Advanced Numerical and Semi-Analytical Methods for Differential Equations

17Hybrid Methods

17.1 Introduction

Laplace transform method, Adomian decomposition method (ADM), and homotopy perturbation method (HPM) have been implemented for solving various types of differential equations in Chapters 2, 11, and 12, respectively. This chapter deals with the methods that combine more than one method. These have been termed as hybrid methods. Two main such methods, which are getting more attention of research, are homotopy perturbation transform method (HPTM) and Laplace Adomian decomposition method (LADM). Although the Laplace transform method is an effective method for solving ordinary and partial differential equations (PDEs), a notable difficulty with this method is about handling nonlinear terms if appearing in the differential equations. This difficulty may be overcome by combining Laplace transform (LT) with the HPM and ADM.

First, we briefly describe the HPTM in the following section.

17.2 Homotopy Perturbation Transform Method

Here, the Laplace transform method and HPM are combined to have a method called the HPTM for solving nonlinear differential equations. It is also called as Laplace homotopy perturbation method (LHPM).

Let us consider a general nonlinear PDE with source term g(x, t) to illustrate the basic idea of HPTM as below [1–3]

(17.1)

subject to initial conditions

(17.2)

where D is the linear differential operator (or ), R is the linear ...