A Banach space solution of an abstract Cauchy problem which has Robin boundary data
In this paper, a problem which involves the diffusion equation coupled by way of Robin boundary data with a nonlinear system of ordinary differential equations is considered. This problem has been proposed as a model to describe the absorption through the skin, the distribution throughout the body, and the metabolism of a substance in a mammal. The problem is set as an abstract Cauchy problem in a Banach space and is shown to have a unique solution. Finite dimensional approximations of the problem are obtained by replacing the spatial partial derivatives with finite differences. The approximate solutions are shown to converge to the exact solution of the original problem. Comparisons of numerical solutions with experimental data are presented.