A Novel Method for Analytical Solutions of Fractional Partial Differential Equations

Mehmet Ali Akinlar,Muhammet Kurulay

doi:10.1155/2013/195708

Abstract

A new solution technique for analytical solutions of fractional partial differential equations (FPDEs) is presented. The solutions are expressed as a finite sum of a vector type functional. By employing MAPLE software, it is shown that the solutions might be extended to an arbitrary degree which makes the present method not only different from the others in the literature but also quite efficient. The method is applied to special Bagley-Torvik and Diethelm fractional differential equations as well as a more general fractional differential equation.

Highlights

By employing MAPLE software, it is shown that the solutions might be extended to an arbitrary degree which makes the present method different from the others in the literature and quite efficient
Fractional calculus is a significantly important and useful branch of mathematics having a broad range of applications at almost any branch of science
We present a new method for the analytical solutions of fractional partial differential equations (FPDEs)

Summary

Introduction

Fractional calculus is a significantly important and useful branch of mathematics having a broad range of applications at almost any branch of science. We have well-known definitions of a fractional derivative of order α > 0 such as Riemann-Liouville, Grunwald-Letnikow, Caputo, and generalized functions approach [9]. For the purpose of this paper, Caputo’s definition of fractional differentiation will be used, taking the advantage of Caputo’s approach that the initial conditions for fractional differential equations with Caputo’s derivatives take on the traditional form as for integer-order differential equations. The Riemann-Liouville fractional derivative is mostly used by mathematicians but this approach is not suitable for the physical problems of the real world since it requires the definition of fractional order initial conditions, which have no physically meaningful explanation yet. Caputo introduced an alternative definition, which has the advantage of defining integer order initial conditions for fractional order differential equations. We illustrate three computational examples as the application of the present method and complete the paper with a discussion section

A Novel Method for Analytical Solutions of FPDEs

Computational Applications

Conclusion