Computational Fractional Dynamical Systems

3Adomian Decomposition Method

3.1 Introduction

In Chapter 1, we have already discussed preliminaries and notations of fractional calculus. The development of recent and robust methods for solving linear and nonlinear ordinary/partial/fractional differential equations has been demonstrated in Chapter 2. In this chapter, we will discuss about Adomian decomposition method (ADM). The ADM was first introduced by Adomian in the early 1980s (Adomian 1990; Wazwaz 1998). It is a semi‐analytical approach for solving linear and nonlinear ordinary/partial/fractional differential equations. It allows us to handle both nonlinear initial and boundary values problems. The method of solution of this method (Evans and Raslan 2005; Momani and Odibat 2006) is based primarily on decomposing the nonlinear operator equation to a set of functions. Each series term is constructed from a polynomial generated by expanding an analytic function into a power series. The theoretical formulation of this technique is usually quite simple, but the actual difficulty arises when calculating the polynomials involved or when proving the convergence of the series of functions. In this chapter, we present the ADM procedure to solve linear and nonlinear fractional partial differential equations (PDEs) along with examples.

In the subsequent sections, firstly, the theories behind the method with respect to fractional order are given. Then the systematic study of the technique, as mentioned earlier, and two problems ...