Solution of Fractional Partial Differential Equations in Fluid Mechanics by Extension of Some Iterative Method

A A Hemeda

doi:10.1155/2013/717540

Abstract

An extension of the so-called new iterative method (NIM) has been used to handle linear and nonlinear fractional partial differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. Therefore, a general framework of the NIM is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation, and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the NIM with those obtained by both Adomian decomposition method (ADM) and the variational iteration method (VIM) reveals that the NIM is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.

Highlights

Recent advances of fractional differential equations are stimulated by new examples of applications in fluid mechanics, viscoelasticity, mathematical biology, electrochemistry, and physics
Throughout this work, fractional partial differential equations are obtained from the corresponding integer order equations by replacing the first-order or the second-order time derivative by a fractional in the Caputo sense [32] of order α with 0 < α ≤ 1 or 1 < α ≤ 2
New Iterative Method (NIM) has been known as a powerful tool for solving many functional equations such as ordinary, partial differential equations, integral equations, integrodifferential equations, and so many other equations

Summary

Introduction

Recent advances of fractional differential equations are stimulated by new examples of applications in fluid mechanics, viscoelasticity, mathematical biology, electrochemistry, and physics. The objective of this work is to extend the application of the NIM to obtain analytical solutions to some fractional partial differential equations in fluid mechanics. These equations include wave equation, Burgers equation, KdV equation, Klein-Gordon equation, and Boussinesq-like equation. The NIM is a computational method that yields analytical solutions and has certain advantages over standard numerical methods. It is free from rounding-off errors as it does not involve discretization and does not require large computer. Throughout this work, fractional partial differential equations are obtained from the corresponding integer order equations by replacing the first-order or the second-order time derivative by a fractional in the Caputo sense [32] of order α with 0 < α ≤ 1 or 1 < α ≤ 2

Preliminaries and Notations

Numerical Examples

Conclusion