**1. Introduction**

The governing equations of physical, biological and economical models often involve features which make it impossible to obtain their exact solution. For instance, problems where we observe "a complicated algebraic equations", "the occurrence of a complicated integral", in case of differential equations (DE), "a varying coefficients or nonlinear term" sometimes problems with an awkwardly shaped boundary are tough to solve with the limited methods for finding analytical solutions. The main purpose of this chapter is to describe the application of perturbation expansion techniques to the solution of DE. Approximate expressions are generated in the form of asymptotic series. These may not and often do not converge but in a truncated form of only two or three terms, provide a useful approximation to the original problem. These analytical techniques provide an alternative to the direct computer solution. Before attempting to solve these DE numerically, one should have an awareness of the perturbation approach. An example of this occurs in boundary layer problems where there are regions of rapid change of quantities such as fluid velocity, temperature or concentration. Appropriate scaling of the boundary layer dimension is required before a numerical solution can be generated which will capture the behavior in the rapidly changing region.

When a large or small parameter occurs in a mathematical model of a process there are various methods of constructing perturbation expansions for the solution of the governing equations. Often the terms in the perturbation expansions are governed by simpler equations for which the exact solution techniques are available. Even if exact solutions cannot be obtained, the numerical methods used to solve the perturbation equations approximately are often easier to construct than the numerical approximation for the original governing equation.

First, we consider a model problem for which an exact solution is available against which the perturbation expansion can be compared. A feature of the perturbation expansions is that they often form divergence series. The concept of an asymptotic expansion will be introduced and the value of a truncated divergent series will be demonstrated.
