Chapter 2 - Embedding-Parameters Perturbation Method

Abstract

In essence, the classical perturbation method depends on the availability of some small (or large) parameters, i.e., the so-called “perturbation variable.” However, in reality not all nonlinear problems have such perturbation variables, so the classical perturbation method has serious limitations. In this chapter, we introduce a novel method for solving nonlinear differential equations, i.e., the embedding-parameters perturbation method. By introducing special embedding-parameter transformations for both the independent and dependent variables, we can embed the specified small parameters into nonlinear differential equations, which then can be solved by a standard perturbation method. Solutions to the original differential equations can be obtained by substituting the asymptotic series solutions into the reversed transformation.