O, then the function

(28) iJ1 (X)

u(x, t)=iJx4 [U(x,t)*f(x)],

(X)

'lf1.'lzere * denotes operation (2), is a solution of boundary value pro­blem (27).

Proof. Under the hypothesis (X)

u(x, t)=U(x, t) *P4)(X)

and it can directly be verified that u(x, t) is a solution of the heat equa­tion ut=uxx in D. The boundary value conditions are satisfied since opera­tion (2) has the properties (f*gXO)=O, and W(f*g)=O for arbitrary f, gECC([O, 1]). 0

Theorem 3.4.7 is only a conditional existence theorem. In each case the existence of the special solution U(x, t) for the case f(X)=:EX3!6-xq(I..~2(2) should be ensured. There easily can be given examples when such a solution does not exist. This is the case especially for some inverse ill-posed problems for the heat equation. The usefulness of representation (28) is in its universality.

Now we shall consider the Samarskiy-Ionkin problem (see [66]) in which the existence of the special solution U(x, t) can easily be proved using the considerations of 3.4.1. This problem is announced to be important for plasma physics. It is a mathematical model of diffusion in turbulent plasma [65, 66].

The Samarskiy-Ionkin problem is said to be the following non-local boundary value problem for the heat equation:

1

(29) nCO, t)=O, fU(~, t)d.~=O, o

u(x, O)=f(x).

in the strip D={(x, t):O<x<l, t>O}. Let us try to find the solution U(x, t) of (24) for x 3

f(x)==L{x} = 6"" 1

-1~ , where Lf(x) = l2f(x)-x I (1-n2f(~)d~ is the right inverse of d2/dx2 ,

o 1

determined by (Lf)(O)=O and f (Lf)(~)d~=O. To this end we apply the o

projector operators (21) to the equation ut = uxx and to the initial condition u(x, 0) =f (x). Denoting un(x, t)=Pnu(x, t), we shall have

un(x, t)=An(t)( -2x cos 2;;'l;nx)+ Bn(t) sin 2;;'l;nx

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174 CHAPTER 3

with unknown functions AnU) and Bn(t). It is easy to see that un(x, t) should be a solution of the heat equation, such that uncx, 0)= fnCx) with fix) = Pnf(x), given by (21). In our special case f(x)== L{xl. we have

fn(x)=.PnL{x} = LPn{x} = - 4!2n2 (-2x cos 2nnx)- 2~n3 sin 2nnx.

For An(t) and BnCt) we get easily the system of ordinary differential equa­tions

A~(t) = -4n2n2AnCt),

B~Ct)= -4n2n2BnCt)+8nnAnCt)

with the initial conditions AnCO)=-lJC4n2n2) and Bn(0)=-lJ(2.n3n3 ). We get An(t) = -exp (- 4n2n2t)/(2nn)2 and Bn(t) = -(1 +4n2n2t) exp (-4n2n2t)/(2n3n3),

n=l, 2, .... Therefore, if there exists a solution of (25) with fCx)==L{x}

= ~3 -1~ , the only candidate for such a solution is

(30) V' 1

U(x, t)= ~ 2n3n3 [nnx cos 2nnx-(1 + 4n2n2 t) sin 2nn x]e-4""n". n=1

The series is uniformly convergent on D. It is easy to verify that U(x, t) satisfies the heat equation u,=uxx in D. It remains only to see that U(x,i)

1

satisfies the boundary value conditions U(O, t)=O, J U(~, t)d~=O and the o

initial value condition U(x, 0)=x3/6-x/12. The boundary value conditions are satisfied since each un(x, t) satisfies them: As for the initial value con­dition, it is satisfied due to (23).

Combining representation (28) with the convolution (20), we can prove the following theorem.

1

Theorem 3.4.8. If f(x)E~2([0, 1]), fCO)=Oand fj(~)d~=O, then pro­o

blem (29) has a classical solution. It can be represented in the form x 1

(31) u(x, t)= -2fQ(x-~, t)f(~)d~-fQ(l+x-~, t)f(~)d~ o x

1

+ JQ(I-X-~, t)f(I~l)sgn~d~ -x

with 00

(32) Q(x, t)=UxxCx, t)= ~ {-2xcos2nnx+8nntsin2nnx}e-4n''''''. n=1

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CONVOLUTIONS CONNECfED WITH SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS 175

Proof. The existence of a classical solution of (29) follows from Theo­rem 3-4.7, since there exists the special solution for j(x)~x3/6-xjl2. In D (28) can be represented in the form

u(x, i) = Uxx(x, t)~ f(x). 1

Then, using convolution (20), we can write at once (31), since JUxxCt, i)dt=O. u

1

for i>O andjf(t)dt=O. 0 o

The restriction f E l{i2([O, 1]) can be replaced by the weaker assump­tion f E l{il([O, 1 D.

The Biisadze-Samarskiy problem (see [5])

uxx+ uyy = 0,

(33) u(O, y)=O, u(l, y)=u(1/2, y),

u(x, 0)=0, u(x, l)=j(x)

in the square domain D={(x, y): O<x<l, 0<y<1} is another example of non-local boundary value problem, well suited to the convolutional approach dev,eloped here.

(34)

L em m a 6. The function

m=-oo

sin (2n/3+4mn)x. sh (2n/3+4mn)y (1 +6m)3 sh (2n/3+4mn)

3 7 is a solution of (33) for f(x)= -~ - 2~ .

Proof. It is easy to see that U(x, t) satisfies the Laplace equation in each interior point of the square. The first three boundary value conditions are satisfied in an obvious way. As for the last condition, its satisfaction follows from Lemma 1.

Theorem 3.4.9. Lei f(x)El{i4([O, 1]) and f(0)=f"(0)=f(1)-f(I/2) =/,,(1)-/,,(1/2)=0. Then ike function

(35) 1 ~ i;

u(x, y)= - f dt{f U(x+t-1], y)f<4) (1])d1]- f U($ - X~-1], y)P4)C\17I) sgn 17d1]}. 1/2 x ~x

where U(x, y) is the special solution (34), is a classical solution of Bi­isadze-Samarskiy boundary value problem (33) on ihe square D.

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176

Proof. Since according to Duhamel-type representation (28) (x)

u(x, y)=U(x, y) * j<4)(X),

(x)

CHAPfER3

then u.u+uyy = {Uxx+Uyy}*!(4)(X)=0. The first three boundary value con­ditions in (33) are satisfied in an obvious way. As for the condition u(x, 1) = lex), its satisfaction follows from Lemma 2.

The Duhamel-type representations (31) and (35) can be used for nume­rical calculation of the solution at points where we need its values with a prescribed accuracy. To this end some quadrature formulas can be used. An advantage of such a numerical method over the difference methods con­sists in the possibility to reach greater accuracy and in the stability of the method.

Duhamel-type representation can be proposed for a broad class of linear partial differential equations in rectangular domains. In [46] the class of par­tial differential equations

m n

2 PiiJjiJtj)u-I Qk()2/iJxk)U=!(Xl"'" xn;tl ,·· .,tm),

j=1 k=l

with polynomials Pj and Qk is considered in domains D=[O, lrX[O, oo)m. The boundary value problems considered there are of the form

xV){(iJl/iJtj)u}=!5l) , [=0, 1, ... , degPj-l, j=l, 2, ... , m ]

with respect to the "time variables" t l , t2, ••• , tm, and

,,2s I v_ll = (S) iJ 2s gk '

X k xk=o

<P~'){(iJ2S/iJx~s)u}=h<,:), s=O, 1, ... ,degQk-1, k=l, 2, ... , n k

with respect to the "space variables" Xl' X2"'" Xw Here XEI), X(2), ••• , x(Jn) and 15(1), (jj(2), ••• , qj(n) are non-zero, but otherwise arbitrary linear functionals on <m'([0, 00)) and <m'1([0, 1]) correspondingly. !5l), g~S) and h<,:) are given boun-dary value functions. In all these boundary value problems the existence problem can be reduced to the existence of a solution of a simpler pro blem with simple boundary value functions. It seems that the potentialities of the convolutional method for linear non-local boundary value problems are great­er than the examples considered here indicate.