Revised Variational Iteration Method for Solving Systems of Nonlinear Fractional-Order Differential Equations

H. Jafari,D. Baleanu,C. Ünlü

doi:10.1155/2013/461837

Abstract

A modification of the variational iteration method (VIM) for solving systems of nonlinear fractional-order differential equations is proposed. The fractional derivatives are described in the Caputo sense. The solutions of fractional differential equations (FDE) obtained using the traditional variational iteration method give good approximations in the neighborhood of the initial position. The main advantage of the present method is that it can accelerate the convergence of the iterative approximate solutions relative to the approximate solutions obtained using the traditional variational iteration method. Illustrative examples are presented to show the validity of this modification.

Highlights

Fractional-order calculus has been studied as an alternative calculus in mathematics
Chemistry, biology, and engineering can be modeled with fractional derivatives [1,2,3,4,5,6,7,8,9,10,11,12]
The objective of this paper is the use of revised variational iteration method (RVIM) for solving systems of nonlinear fractional-order differential equations

Summary

Introduction

Fractional-order calculus has been studied as an alternative calculus in mathematics. Since most fractional differential equations do not have exact analytic solutions, approximate and numerical techniques, are used extensively. The variational iteration method is relatively new approaches to provide approximate solutions to linear and nonlinear problems. The variational iteration method, which is proposed by He [19], was successfully applied to find the solutions of several classes of variational problems. Daftardar-Gejji and Jafari [28] have explored the Adomian decomposition method to obtain solution of a system of linear and nonlinear fractional differential equations. Further in [29], they have suggested a modification (termed as “revised ADM”) of this method and applied revised method for solving systems of linear/nonlinear ordinary and fractional differential equations [30]. The objective of this paper is the use of revised variational iteration method (RVIM) for solving systems of nonlinear fractional-order differential equations.

Definitions and Preliminaries

The VIM for FDE

The System of FDE and Revised VIM

Test Examples

Conclusion