Advanced Numerical and Semi-Analytical Methods for Differential Equations

11Adomian Decomposition Method

11.1 Introduction

The Adomian Decomposition Method (ADM) was first introduced by Adomian in the early 1980s [1,2]. It is an efficient semi‐analytical technique used for solving linear and nonlinear differential equations. It permits us to handle both nonlinear initial value problems (IVPs) and boundary value problems (BVPs).

The solution technique of this method [3,4] is mainly based on decomposing the solution of nonlinear operator equation to a series of functions. Each presented term of the obtained series is developed from a polynomial generated in the expansion of an analytic function into a power series. Generally, the abstract formulation of this technique is very simple, but the actual difficulty arises while calculating the required polynomials and also in proving the convergence of the series of functions. In this chapter, we present procedures for solving linear as well as nonlinear ordinary/partial differential equations by the ADM along with example problems for clear understanding.

The following section addresses the ADM for handling ordinary differential equations (ODEs).

11.2 ADM for ODEs

The ADM mainly depends on decomposing the governing differential equation 5,6] viz. F(x, p(x), p′(x)) = 0 into two components

(11.1)

where and ...