On Solution Representation of Generalized Abel Integral Equation

Arsen Pskhu

doi:10.1155/2013/106251

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.

Highlights

Consider the equation m∑λkD0−xαk u (x) = f (x), (1)k=0 where αk ≥ 0, λk ∈ R, x ∈ (0, l), and D0βx is the RiemannLiouville operator of fractional integrodifferentiation of order β, which is defined as follows: D0βxφ (x) = x ∫ φ (t) (x −t)−β−1dt for β < 0, D00xφ (x) = φ (x), D0βxφ (x) = dn dxn
Generalized Abel integral equations of the second kind were investigated by operational method in [6]
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

Summary

Introduction

K=0 where αk ≥ 0, λk ∈ R, x ∈ (0, l), and D0βx is the RiemannLiouville operator of fractional integrodifferentiation of order β, which is defined as follows: D0βxφ (x). Generalized Abel integral equations of the second kind were investigated by operational method in [6]. We obtain a representation for solution of (1) The results cover both cases, the solution of equation of the first kind (α0 > 0) and that of the second kind (α0 = 0). 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

Preliminaries

Auxiliary Results

Main Result