Computational Fractional Dynamical Systems

12Residual Power Series Method

12.1 Introduction

In Chapter 11, we have already discussed boundary characteristics orthogonal polynomials‐based solution of linear and nonlinear ordinary/partial/fractional differential equations where the orthogonal polynomials are generated by using the Gram–Schmidt orthogonalization procedure. In this chapter, we will discuss about residual power series method (RPSM). The RPSM was first developed in 2013 by Omar Abu Arqub, a Jordanian mathematician, to determine the values of the coefficients of power series solution for the first‐ and second‐order fuzzy differential equations (Arqub 2013). RPSM is an intuitive and reliable to construct power series solutions for linear and nonlinear equations without linearization, perturbation, or discretization. Over the last few years, the RPSM has been used to solve different nonlinear ordinary and partial differential equations (PDEs) of various forms, classifications, and orders. One may find the successful implementation of this method in the solutions of the generalized Lane–Emden equation (Arqub et al. 2013), higher‐order ordinary differential equations (Arqub et al. 2013), fractional coupled physical equations arising in fluids flow (Arafa and Elmahdy 2018), solitary pattern solutions for nonlinear time‐fractional dispersive PDEs (Arqub et al. 2015), and predicting and representing the multiplicity of solutions to boundary value problems of fractional order (Arqub et al. 2014), etc. The RPSM distinguishes ...