The Semianalytical Solutions for Stiff Systems of Ordinary Differential Equations by Using Variational Iteration Method and Modified Variational Iteration Method with Comparison to Exact Solutions

Mehmet Tarik Atay,Okan Kilic

doi:10.1155/2013/143915

Abstract

The Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM) are used to find solutions of systems of stiff ordinary differential equations for both linear and nonlinear problems. Some examples are given to illustrate the accuracy and effectiveness of these methods. We compare our results with exact results. In some studies related to stiff ordinary differential equations, problems were solved by Adomian Decomposition Method and VIM and Homotopy Perturbation Method. Comparisons with exact solutions reveal that the Variational Iteration Method (VIM) and the Modified Variational Iteration Method (MVIM) are easier to implement. In fact, these methods are promising methods for various systems of linear and nonlinear stiff ordinary differential equations. Furthermore, VIM, or in some cases MVIM, is giving exact solutions in linear cases and very satisfactory solutions when compared to exact solutions for nonlinear cases depending on the stiffness ratio of the stiff system to be solved.

Highlights

Stiff system of ordinary differential equations (ODEs) can be seen in modeling various real-world problems [1]
The purpose of this paper is to show the merits of Variation Iteration Method (VIM) and Modified Variation Iteration Method (MVIM) in solving some stiff systems of ordinary differential equations
The results of the Variation Iteration Method (VIM) and Modified Variation Iteration Method (MVIM) are compared with the exact solutions of the problems to show the efficiency of these methods

Summary

Introduction

Stiff system of ordinary differential equations (ODEs) can be seen in modeling various real-world problems [1]. Solving such a system of ODEs is important, and many numerical and analytical methods have been developed [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. The Variation Iteration Method is useful for obtaining exact and approximate solutions of linear and nonlinear differential equations. The results of the Variation Iteration Method (VIM) and Modified Variation Iteration Method (MVIM) are compared with the exact solutions of the problems to show the efficiency of these methods

Variation Iteration Method

Stiffness of a System of Ordinary Differential Equations

Applications

Discussion and Conclusions