Computational Fractional Dynamical Systems

7Fractional Differential Transform Method

7.1 Introduction

In Chapter 6, we have already discussed the hybrid method, which combines homotopy perturbation method (HPM) and various transform methods. DTM is a semi‐analytical method based on the Taylor series expansion, which constructs an analytical solution in the form of a polynomial. Zhou (1986) first proposed the differential transform method (DTM) and initially applied it to initial value problems used in electrical circuits. Jang et al. (2001) claimed that DTM is an iterative process to obtain the solution of differential equations in the Taylor series form. Taylor's standard high‐order series method involves symbolic computation. Although the Taylor series method needs more computational work for higher orders, this method reduces the computational domain size and applies to several problems (Fatma 2004; Hassan 2004; Erturk and Momani 2008). In this chapter, we use DTM with a combination of Caputo fractional derivatives, which is called as fractional differential transform method (FDTM). The present method is based on the combination of the classical one‐dimensional FDTM and generalized Taylor's formula. The concerned authors may follow (Arikoglu and Ozkol 2006, 2007; Nazari and Shahmorad 2010; Methi 2016) for more details.

7.2 Fractional Differential Transform Method

There are various ways to generalize the notion of differentiation of fractional orders. The fractional differentiation in Riemann–Liouville sense is ...