Advanced Numerical and Semi-Analytical Methods for Differential Equations

13Variational Iteration Method

13.1 Introduction

The variational iteration method (VIM) is one of the well‐known semi‐analytical methods for solving linear and nonlinear ordinary as well as partial differential equations. He [1,2] developed the VIM and successfully applied to ordinary and partial differential equations. Further, the method was used by many researchers for solving linear, nonlinear, homogeneous, and inhomogeneous differential equations. The main advantage of the method lies in its flexibility and ability to solve nonlinear equations easily. The method can be used in bounded and unbounded domains as well. By this method one can find the convergent successive approximations of the exact solution of the differential equations if such a solution exists. Wazwaz [3] used the VIM for solving the linear and nonlinear Volterra integral and integro‐differential equations and explained clearly how to use this method for solving homogenous and inhomogeneous partial differential equations in Ref. [4].

13.2 VIM Procedure

In order to illustrate the VIM, we address the general nonlinear system [ 1 , 2 ,5] as

(13.1)

where L is the linear operator, N is the nonlinear operator, and g(x, t) is the given continuous function.

The correction functional for Eq. (13.1) may directly be written as [ 1 , 2 , 5 ]

(13.2)

where λ is Lagrange multiplier, which can be identified optimally via ...