Principles of Homotopy Analysis

In this chapter we shall introduce the basic principles of the homotopy analysis method for nonlinear differential equations. In Section 2.1, we shall review the topological principle of homotopy and then discuss how this relates to the homotopy analysis method for constructing solutions to nonlinear differential equations. As with any perturbation method, we shall need a way to compute higher order corrections, and this is outlined in Section 2.2, where we discuss the higher order deformation equations. From here, we are then able to outline the general method of constructing series solutions to nonlinear differential equations and relevant initial or boundary value problems in Section 2.3. Often times when dealing with nonlinear differential equations, the question of whether solutions exist and are unique is of importance. Accordingly, in Section 2.4, we discuss the existence and uniqueness of solutions obtained via the homotopy analysis method. Throughout mathematics, there are often multiple ways to solve a given problem. In obtaining perturbation solutions to differential equations, there are multiple iterative routines one may employ. In Section 2.5, we compare the method of homotopy analysis to some other perturbation schemes, and highlight one primary benefit of the method — namely, that the homotopy analysis method solution does not require small model parameters. From here, we will be in a position to introduce more advanced methods which permit the control of convergence of solutions in Chapter 3 and advanced treatments for more complicated systems of equations in Chapter 4.