Invariant Imbedding and Partial Differential Equations

Abstract

The usefulness of much of the theoretical results of mathematical physics rests on our ability to obtain numerical solutions of partial differential equations. The application of standard techniques developed for ordinary differential equations is complicated by two factors. First the dimension of the problem has increased many-fold. As we shall see, the conversion of partial differential equations to approximate systems of ordinary differential or difference equations can result in systems of such high dimension as to strain the memories of even the largest digitalcomputers. Second, we will be dealing with problems where conditions are specified on the boundary of a region. Hence, initial value methods cannot be applied directly.