Variational Iteration Method For Differential Equation With Piecewise Continuous Arguments

Variational iteration method is a very effective approach,which can give approximate or exact solutions to both linear and nonlinear problems.Moreover,this technique reduces the volume of calculations because it has no need of discretization of the variables,linearization or small perturbations.So far,it has been applied to oscillation equations,wave equations,delay differential equations and fractional differential equations,and so on.In this paper,variational iteration method is applied to initial value problems of differential equation with piecewise continuous arguments.The main contents and results of this dissertation are listed as follows.In Chapter 1,the background of the variational iteration method is introduced,and we overview the development and applications of this method in recent years.Then the main contents of this dissertation is proposed.In Chapter 2,the variational iteration method is applied to obtain approximately analytical solutions of initial value problem of linear differential equation with piecewise continuous arguments.First,we construct the iteration scheme,substitute the initial value,and an iterative sequence is obtained.Then we prove that the iterative sequence converges to the exact solution of the problem.The numerical results show that our theoretical analysis are right and the variational iteration method is a powerful method for solving this kind of problem.In Chapter 3,the variational iteration method is applied to obtain approximately analytical solutions of initial value problem of nonlinear differential equation with piecewise continuous arguments.Firstly,the Lagrangian multiplier is obtained by using the basic theory of the variational iteration method.Then the iterative initial value is selected and the iterative scheme is obtained in the iterative scheme.The sequence is proved to be convergent.An example shows that the result is correct.In Chapter 4,the variational iteration method is applied to obtain approximately analytical solutions of initial value problem of partial differential equation with piecewise continuous arguments.First,the Lagrangian multiplier is determined,and then the Lagrangian multiplier and the selected initial value are substituted into the iterative scheme.Then an iterative solution sequence is obtained.Finally,the convergence of the iterative solution sequence is proved,and the correctness of the conclusion is verified by an example.