Foreword by Felix E. Browder

Summary

The one-dimensional heat equation, first studied by Fourier at the beginning of the 19th century in his celebrated volume on the analytical theory of heat, has become during the intervening century and a half the paradigm for the very extensive study of parabolic partial differential equations, linear and nonlinear. The present volume is a systematic development of a variety of aspects of this paradigm, of which many have not yet received an extension to the multidimensional space-variable case. Of particular interest are the discussions of free-boundary-value problems such as the one-phase Stefan problem, inverse problems, and some classes of not-well-posed problems.

This type of treatment using concrete analytic machinery for the detailed study of this very familiar and widely applicable partial differential equation should prove valuable as a textbook for courses that try to present basic aspects of partial differential equations in simple but useful cases (like the heat equation in one dimension), where the basic concepts are relatively unobscured by the technical problems and complications encountered in the more general classes of equations. The treatment is reasonably complete and can be followed by scientists who do not necessarily have the mathematical experience necessary for some of the more elaborate treatises on general parabolic equations. In addition, the relative completeness of the presentation for this case makes the volume suitable as a reference book for specific results in this area, for which a reference by specialization of more general results is inappropriate.