Advanced Numerical and Semi-Analytical Methods for Differential Equations

9Akbari–Ganji's Method

9.1 Introduction

There exist a variety of analytical and numerical methods for solving linear differential equations as discussed in Chapters 1–8, but methods for solving nonlinear differential equations 1,2] are quite challenging. In this regard, this chapter focuses on an innovative algebraic approach proposed by Akbari et al. [3] for solving nonlinear differential equations. However, in Ref. [4], it is also termed as a semi‐analytic method. The algebraic approach was initially referred to as Algebraic Method and then renamed as Akbari–Ganji's method (AGM). Akbari et al. [5] gave a detailed discussion on solving nonlinear, non‐vibrational, vibrational, and integro‐differential equations using the AGM along with Maple codes.

There exist various other semi‐analytical approaches for solving nonlinear differential equations viz. Lyapunov's small parameter method, Adomian decomposition method (ADM), etc. Later in Chapter >11, the ADM approach is discussed for solving nonlinear (ordinary or partial) differential equations. There also exist other new techniques viz. homotopy perturbation method (HPM), variational iteration method (VIM), homotopy analysis method (HAM), etc. which are further considered in Chapters 12–14. In this chapter we will mainly focus on solving nonlinear ordinary differential equations only using the AGM. The chapter is organized such that the next section consists of preliminaries and AGM approach for solving nonlinear ordinary differential ...