The Difference Schemes for the Diffusion Equation

Abstract

Contemporary numerical methods for solving problems of the mathematical chemistry are gaining increasing popularity. The aim of this chapter is to introduce the reader with the relevant facts about the basic concepts of the theory of the difference schemes for the linear diffusion equations. The linear diffusion equations play an important and crucial role in most models of a biosensor theory. The selection of the difference methods for the solution of these equations is motivated by the following two arguments:

• The geometry (size) of a biosensor really does not change during the measurements;

• The simplicity and efficiency of the difference method.

The most popular simple and together effective difference schemes are presented here. These difference schemes are extensively applied to the solution of a biosensor problems in the next chapter. This method is being frequently used in solving applied problems not only by professional mathematicians, but also by laymen. The concepts presented below are of a primary nature and are sufficient for the solution of the problems of the biosensor. In this book the notations of [222] are mainly applied. The many aspects of the numerical methods for the solution of the partial differential equations are presented in [5, 12, 187, 222].