Research Article | Open Access

# On Solution Representation of Generalized Abel Integral Equation

**Academic Editor:**Mike Tsionas

#### 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.

#### References

- N. H. Abel, “Auflösung einer mechanischen aufgabe,”
*Journal für Die Reine und Angewandte Mathematik*, vol. 1, pp. 153–157, 1826. View at: Google Scholar - J. D. Tamarkin, “On integrable solutions of Abel's integral equation,”
*Annals of Mathematics*, vol. 31, no. 2, pp. 219–229, 1930. View at: Publisher Site | Google Scholar | MathSciNet - E. Hille and J. D. Tamarkin, “On the theory of linear integral equations,”
*Annals of Mathematics*, vol. 31, no. 3, pp. 479–528, 1930. View at: Publisher Site | Google Scholar | MathSciNet - V. Kostitzin, “Sur une généralisation de l'équation intégrale d'Abel,”
*Comptes Rendus de l'Académie des Sciences*, vol. 224, pp. 885–887, 1947. View at: Google Scholar | Zentralblatt MATH | MathSciNet - P. Rieder, “Die Auflösung einer Klasse verallgemeinerter Abelscher Integralgleichungen,”
*Mathematische Zeitschrift*, vol. 109, pp. 29–52, 1969. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. Gorenflo and Y. Luchko, “Operational method for solving generalized Abel integral equation of second kind,”
*Integral Transforms and Special Functions*, vol. 5, no. 1-2, pp. 47–58, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. Gorenflo and S. Vessella,
*Abel Integral Equations: Analysis and Applications*, vol. 1461, Springer, Berlin, Germany, 1991. View at: MathSciNet - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives: Theory and Applications*, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993. View at: MathSciNet - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204 of*North-Holland Mathematics Studies*, Elsevier, Amsterdam, The Netherlands, 2006. - E. M. Wright, “On the coefficients of power series having exponential singularities,”
*The Journal of the London Mathematical Society*, vol. 8, no. 1, pp. 71–79, 1933. View at: Publisher Site | Google Scholar | MathSciNet - A. V. Pskhu, “Initial-value problem for a linear ordinary differential equation of noninteger order,”
*Sbornik: Mathematics*, vol. 202, no. 4, pp. 111–122, 2011. View at: Google Scholar

#### Copyright

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.