Convergence Of Variational Iteration Method For Caputo Fractional Differential Equations And Neutral Differential Equations With Pantograph Delay

In recent decades, fractional derivatives and fractional integrals have been widely used in many fields such as colored noise, control theory, chaos etc. The exact solutions of many fractional differential equations are expressed in some special functions, and so these make them difficult to be calculated out. There are even many equations that can not be solved analytically, so it is necessary to solve them for the approximation analytical solutions or the numerical solutions. Neutral differential equations with proportional delay is widely used in the fields of physics, biology etc. It can simulate many phenomena in the real world. In recent years, the problems of its numerical solutions have drawn the attention of many researchers.Although the variational iteration method (VIM) has been successfully used to calculate the analytical solutions or the approximation analytical solutions of many kinds of differential equations, integral equations and integro-differential equations, there is a little research on the convergence of VIM for these problems. In this paper, the variational iteration method is applied to solve initial value problems of Caputo fractional differential equations and initial value problems of neutral differential equations with pantograph delay. The convergence of the variational iteration method for solving the initial value problems of these two kinds of equations has been proved. The numerical examples show the efficiency of the variational iteration method for solving the initial value problems of these two kinds of equations.