Fundamental Solutions for Differential Operators and Applications

Overview Many problems in mathematical physics and applied mathematics can be reduced to boundary value problems for differential, and in some cases, inte­ grodifferential equations. These equations are solved by using methods from the theory of ordinary and partial differential equations, variational calculus, operational calculus, function theory, functional analysis, probability theory, numerical analysis and computational techniques. Mathematical models of quantum physics require new areas such as generalized functions, theory of distributions, functions of several complex variables, and topological and al­ gebraic methods. The main purpose of this book is to provide a self contained and system­ atic introduction to just one aspect of analysis which deals with the theory of fundamental solutions for differential operators and their applications to boundary value problems of mathematical physics, applied mathematics, and engineering, with the related applicable and computational features. The sub­ ject matter of this book has its own deep rooted theoretical importance since it is related to Green's functions which are associated with most boundary value problems. The application of fundamental solutions to a recently devel­ oped area of boundary element methods has provided a distinct advantage in that an integral equation representation of a boundary value problem is often x PREFACE more easily solved by numerical methods than a differential equation with specified boundary and initial conditions. This situation makes the subject more attractive to those whose interest is primarily in numerical methods.