Numerical Methods for Fractional Order Singular Partial Differential Equations with Variable Coefficients

Asma Ali Elbeleze,Bachok M Taib,Adem Kılıçman

doi:10.1155/2014/398286

Abstract

We implement relatively analytical methods, the homotopy perturbation method and the variational iteration method, for solving singular fractional partial differential equations of fractional order. The process of the methods which produce solutions in terms of convergent series is explained. The fractional derivatives are described in Caputo sense. Some examples are given to show the accurate and easily implemented of these methods even with the presence of singularities.

Highlights

In the last decades, fractional calculus found many applications in various fields of engineering and physical sciences such as physics, chemistry, biology, economy, viscoelasticity, electrochemistry, electromagnetic, relaxation processes, diffusion, control, porous media, and many more; see, for example, [1,2,3,4,5,6].The numerical solution of singular differential equations of integer order has been a hot topic in numerical and computational mathematics for a long time [7, 8]
Considerable attention has been given to the solution of singular partial differential equations of fractional order
The objective of the present paper is to extend the application of homotopy perturbation method to provide approximate solutions and to make comparison with that obtained by the variational iteration method for singular partial differential equations of fractional order:

Summary

Introduction

The numerical solution of singular differential equations of integer order has been a hot topic in numerical and computational mathematics for a long time [7, 8]. Singular partial differential equations of fractional order, as generalizations of classical singular partial differential equations of integer order, are increasingly used to model problems in physics and engineering. Considerable attention has been given to the solution of singular partial differential equations of fractional order. In most of these equations analytical solutions are either quite difficult or impossible to achieve, so approximations and numerical techniques must be used. Several methods have been used to solve singular differential equations such as variational iteration method [8], homotopy perturbation method [7], and homotopy analysis method [9]

Objectives

Methods

Conclusion