Analytical Solution for the Time Fractional BBM-Burger Equation by Using Modified Residual Power Series Method

Jianke Zhang,Yuelei Xiao,Zhirou Wei,Longquan Yong

doi:10.1155/2018/2891373

Abstract

In this study, a generalized Taylor series formula together with residual error function, which is named the residual power series method (RPSM), is used for finding the series solution of the time fractional Benjamin-Bona-Mahony-Burger (BBM-Burger) equation. The BBM-Burger equation is useful in describing approximately the unidirectional propagation of long waves in certain nonlinear dispersive systems. The numerical solution of the BBM-Burger equation is calculated by Maple. The numerical results show that the RPSM is reliable and powerful in solving the numerical solutions of the BBM-Burger equation compared with the exact solutions as well as the solutions obtained by homotopy analysis transform method through different graphical representations and tables.

Highlights

Today, fractional differential equations are more and more important in many fields, such as mathematics and dynamic systems [1, 2]
We can compare the exact solution of the BBM-Burger equation with the analytical approximate solution by graphics and charts
We discuss the analytical solution of the time fractional BBM-Burger equation by using residual power series method (RPSM)

Summary

Introduction

Fractional differential equations are more and more important in many fields, such as mathematics and dynamic systems [1, 2]. The persons who firstly proposed fractional differential equations were Leibniz and L’Hopital in 1695. Lakshmikantham and Vatsala [3] discussed the basic theory for the initial value problem involving Riemann-Liouville differential operators by fractional differential equations. Diethelm and Ford [4] proposed the analytical questions of existence and uniqueness of solutions by fractional differential equations. Many other academics studied different theories in fractional differential equations. Most of the problems do not possess analytical solution, and a lot of numerical methods have been developed to solve these fractional differential equations

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Conclusion