Some Research On Variational Iteration Method

This paper is mainly about some application of VIM in the numerical solution of integral equations.There are four chapters in all.First,we make a simple introduction about the common methods of the numerical solution to integral equations,including iterative methods and projective methods.All these methods make a basis of VIM.At the same time,we give the knowledge of variation,which is the theory basis of VIM.In chapter 2,we give an elementary introduction to the concept of VIM. First,the main concepts in VIM,such as general Lagrange multiplier,restricted variation,correction functional,are explained heuristically.Subsequently we give the procedure of VIM and a simple statement about the convergence.In chapter 3,the VIM is applied to solve the integral equations system of the second kind,including the Fredholm integral equations system of the second kind and Volterra integral equations system of the second kind.We give the concrete VIM for the integral equations system of the second kind.The key lies in the derivation of the equations,which makes the VIM feasible.Then,we can give the iterative formula without difficulty.Last,two numerical examples show our method is efficient and having an obvious advantage compared with the related reference.Chapter 4 is mainly about the application of VIM in high-order integro-differential equations.We give an indirect VIM to solve high-order Volterra integro-differential equations,that is,reducing the high-order Volterra integro-differential equations to an ordinary system of equations,then making an iterative formula for each variables.This method gives a simple expression of Lagrange multiplierλ,i.eλ=-1,which saves the procedure to computeλ.Last,the numerical results obtained with minimum amount of computation are compared with the exact solutions and the results from related reference to show the efficiency of the method.The results show that our method is of high accuracy and efficient for solving high-order Volterra integro-differential equations.