Asymptotic expansions domain perturbation

We could obtain a general solution of Eq. (4-94). However, there is no obvious way to apply the boundary condition at the channel wall, at least in the general form (4-95) or (4-96). The method of domain perturbations provides an approximate way to solve this problem for e << 1. The basic idea is to replace the exact boundary condition, (4-96), with an approximate boundary condition that is asymptotically equivalent for e 1 but now applied at the coordinate surface y = d/2. The method of domain perturbations leads to a regular perturbation expansion in the small parameter s. 

The main point here is that the solution procedure for this particular problem of a singular (or matched) asymptotic expansion follows a very generic routine. Given that there are two sub-domains in the solution domain, which overlap so that matching is possible (the sub-domains here are the core and the boundary-layer regions), the solution of a singular perturbation problem usually proceeds sequentially back and forth as we add higher order... 

The first approximation of the deviation from sphericity is contained in the function f>(, t), and we have anticipated, from the preceding arguments, that the magnitude of term is proportional to Ca. The presumption that the expansion for /(x) proceeds in whose powers of Ca is based on the fact that domain perturbation techniques almost always lead to a regular (rather than singular) asymptotic structure. 

There are a number of variations for the perturbation technique. Among them, the method of matched asymptotic expansion is the easiest to apply. The method is useful for obtaining expansions from separate domains of validity. 

Beginning with this section, we will now consider partial differential equations. Let us consider a general approach for constructing asymptotic expansions of the solutions of singularly perturbed linear partial differential equations, which was proposed in the well-known fundamental work of Vishik and Lyusternik [27]. We will illustrate the idea of this approach (known in the literature as the method of Vishik-Lyusternik) on a simple example of an elliptic equation in a bounded domain with smooth boundary. 

---