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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

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 [68].

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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