Analytical study for time and time-space fractional Burgers\u2019 equation

Km Saad,Eman Hf Al-Sharif

doi:10.1186/s13662-017-1358-0

Abstract

In this paper, the variational iteration method (VIM) is applied to solve the time and space-time fractional Burgers’ equation for various initial conditions. VIM solutions are computed for the fractional Burgers’ equation to show the behavior of VIM solutions as the fractional derivative parameter is changed. The results obtained by VIM are compared with exact solutions and also with expansions of the exact solutions. VIM solutions are found to be in excellent agreement with these exact solutions.

Highlights

1 Introduction In recent years fractional calculus has been utilized to find solutions of equations governing the modeling of real materials in engineering and physics
The use of fractional calculus has gained the attention of scientists and engineers as, though counter-intuitive, the fractional derivative does arise in physical problems
5 The space-time fractional Burgers’ equation 5.1 Tanh initial condition we study variational iteration method (VIM) solutions of the space-time fractional Burgers’ equation

Summary

Introduction

In recent years fractional calculus has been utilized to find solutions of equations governing the modeling of real materials in engineering and physics. The use of fractional calculus has gained the attention of scientists and engineers as, though counter-intuitive, the fractional derivative does arise in physical problems. In this connection there is the work of [ – ]. Due to the difficulty of obtaining exact solutions of equations involving a fractional derivative, approximate and numerical techniques have tended to be used instead. In this paper we study approximate solutions using VIM for the time and space-time fractional Burgers’ equation. The fourth and fifth sections are devoted to the application of VIM to evaluate solutions of the time and space-time fractional Burgers’ equation, respectively.

The time-fractional Burgers’ equation

Polynomial initial condition

Conclusions