Computational Solution of a Fractional Integro-Differential Equation

Muhammet Kurulay,Ranis Ibragimov,Mehmet Ali Akinlar

doi:10.1155/2013/865952

Abstract

Although differential transform method (DTM) is a highly efficient technique in the approximate analytical solutions of fractional differential equations, applicability of this method to the system of fractional integro-differential equations in higher dimensions has not been studied in detail in the literature. The major goal of this paper is to investigate the applicability of this method to the system of two-dimensional fractional integral equations, in particular to the two-dimensional fractional integro-Volterra equations. We deal with two different types of systems of fractional integral equations having some initial conditions. Computational results indicate that the results obtained by DTM are quite close to the exact solutions, which proves the power of DTM in the solutions of these sorts of systems of fractional integral equations.

Highlights

The subject of the present paper is to investigate the applicability of the differential transform method to the systems of the two-dimensional Volterra integro-differential equations of the second kind
Using the properties of DTM, it is not hard to show that the function u(x, y) can be represented as
The method was used in a direct way without using linearization, perturbation, or restrictive assumptions

Summary

Introduction

The subject of the present paper is to investigate the applicability of the differential transform method to the systems of the two-dimensional Volterra integro-differential equations of the second kind. To the best of our knowledge, the Volterra-integro differential equations considered in this paper was not studied with any method in the literature. Solving a new equation with differential transform method is our main purpose in this paper. For this propose, we consider the system of two-dimensional fractional Volterra integro-differential equations in the form of. The Volterra integral equations find application in many different areas including sorption kinetics, demography, viscoelastic materials, oscillation of a spring, financial mathematics, stochastic dynamical systems, and mathematical biology. Having defined the problem previously, we describe the differential transform method shortly

Differential Transform Method

Computational Applications

Conclusions