Asymptotic expansion method in a singularly perturbed mckendrick problem

Abstract

In this chapter we generalize the examples from Chap. 2 by allowing for a continuous age structure of the population. This leads to the McKendrick model which is a system of partial differential equations with nonlocal boundary conditions. Here, the Tikhonov theorem cannot be applied and we use the asymptotic expansion introduced in Chap. 2. The proof of the convergence of the approximation must be adopted to this specific model and becomes quite complex, involving the analysis of initial, boundary and corner layers. For this we need some sophisticated tools from functional analysis and semigroups of operators theory, the rudiments of which are presented in the introductory sections of the chapter.