Exact solutions for some of the fractional integro-differential equations with the nonlocal boundary conditions by using the modification of He’s variational iteration method

Abstract

AbstractIn this paper, the modiflcation of He’s variational iteration method (MVIM) is developed to solve fractional integro-difierential equations with nonlocal boundary conditions. It is shown that by choosing suitable initial approximation,the exact solution obtains by one iteration. It is illustrated that the propose method is efiective and has high con-vergency rate. Keywords : Fractional integro-difierential equations, nonlocal boundary condition, modiflcation of He’s variational iteration method. 1 Introduction The variational iteration method was flrst proposed by He [7, 8, 9] and has been worked out over a number of yearsby many authors. This method has been shown to efiectively, easily and accurately solve a large class of nonlinearproblems. Generally, one or two iterations lead to high accurate solutions.This method is, in fact, a modiflcation ofthe general Lagrange multiplier method into an iteration method, which is called correction functional. Applicationsof the method have been enlarged due to its °exibility, convenience and e–ciency. The convergence of the methodis systematically discussed by Tatari and Dehghan [17] , Odibat [14]. There are several modiflcations of He’s VIM[6, 7, 8, 9, 10]. In this paper we propose the reliable modiflcation of He’s VIM (MVIM) that was introdused byGhorbani et al. [4] for solving the fractional order integro-difierential equations with nonlocal boundary conditionsby constructing an initial trial-function without unknown parameters, so that one iteration leads to exact solution.Consider fractional order integro-difierential equation of the form