Applications Of Modified Adomian Decomposition Method To Nonlinear Equations

Whether in the field of natural sciences or social sciences,nonlinear problems have attracted much attention.Adomian decomposition method(namely ADM)is a popular method used to solve the approximate analytic solution of linear and nonlinear problems.The Adomian decomposition method was first introduced by Adomian since the beginning of the 1980's,The main idea of the method are firstly,decompose the equation in several parts,linear and nonlinear parts;secondly,The solution of the equation is resolved into infinite components;thirdly,The special polynomial instead of the nonlinear parts;finally,components are gradually calculated from low order to high order.Since Adomian was proposed,it has been optimized and improved by many experts and scholars.In 2010,Wazwaz proposed a new method(LADM),which combines Laplace transform with Adomian decomposition method,and the method is developed for analytic treatment of the nonlinear Volterra integro–differential equations.Zhang Xiaolong et al proposed a new analytic approximation method with a convergence acceleration parameter c.It turns out that the convergence region and rate can be greatly enlarged by choosing a proper value of c.In this paper,a modified ADM is presented.First,a convergence parameter is introduced into the equation.Then,the ADM with parameters is combined with Laplace transform.And the modified method reduces to the traditional LADM when c = 1.It is demonstrated by examples that solutions given by the modified method are more accurate than solutions given by the LADM when the parameters are optimal.This paper is organized as follows.In chapter 1,the thesis elaborated the background of the research on nonlinear problems and ADM.In chapter 2,several definitions of fractional calculus are briefly introduced,and the basic principle and some improvements of ADM are emphatically introduced.In chapter 3,the modified ADM is developed,and the method is applied to construct the analytic solutions of nonlinear fractional Riccati equation.In chapter 4,some nonlinear equations are solved with this new method.In the last section,some concluding remarks are given.