HPM for Solving the Time-fractional Coupled Burgers Equations

Khadijah Abu Alnaja

doi:10.24297/jam.v12i4.350

Abstract

In this paper, the homotopy perturbation method is implemented to derive the explicit approximate solutions for the time-fractional coupled Burger's equations. The including fractional derivative is in the Caputo sense. Special attention is given to prove the convergence of the method. The results are compared with those obtained by the exact at special cases of the fractional derivatives. The results reveal that the proposed method is very effective and simple.

Highlights

In this paper, we will implement one of these methods, namely, homotopy perturbation method (HPM) ([7]-[10]), to solve the propsed problem ([1], [3], [6], [12])
The applications of homotopy theory among scientists appeared, and the homotopy theory becomes a powerful mathematical tool, when it is successfully coupled with perturbation theory
We will illustrate how implement HPM to obtain the approximate solutions of the fraction non-linear coupled system of Burger's equations of the following form: D u uxx 2uuxx 0, 0 1, t > 0, (5)

Summary

INTRODUCTION

We will implement one of these methods, namely, homotopy perturbation method (HPM) ([7]-[10]), to solve the propsed problem ([1], [3], [6], [12]). Unlike the traditional numerical methods, such as, the finite element method [5] and the finite difference method [19], HPM does not need, discretization or linearization. The applications of homotopy theory among scientists appeared, and the homotopy theory becomes a powerful mathematical tool, when it is successfully coupled with perturbation theory. The Caputo fractional derivative of the function g(t) is defined as follows: D g (t ). The Caputo fractional derivative operator is satisfied the linear relationship and the so-called Leibnitz rule, that's:. For the Caputo's derivative we have D C 0, C is a constant, and [18]. (4) For more details on fractional derivatives definitions and their properties see ([13], [18])

Approximate Solution using HPM

Conclusion and General Remarks