Operational Method for Solving Integral Equations with ... - CiteSeerX

Operational Method for Solving Integral Equations with Gauss’s Hypergeometric Function as a Kernel R. Gorenflo∗ , Yu.F. Luchko∗ , and H.M. Srivastava∗∗ ∗ Department

of Mathematics and Computer Science, Free University of Berlin Arnimallee 2-6, D-14195 Berlin, Germany E-Mail: [email protected]; [email protected] ∗∗ Department

of Mathematics and Statistics, University of Victoria Victoria, British Columbia V8W 3P4, Canada E-Mail: [email protected]

Abstract In the present paper, the authors first develop an operational calculus for the integral operator with the well-known Gauss hypergeometric function as a kernel. This operational calculus is then used for finding the exact solution of integral equations of Volterra-type with Gauss’s hypergeometric function in the kernel. These analytical solutions are expressed in terms of integral operators with the generalized Mittag-Leffler type function and the generalized Wright function as a kernel.

Key words: Operational calculus, integral equations, Gauss’s hypergeometric function, generalized Mittag-Leffler type function, generalized Wright function, Riemann-Liouville and Erd´elyi-Kober fractional derivative operators. 1991 Mathematics Subject Classification: 44A40, 44A10, 45E10, 44A35, 33C10, 33C40.

1. Introduction It is well known that Mikusi´ nski’s operational calculus can be used not only for the solution of ordinary and partial differential equations, but for finding the analytical solutions of some integral equations as well. This idea was realized, in particular, in [2], [5], and [9] for the case of integral equations of the first kind. In recent years Mikusi´ nski’s scheme was applied by sev1

eral mathematicians for the development of operational calculi for operators which are more involved than the familiar operator of differentiation. In some cases, these operational calculi were used for solution of integral or integrodifferential equations. In particular, Al-Bassam and Luchko [1] have considered operational method of solution of integro-differential equations with multiple Erd´elyi-Kober fractional derivative operator, Gorenflo and Luchko [6] have used operational calculus for the Riemann-Liouville fractional derivative operator for finding the analytical solutions of the generalized Abel’s integral equations of the second kind, and Luchko and Srivastava [7] have found (by means of the same operational calculus) the exact solutions of the integro-differential equations with the Riemann-Liouville fractional derivative operator. In the present paper, we develop an operational calculus for integral operator with Gauss’s hypergeometric function as a kernel. This operational calculus is based on the interpretation of the convolution obtained here as the operation of multiplication in some space of functions. Finally, we present the applications of the developed operational calculus to the solution of the integral equations of the second kind with Gauss’s hypergeometric function as a kernel.

2. Integral Operator with Gauss’s Hypergeometric Function as a Kernel We begin by defining the functional space Cα (α ∈ R), which was at first introduced in [3] and which will be used in our present investigation. Definition 1 A function f (x) (x > 0) is said to be in the space Cα (α ∈ R) if it can be represented in the form: f (x) = xp f1 (x) for some number p > α, where the function f1 (x) is continuous in the interval [0, ∞). Clearly, the function space Cα is a linear space and the following inclusion is true for the set of spaces Cα (α ∈ R): Cα ⊆ Cβ (α ≥ β).

(2.1)

In this paper we will deal with the operator xa
x t (x − t)2a−1 F (a, b; 2a; 1 − )t−2a f (t)dt, a > 0 (2.2) (La,b f )(x) = Γ(2a) 0 x 2

with the Gauss hypergeometric function F (a, b; c; z) =

∞ 

(a)n (b)n z n (c)n n! n=0

in the kernel. Such operators were investigated in many articles (see, for references, [11]) and there are many interesting particular cases of these operators with various special functions in the kernels, which we will consider in our paper later. Theorem 1 Let α = max{2a − 1, a + b − 1}. Then the operator La,b is a linear map of the space Cα into itself: La,b : Cα → Cα+a ⊂ Cα . Proof. Setting t = xτ in (2.2), we obtain xp+a
1 (1 − τ )2a−1 F (a, b; 2a; 1 − τ )tp−2a f1 (xτ )dτ = xp+a f2 (x), (La,b f )(x) = Γ(2a) 0 (2.3) where p > α and f1 (x) ∈ C[0, ∞). Using asymptotic properties of Gauss’s hypergeometric function (see, for example, [10]) and the expression for the parameter α, we have   

τ p−2a , a > b, 2a−1 p−2a 2a−1 p−2a F (a, b; 2a; 1 − τ )t | ≤ M(1 − τ ) · τ | ln τ |, a = b, |(1 − τ )   p−a−b , a 0, and c > 0. It means that the integral in the representation (2.3) is uniformly convergent with respect to x in every finite closed interval [0, X], X > 0, and, consequently, we have f2 (x) ∈ C[0, ∞), which proves Theorem 1. It follows from Theorem 1 that all the functions f (x), (La,b f )(x), (La,b La,b f )(x), (La,b La,b . . . La,b f )(x) := (Lna,b f )(x) 

 n



belong to the space Cα in the case f ∈ Cα and α = max{2a − 1, a + b − 1}. The exact form of the operator Lna,b is given by 3

Theorem 2 The operator Lna,b (composition of n operators La,b ) has the form: (Lna,b f )(x)

x(2−n)a = Γ(2na)


x 0

t (x − t)2na−1 F (na, (n − 1)a + b; 2na; 1 − )t−2a f (t)dt, x (2.4) a > 0, n ∈ N.

Proof. The assertion of this theorem is obtained by using mathematical induction with respect to n, changing the order of integration in the corresponding repeated integrals and using the reference [10] for analytical values of the integrals containing the product of two Gauss’s hypergeometric functions. Since this proof has many purely technical evaluations, we omit it here.

3. The Convolutional Ring Cα The general definition of convolution of a linear operator was introduced in [3] and is given by Definition 2 Let Λ be a linear space and let L : Λ → Λ be a linear operator. A bilinear, commutative, and associative operation ∗: Λ×Λ →Λ is said to be convolution of the linear operator L if and only if L(f ∗ g) = (Lf ∗ g), ∀f, g ∈ Λ. In the sense of this definition, the Laplace convolution, for example, (f ∗ g)(x) =


x 0

f (x − t)g(t)dt, x > 0

is a convolution of the Volterra integral operator (V f )(x) =


x 0

f (t)dt.

This definition of convolution is especially suitable for development of operational calculus of Mikusi´ nski type (see [1], [3], and [7]).

4

Theorem 3 In the sense of Definition 2, the operation ∗a,b , a > 0, given by (f ∗a,b g)(x) = (Ia,b (f ◦ g))(x),

(3.1)

 xa+b  x  (x − t)b−a−1 t1−2a−2b f (t)dt,   Γ(b−a) 0     

(Ia,b f )(x) =  x1−2a f (x),       

x2a Γ(a−b)

x 0

b > a, b = a,

(x − t)a−b−1 t1−4a f (t)dt,

b < a,

where (f ◦ g)(x) =


1
1 0

0

(u1 (1 − u1 ))−2a f (xu1 u2 )g(x(1 − u1 )(1 − u2 ))du1 du2, (u2(1 − u2 ))a+b

is the convolution (without divisors of zero) of the linear operator La,b in the space Cα , α = max{2a − 1, a + b − 1}. Proof. First of all, it is easily seen that the operation ∗a,b is bilinear and commutative. Let f, g ∈ Cα , α = max{2a − 1, a + b − 1}. Then (f ◦g)(x) = xp+q


1
1 0

0

(u1 (1−u1 ))−2a (u2 (1−u2 ))−a−b (u1 u2 )p (1−u1 )q (1−u2 )q

·f1 (xu1 u2 )g1 (x(1 − u1 )(1 − u2))du1 du2 = xp+q φ(x), p > α, q > α, f1 , g1 ∈ C[0, ∞). Using the equality α = max{2a − 1, a + b − 1} we readily find that the last integral is uniformly convergent with respect to x in every finite closed interval and φ(x) ∈ C[0, ∞). It gives us the inclusion (f ◦ g)(x) ∈ C2α . Let h ∈ C2α , α = max{2a − 1, a + b − 1} and we consider the function (Ia,b h)(x):  1−2a+p  1 x  (1 − t)b−a−1 t1−2a−2b+p h1 (xt)dt   Γ(b−a) 0     

(Ia,b h)(x) =

= x1−2a+p h2 (x),

x1−2a+p h (x),

1          x−a−b+1+p 1 (1 − t)a−b−1 t1−4a+p h Γ(a−b)

1 (xt)dt

0

p > 2α, h1 ∈ C[0, ∞). 5

= x−a−b+1+p h3 (x),

b > a, b = a, b < a,

Since the functions h2 (x) and h3 (x) are determined by means of integrals which are uniformly convergent with respect to x in every finite closed interval, we have h2 , h3 ∈ C[0, ∞). For p > 2α, α = max{2a − 1, a + b − 1}, we obtain also the inequalities 1 − 2a + p > α and −a − b + 1 + p > α. These two facts and Definition 1 give us the property ∗a,b : Cα × Cα → Cα , α = max{2a − 1, a + b − 1}. Next, for the associativity of the operation ∗a,b , we evaluate, for the functions f, g, h ∈ Cα of the form: f (x) = xp1 , g(x) = xp2 , h(x) = xp3 , min{p1 , p2 , p3 } > α = max{2a−1, a+b−1}, the expression a,b

a,b

(f ∗ g)∗ h =

        

3

Γ(1−2a+p )Γ(1−a−b+p1 )

i i=1 x2−4a+p1 +p2 +p3 Γ(3−6a+p , 1 +p2 +p3 )Γ(3−5a−b+p1 +p2 +p3 )

3

b ≥ a,

Γ(1−2a+p )Γ(1−a−b+p1 )

i i=1 x2−2a−2b+p1 +p2 +p3 Γ(3−4a−2b+p , 1 +p2 +p3 )Γ(3−3a−3b+p1 +p2 +p3 )

b≤a

= f ∗a,b (g ∗a,b h). The associativity of the operation ∗a,b for all functions f , g, and h from Cα , α = max{2a−1, a+ b−1}, follows now from the last relation, bilinearity of the operation ∗a,b and the Weierstrass approximation theorem (see [3] for details). Further, we have (f (x) = xp1 , g(x) = xp2 , p1 , p2 > α) ((La,b f ) ∗

a,b

    

g)(x) =

   

1−a+p1 +p2

x

1−b+p1 +p2

x

2

Γ(1−2a+pi )Γ(1−a−b+pi ) i=1 , Γ(2−3a+p1 +p2 )Γ(2−2a−b+p1 +p2 )

2

Γ(1−2a+pi )Γ(1−a−b+pi ) i=1 Γ(2−2a−b+p1 +p2 )Γ(2−a−2b+p1 +p2 )

b≥a b≤a

= (La,b (f ∗a,b g))(x), which, in view of the Weierstrass approximation theorem, linearity of the operator La,b and the bilinearity of the operation ∗a,b , yields ((La,b f )∗a,b g)(x) = (La,b (f ∗a,b g))(x), ∀f, g ∈ Cα , α = max{2a−1, a+b−1}. Finally, the absence of divisors of zero for the operation ∗a,b follows from two facts. Firstly, we note that the operator Ia,b is the modified RiemannLiouville fractional integral operator (see [11]) which has a left inverse operator and consequently zero is not an eigenvalue of this operator in the space Cα . Secondly, the operation f ◦ g is a multiple Laplace convolution and has no divisors of zero due to the theorem from [8]. As a consequence of Theorem 3, we obtain 6

Theorem 4 The space Cα , α = max{2a − 1, a + b − 1}, with the operations ∗a,b and ordinary addition, becomes a commutative ring (Cα , ∗a,b , +) without divisors of zero. By direct calculation, we prove the following important result for the development of operational calculus. Theorem 5 The operator La,b has the convolutional representation: (La,b f )(x) = (f ∗

a,b

 2a+b−1 x    Γ(a)Γ(b) ,

h)(x), h(x) =   

0 < b ≤ a, (3.2)

x3a−1 , Γ(a)Γ(2a−b)

a ≤ b < 2a,

f ∈ Cα , α = max{2a − 1, a + b − 1}. Repeatedly using the representation (3.2), we readily obtain for the operator Lna,b given by (2.4) the convolutional representation in the functional space Cα , α = max{2a − 1, a + b − 1}: a,b . . . ∗a,b h )(x) = (f ∗a,b hn )(x) (Lna,b f )(x) = (f ∗a,b h  ∗ h

(3.3)

n

with n

h (x) :=

 x(n+1)a+b−1    Γ(na)Γ((n−1)a+b) ,   

x(n+2)a−1 , Γ(na)Γ((n+1)a−b)

0 < b ≤ a, n ∈ N.

(3.4)

a ≤ b < 2a,

3. Operational Calculus for the Operator La,b In what follows we will always suppose that α = max{2a−1, a+b−1} and 0 < a, 0 < b < 2a. Following a line similar to Mikusi´ nski’s, we can extend a,b the ring (Cα , ∗ , +) to the quotient field M by means of the factorization of the set Cα × (Cα − {0}) with respect to the equivalence relation (f, g) ∼ (f1 , g1) ⇔ (f ∗a,b g1 )(x) ≡ (g ∗a,b f1 )(x). We can consider the elements of M as convolution quotient f /g and define the operation in M, as usual, by f1 f (f ∗a,b g1 ) + (g ∗a,b f1 ) + = g g1 (g ∗a,b g1 ) 7

(4.1)

and

(f ∗a,b f1 ) f f1 = · . g g1 (g ∗a,b g1 )

(4.2)

It follows from these definitions and Theorems 3 and 4 that the set M is a commutative field with respect to the operations (4.1) and (4.2). The ring Cα can be embedded in the field M by the map (f ∗a,b h) , f→ h

(4.3)

where h(x) is given by (3.2). Moreover, the field C of complex numbers can also be embedded in M by the map: zh . h

z→

(4.4)

It is easily verified that the element I = h/h of the field M is unity of this field with respect to the multiplication (4.2) and this element is not reduced to a function from the ring Cα . We introduce now one more element of the field M, possessing this property, which will play an important role in the applications of our operational calculus. Definition 3 The algebraic inverse of the operator La,b is said to be the element S of the field M, which is reciprocal to the element h in the field M with respect to the multiplication operation: S=

I h h ≡ ≡ 2. a,b h (h ∗ h) h

(4.5)

For many applications it is important to determine the elements of the field M which can be represented by means of the elements of the ring Cα . One useful class of such functions is given by the following theorem. Theorem 6 Suppose that the power series ∞ 

αk z k (z, αk ∈ C),

k=0

is convergent at a point z0 = 0. Then the power series ∞  k=1

αk S −k =

∞ 

αk hk (x),

k=1

where hk (x) is given by (3.4), defines an element of the ring Cα . 8

Proof. Using the relation (3.4), we have f (x) :=

∞  k=1

  

λ=

 

αk hk (x) = xa−λ f1 (x),

1 − a − b, 0 < b ≤ a, 1 − 2a,

a ≤ b < 2a,

f1 (x) :=

∞ 

αk+1 xak . k=0 Γ(1 − a − λ + ak)Γ(1 − b − λ + ak)

It is easy to check that a − λ > α = max{2a − 1, a + b − 1}. Let us prove that f1 ∈ C[0, ∞). Since the series ∞  k=0

αk z0k , z0 = 0,

is convergent, we obtain the inequality sup |αk z0k | = M0 < ∞, k≥0

and, consequently, |αk | ≤

M0 , 0 < M0 < ∞, k ∈ N0 := N ∪ {0}. |z0 |k

(4.6)

For sufficiently large k we have also γ k kδ 1 ≤ 2k , Γ(1 − a − λ + ak)Γ(1 − b − λ + ak) k

(4.7)

where k ≥ k0 , γ and δ are constants. Finally, combining the inequalities (4.6) and (4.7), we find for x = X, 0 < X < ∞, that

∞





∞

 αk+1 X ak M0 X ak γ k k δ

= M < ∞.

≤ M+ k+1 k 2k

k=0 Γ(1 − a − λ + ak)Γ(1 − b − λ + ak) k=k0 |z0 |

According to the Abel theorem, the series representing f1 (x) is uniformly convergent on any finite closed interval 0 ≤ x ≤ X, 0 < X < ∞, and we have f1 ∈ C[0, ∞).

9

Corollary 1 The following operational relations hold true: h I = = h(I + ρh + ρ2 h2 + . . .) S −ρ I − ρh

(4.8)

= xa−λ Φa,a (ρxa ; 1 − a − λ, 1 − b − λ),   

λ=

 

1 − a − b, 0 < b ≤ a, 1 − 2a,

a ≤ b < 2a,

where Φα1 ,α2 (z; β1 , β2 ) is the generalized Mittag-Leffler type function defined by (cf. [4]) Φα1 ,α2 (z; β1 , β2 ) =

∞ 

zk , k=0 Γ(β1 + α1 k)Γ(β2 + α2 k)

(4.9)

0 < α1 < ∞, 0 < α2 < ∞, −∞ < β1 < ∞, −∞ < β2 < ∞. This is an example of an entire function of order

α1 + α2 1 and type . 1 α1 + α2 (α1α1 α2α2 ) α1 +α2

Among others, the Bessel function can be expressed in terms of the generalized Mittag-Leffler type function: 

 ν

z Jν (z) = 2

Φ1,1



z2 − ; 1, ν + 1 . 4

(4.10)

It follows from the operational relation (4.8) that xma−λ I = (S − ρ)m (m − 1)! 

· 1 Ψ2



(m, 1) ; ρxa , (1 − a(2 − m) − λ, a), (1 − a(1 − m) − b − λ, a)   

1 − a − b, 0 < b ≤ a,



1 − 2a,

m ∈ N, λ =  where p Ψq



(a1 ,A1 ),...,(ap ,Ap ) ;z (b1 ,B1 )...(bq ,Bq )

 p Ψq

(4.11)



a ≤ b < 2a,

is the generalized Wright function (see [12]): 



p ∞ k  (a1 , A1 ), . . . , (ap , Ap ) i=1 Γ(ai + Ai k) z ;z = . q (b1 , B1 ) . . . (bq , Bq ) i=1 Γ(bi + Bi k) k! k=0

10

(4.12)

Among particular cases of this function, we note the Wright function 

W (z; α, β) =

0 Ψ1



− ;z , (β, α)

(4.13)

and the generalized Wright function µ (z) Jν,λ



 ν+2λ

z = 2

1 Ψ2



(1, 1) z2 ;− . 4 (ν + λ + 1, µ), (λ + 1, 1)

(4.14)

The function (4.9) is also a particular case of the function (4.12): 

Φα1 ,α2 (z; β1 , β2 ) =

1 Ψ2



(1, 1) ;z . (β1 , α1 ), (β2 , α2 )

The generalized Wright function (4.12) can be expressed in terms of more general Fox’s H-function as well (see [12]). Many others operational relations can be obtained using Theorem 6. In particular, any rational function of S with the power of numerator less than the power of denominator can be represented as a finite sum of the partial fractions and reduced to some element of the ring Cα by using the operational relations (4.8) and (4.11).

5. Applications Involving Integral Equations with Gauss’s Hypergeometric Function as a Kernel In this part of our paper, we will solve integral equations with operators (2.2) and (2.4) using the already developed operational calculus. Theorem 7 The integral operator La,b defined by (2.2) may be represented in the field M in the form: (La,b f )(x) =

I · f. S

(5.1)

Proof. From Definition 3 we see that the element S of the field M is reciprocal to the element h, and conversely, the element h is reciprocal to the element S, that is, I h= . S To obtain the representation (4.1) we use now the embedding (4.3) of the ring Cα in the field M and Theorem 5. As a consequence of Theorem 7 and the representation (3.3), we get 11

Corollary 2 (Lna,b f )(x) =

I · f. Sn

(5.2)

Theorem 8 The unique solution of the integral equation of the second kind y(x) − λ(La,b y)(x) = f (x), f ∈ Cα , x > 0

(5.3)

in the space Cα is represented in the form: a

y(x) = f (x) + λx


1
1 0

0

((1 − u)(1 − v))a−1 ua+b v 2a

(5.4)

·Φa,a (λxa ((1 − u)(1 − v))a ; a, a)f (xuv)dudv, where Φα1 ,α2 (z; β1 , β2 ) is the generalized Mittag-Leffler type function (4.9). Proof. Using the embedding (4.3) of the ring Cα in the field M and Theorem 7, we rewrite the integral equation (5.3) in the functional space Cα as an algebraic equation in the field M: y−λ

I · y = f, S

(5.5)

whose unique solution in the field M has the form: y=

I S ·f =f +λ · f. S−λ S−λ

(5.6)

It means that the integral equation (5.3) may have only unique solution in the space Cα , if the solution (5.6) of the equation (5.5) corresponds to some element of the ring Cα . Using the operational relation (4.8), embedding (4.3) and, finally, Theorem 3, we find that the solution (5.6) is an element of Cα and it has the form: y(x) = f + λ(xa−c Φa,a (ρxa ; 1 − a − c, 1 − b − c) ∗a,b f )(x),   

c=

 

1 − a − b, 0 < b ≤ a, 1 − 2a,

a ≤ b < 2a.

Using the representation (3.1) of the convolution ∗a,b and evaluating one of the integrals, we get the representation (5.4). As a natural generalization of Theorem 8, we have

12

Theorem 9 The unique solution of the integral equation of the second kind y(x) −

m  i=1

λi (Lia,b y)(x) = f (x), f ∈ Cα , x > 0

(5.7)

in the space Cα is represented in the form: y(x) = f (x) + (K ∗a,b f )(x), where K(x) = 

· 1 Ψ2

ni m   i=1 j=1

dij

(5.8)

xaj−λ (j − 1)!



(j, 1) ; wixa , (1 − a(2 − j) − λ, a), (1 − a(1 − m) − b − λ, a)

the constants wi (1 ≤ i ≤ m) and dij (1 ≤ i ≤ m, 1 ≤ j ≤ ni ) are determined by the representation of the rational function as the sum of partial fractions: ni m  λ1 z m−1 + λ2 z m−2 + . . . + λm  1 = dij , m m−1 z − λ1 z − . . . − λm (z − wi )j i=1 j=1

associated with the equation (5.7). Proof. To prove this theorem, we use the same arguments as in the proof of Theorem 8, using the representation (5.2) instead of (5.1), and the operational relation (4.11) instead of (4.8). The Gauss hypergeometric function is a rather general function which has many elementary and special functions as particular cases (see [10]). We consider some examples of the integral equation (5.3) and its solution (5.4) for particular values of the parameters a and b. Example 1 Let a = 1, b = 1. Then the equation (5.3) has the form: 2

y(x) − λx


x 0

(ln x − ln u)y(u)

du = f (x), f ∈ C1 , x > 0, u2

(5.9)

and its unique solution in the space C1 is given by y(x) = f (x) + λx


1
1 0

0



I0 (2 λx(1 − u)(1 − v))f (xuv)

du dv , u2 v 2

(5.10)

where I0 (z) is the modified Bessel function of the zero index. We used here Theorem 8, the formula (4.10), and the fact that 1 F (1, 1; 2; z) = − ln(1 − z). z 13

Example 2 Let a = b = 12 . We have then (see [10]) 2 √ 1 1 F ( , ; 1; z) = K( z), 2 2 π where K(z) is the complete elliptic integral of the first kind. The equation (5.3) has in this case the form: 2 √ y(x) − λ x π




x

K

0



u du = f (x), f ∈ C0 , x > 0, 1− y(u) x u

and its solution takes the form: √
1
y(x) = f (x) + λ x 0

1

0



Φ1/2,1/2 (λ x(1 − u)(1 − v); 1/2, 1/2) √ √ f (xuv)du dv, uv 1 − u 1 − v

where Φα1 ,α2 (z; β1 , β2 ) is the generalized Mittag-Leffler type function defined by (4.9). Example 3 In the case a = 1, 0 < b < 2, b = 1, α = max{1, b}, we have the integral equation y(x) −

λ b+1 x 1−b


x 0

(x1−b − u1−b )y(u)

and its solution y(x) = f (x) + λ


1
1 I (2 0 0

du = f (x), f ∈ Cα , x > 0, u2



λx(1 − u)(1 − v)) u1+b v 2

0

f (xuv)du dv.

We used here the formula (see [10]) F (1, b; 2; z) =

1 [(1 − z)1−b − 1]. (b − 1)z

Example 4 The formula 1 1 F (a, a + ; 2a; z) = √ 2 1−z



2 √ 1+ 1−z

2a−1

gives us the unique solution in the space C2a− 1 of the integral equation 2

y(x) −

2a

λ(2x) 2Γ(2a)


x 0

√ √ du 1 ( x − u)2a−1 y(u) 2a+ 1 = f (x), a > , x > 0 2 u 2

of the form: a

y(x) = f (x)+λx


1
1 0

0

((1 − u)(1 − v))a−1 1 u2a+ 2 v 2a

14

Φa,a (λ(x(1−u)(1−v))a )f (xuv)du dv.

Acknowledgments The research carried out in this paper was supported by the Research Commission of the Free University of Berlin and, in the case of H.M. Srivastava, by DAAD (Deutscher Akademischer Austauschdienst).

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References [1] Al-Bassam, M.A and Luchko, Yu.F., On generalized fractional calculus and its application to the solution of integro-differential equations, J. Fract. Calc., 7 (1995), 69-88. [2] Buschman, R.G., Decomposition of an integral operator by use of Mikusi´ nski calculus, SIAM J. Math. Anal., 3 (1972), 83-85. [3] Dimovski, I.H., Convolutional Calculus, Vol. 2, Publ. House Bulgarian Acad. Sci., Sofia, 1982; Second ed.: Kluwer Acad. Publ., East Europ. Ser., Vol. 43, Dordrecht, 1990. [4] Dˇzrbashjan, M.M., On the integral transformations generated by the generalized Mittag-Leffler function, Izv. Akad. Nauk Arm. SSR, 13(3) (1960), 21-63 [in Russian]. [5] Erd´ elyi, A., Operational Calculus and Generalized Functions, Holt, Rinehart and Winston, New York, 1962. [6] Gorenflo, R. and Luchko, Yu., Operational method for solving generalized Abel integral equation of second kind, Integral Transforms and Special Functions, 5 (1997), 47-58. [7] Luchko, Yu.F. and Srivastava, H.M., The exact solution of certain differential equations of fractional order by using operational calculus, Comput. Math. Appl., 29 (1995), 73-85. [8] Mikusi´ nski, J. and Ryll-Nardzewski, G., Un theoreme sur le product de composition des fonctions de plusieurs variables, Studia Math., 13 (1953), 62-68. [9] Nicoliˇc-Despotoviˇc, D., Two integral equations in the field of Mikusi´ nski operators, Publ. Inst. Math. (Beograd) (N.S.), 18(32) (1975), 149-155.

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[10] Prudnikov, A.P., Brychkov, Yu.A. and Marichev, O.I., Integrals and Series. 3: More Special Functions, Gordon and Breach, New York, London, and Paris, 1989. [11] Samko, S.G., Kilbas, A.A. and Marichev, O.I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, London, and Paris, 1993. [12] Srivastava, H.M., Gupta, K.C. and Goyal, S.P., The H-Functions of One and Two Variables with Applications, South Asian Publ., New Delhi and Madras, 1982.

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