Homotopy Perturbation Method for Fractional Gas Dynamics Equation Using Sumudu Transform

Jagdev Singh,A Kılıçman,Devendra Kumar

doi:10.1155/2013/934060

Abstract

A user friendly algorithm based on new homotopy perturbation Sumudu transform method (HPSTM) is proposed to solve nonlinear fractional gas dynamics equation. The fractional derivative is considered in the Caputo sense. Further, the same problem is solved by Adomian decomposition method (ADM). The results obtained by the two methods are in agreement and hence this technique may be considered an alternative and efficient method for finding approximate solutions of both linear and nonlinear fractional differential equations. The HPSTM is a combined form of Sumudu transform, homotopy perturbation method, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The numerical solutions obtained by the proposed method show that the approach is easy to implement and computationally very attractive.

Highlights

Fractional calculus is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders
In Eltayeb et al [46], the Sumudu transform was extended to the distributions and some of their properties were studied in Kılıcman and Eltayeb [47]
The Sumudu transform sends combinations, C(m, n), into permutations, P(m, n), and it will be useful in the discrete systems

Summary

Introduction

Fractional calculus is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders. Singh et al [35] have paid attention to study the solutions of linear and nonlinear partial differential equations by using the homotopy perturbation Sumudu transform method (HPSTM). The objective of the present paper is to extend the application of the HPSTM to obtain analytic and approximate solutions to the time-fractional gas dynamics equation. The advantage of the HPSTM is its capability of combining two powerful methods for obtaining exact and approximate analytical solutions for nonlinear equations. It provides the solutions in terms of convergent series with computable components in a direct way without using linearization, perturbation, or restrictive assumptions. It is worth mentioning that the HPSTM is capable of reducing the volume of the computational work as compared to the classical methods while still maintaining the high accuracy of the numerical result; the size reduction amounts to an improvement of the performance of the approach

Sumudu Transform

Basic Definitions of Fractional Calculus

Conclusions