Advanced Numerical and Semi-Analytical Methods for Differential Equations

14Homotopy Analysis Method

14.1 Introduction

Homotopy analysis method (HAM) [1–7] is one of the well‐known semi‐analytical methods for solving various types of linear and nonlinear differential equations (ordinary as well as partial). This method is based on coupling of the traditional perturbation method and homotopy in topology. By this method one may get exact solution or a power series solution which converges in general to exact solution. The HAM consists of parameter ℏ ≠ 0 called as convergence control parameter, which controls the convergent region and rate of convergence of the series solution. This method was first proposed by Liao [4]. The same was successfully employed to solve many types of problems in science and engineering [ 1 –8] and the references mentioned therein.

14.2 HAM Procedure

To illustrate the idea of HAM, we consider the following differential equation 1– 7 ] in general

(14.1)

where N is the operator (linear or nonlinear) and u is the unknown function in the domain Ω.

The method begins by defining homotopy operator H as below [7] ,

(14.2)

where p ∈ [0, 1] is an embedding parameter and ℏ ≠ 0 is the convergence control parameter [3, 7 ], u₀ is an initial approximation of the solution of Eq. (14.1), φ is an unknown function, and L is the auxiliary linear ...