Existence and Uniqueness for Initial Value Problems

Abstract

In the previous chapter we considered some simple mathematical models of time-dependent processes that led to ordinary differential equations (ODEs) or differential-algebraic equations (DAEs). At that time we gave no thought to the question of whether solutions of such models exist at all or, if they do, whether they are unique. In each case this appeared to be obvious in the particular scientific context. But such an assertion is likely to be valid only for long-established models that describe well-known phenomena. When newer situations are modeled some caution is needed; indeed, the differential equation model might well fail to describe reality correctly, and while reality allows for a unique characterization, the simplified model may not do so. But even apparently traditional differential equation models can exhibit peculiarities that prevent the existence, at least locally, of a unique solution. We shall encounter several such cases in this chapter. Beyond this, most efficient and reliable algorithms turn out to contain in their essence the structure of a corresponding uniqueness theorem. This phenomenon is found already in the relatively simple example of the solution of linear systems of equations (see, e.g., [58]). For these reasons it is also important for a computational scientist or a numerical analyst to gain a clear understanding about the mathematical conditions under which the existence and uniqueness of solutions can be guaranteed.