The Time-Fractional Coupled-Korteweg-de-Vries Equations

Abdon Atangana,Aydin Secer

doi:10.1155/2013/947986

Abstract

We put into practice a relatively new analytical technique, the homotopy decomposition method, for solving the nonlinear fractional coupled-Korteweg-de-Vries equations. Numerical solutions are given, and some properties exhibit reasonable dependence on the fractional-order derivatives’ values. The fractional derivatives are described in the Caputo sense. The reliability of HDM and the reduction in computations give HDM a wider applicability. In addition, the calculations involved in HDM are very simple and straightforward. It is demonstrated that HDM is a powerful and efficient tool for FPDEs. It was also demonstrated that HDM is more efficient than the adomian decomposition method (ADM), variational iteration method (VIM), homotopy analysis method (HAM), and homotopy perturbation method (HPM).

Highlights

Fractional calculus has been used to model physical and engineering processes, which are found to be best described by fractional differential equations
Since only limited classes of equations are solved by analytical means, numerical solution of these nonlinear partial differential equations is of practical importance
We present the basic ideal of the homotopy decomposition method for solving high-order nonlinear fractional partial differential equations

Summary

Introduction

Fractional calculus has been used to model physical and engineering processes, which are found to be best described by fractional differential equations. In [1], homotopy analysis method is applied to obtain approximate analytical solution of the modified Kuramoto-Sivashinsky equation. We extend the application of the homotopy decomposition method (HDM) in order to derive analytical approximate solutions to nonlinear time-fractional coupledKDV equations. This coupled system is used to describe iterations of water waves proposed by Hirota and Satsuma [12]. We present the basic ideal of the homotopy decomposition method for solving high-order nonlinear fractional partial differential equations.

Fractional Derivative Order

Properties and Definitions

Basic Idea of the HDM

Convergence of the Method and Unicity of the Solution

Application

Conclusions