Computational Fractional Dynamical Systems

4Adomian Decomposition Transform Method

4.1 Introduction

In Chapter 3, we have already discussed the Adomian decomposition method (ADM), which is a semi‐analytical approach for solving differential equations. In this chapter, we will discuss the hybrid methods, which are the coupling of ADM with various transform methods, viz. Laplace transform (LT), Sumudu transform (ST), Elzaki transform (ET), and Aboodh transform (AT). With the combination of these transform methods, ADM is called the Adomian decomposition transform method (ADTM) (Mohammed and Salim 2018; Ahmed et al. 2019; Thabet and Kendre 2019; Khalouta and Abdelouahab 2020). Although these four transform methods are helpful for solving fractional differential equations, these methods sometimes fail to handle nonlinear terms arising in the fractional differential equations. These difficulties may be overcome by coupling these transform methods with ADM. The theories behind the four transform methods with respect to fractional order are introduced in the subsequent sections. It is worth mentioning that two simple application example problems of fractional differential equations are investigated using all four methods.

4.2 Transform Methods for the Caputo Sense Derivatives

Definition 4.1

The LT of the Caputo fractional derivative is defined as (Baleanu and Jassim 2019; Faraz et al. 2010):

(4.1)

Definition 4.2 ...