Boundary Value Problems of Mathematical Physics: Volume 2

Excerpt

The first volume of Boundary Value Problems of Mathematical Physics, published in 1967, develops the mathematical foundations required for the study of linear partial differential equations, the subject matter of the present volume.

The field of partial differential equations has grown to such an extent in recent years that it would be impossible to cope adequately with all its aspects in a book of manageable size. Since I am addressing myself principally to graduate students in engineering and the physical sciences, I have emphasized methods for finding solutions in useful form. Even so, the more abstract questions of uniqueness, existence, and stability are not entirely neglected because of the qualitative insight they often provide.

The starting point chosen here for the study of partial differential equations is the notion of a fundamental solution, that is, the response of the physical system to a concentrated or impulsive forcing function. The mathematical formulation requires a discussion of the theory of distributions and generalized solutions of differential equations. In connection with boundary value problems, the Green's function, a fundamental solution satisfying specific boundary conditions, plays a central role. Much effort is devoted to methods for constructing the Green's function. The principal approaches use eigenfunction expansions, transform methods, and integral equations; the interplay among these techniques is stressed and often a problem is solved in alternative ways.

These methods are mainly applied to Laplace's equation, the diffusion equation, and the wave equation, but the theory is formulated in terms broad enough for more general use.

Special features include the treatment of scattering theory, Wiener-Hopf equations, and variational principles.

In conclusion I wish to thank my students and colleagues for their encouragement and Mrs. J. Prangley for her skillful typing of the manuscript.