Journal of Mathematics

Volume 2013 (2013), Article ID 106251, 5 pages

http://dx.doi.org/10.1155/2013/106251

Research Article

On Solution Representation of Generalized Abel Integral Equation

Arsen Pskhu

Scientific Research Institute of Applied Mathematics and Automation, Kabardino-Balkar Scientific Centre, Russian Academy of Sciences, 89A Shortanov Street, Nalchik 360000, Russia

Received 6 December 2012; Accepted 31 January 2013

Academic Editor: Mike Tsionas

Copyright © 2013 Arsen Pskhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

An explicit representation for solution of the generalized Abel integral equation , where is the Riemann-Liouville fractional integral, in terms of the Wright function, is constructed.

1. Introduction

Consider the equation

where ,  ,  , and   is the Riemann-Liouville operator of fractional integrodifferentiation of order , which is defined as follows:

For suitable function , (1) with and ,  ; that is, the Abel integral equation of the first kind (see [1])

may be inverted by the formula

(see [2]). For ,  ,  ,  , and  , (1) has the following form

which is the Abel equation of the second kind, and its solution may be given by the Hille-Tamarkin formula

(see [3]), which can be rewritten as

Here,

is the Mittag-Leffler function.

Equations of type (1) with rational parameters were considered in [4, 5]. Generalized Abel integral equations of the second kind were investigated by operational method in [6]. For a more complete survey on results relating Abel type integral equations and fractional order integral and differential equations and their applications, we refer to [6–8].

In this paper, we obtain a representation for solution of (1). The results cover both cases, the solution of equation of the first kind () and that of the second kind (). The solution is constructed in terms of the Wright function. It should be noted that (1) can be reduced to the generalized Abel integral equation of the second kind, and the method developed in [6] can be applied for (1). This provides an alternative approach to the equation under study.

2. Preliminaries

Now, we recall several properties of the operators of fractional integration and differentiation and special functions which are required for what follows.

The following three relations are valid:

for and or and ;

for and , ;

for and or and (e.g., see [2] and [9, Section 2.1]).

By , we denote the Laplace convolution of functions and :

Operator may be written in the form of convolution

Using (11) and (13) and taking into account the associativity and commutativity of convolution, we get

where ,  , and

with and . The relation

is a particular case of (15).

Recall that the Wright function

satisfies the relations

(see [10]) and

For , relation (19) may be proved by termwise applying formula (11) to the representation of the Wright function as a series (17). Combining (18) and (19), we obtain

3. Auxiliary Results

Here, we investigate properties of functions in terms of which a solution of integral equation (1) will be constructed.

Consider the function

(see [11]), where

In what follows, the parameters and arguments of function (21) satisfy

Remark 1. The function is independent of the distribution of parameters but depends only on their sum . Indeed, let and be the values of functions corresponding to the sets and , with same sets of , , and , and

Let

Then, using (15) and (19), we obtain

Lemma 2. If

then

with positive constants and independent of and .

Proof. Using the asymptotic formula of the Wright function (see [10]), we have

where and are positive constants independent of and . If we recall the definition (21), we get

Using (14), we obtain (28).

Lemma 3. If , then

Proof. If , then combining (15), (19), and (21), we get

Now, let ,  . As follows from Remark 1, in (21), we can take a parameter such that . Therefore, in view of the equality

formula (19), and the relation

we see that

This completes the proof.

Now, we set

The convergence of the integral in (36) follows from (28).

Lemma 4. The function satisfies the relations

Proof. From (28) and (36), we obtain

where

and and are positive constants independent of , , and . By the relation

which holds for and , (39) yields (37).

Relation (38) follows from (28), (31), and definition (36).

Lemma 5. Let , for , and . Denote that

Then, the function is a solution of the integral equation

Proof. Let

Using (20), (21), and (31) and bearing in mind Remark 1, we get

Hence, in view of (36) and (42),

Taking account of the equality

and (14), the relation (46) yields (43).

4. Main Result

Theorem 6. Let , for , , and there exists a function such that . Then, (1) has a unique integrable solution , which for each can be represented as follows:

where is defined by (42).

Proof. Let be a solution of (1). Taking into account (16) and (43), we obtain

or, in view of (13),

Applying the operator to both sides of this equality, bearing in mind (9), we obtain (48).

Now, we prove that a function that is defined by (48) is a solution of (1). Let ,  . Consider the function

Bearing in mind (16), (38), and (42), we get

Using (33), (37), and (38), we obtain

Taking this and relation (37) into account, we may conclude that the function is absolutely continuous and

for , and

Using (10) and (54), we obtain

Then, it follows that

This completes the proof.

Remark 7. Let , , and for . In this case, (1) will appear like (3), and we get

In addition, it follows from the equalities

that if ,  ,  ,  , and , then

Thus, formula (48) coincides with (4) and (7) for the cases of (3) and (5) (i.e., for and ), respectively.

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