For more than two decades, part of my research has been directed at various questions involving the heat equation. In this volume I have interwoven much of my research and the research of others with the classical material, at a presentation level suitable for upper-division and beginning graduate mathematics, engineering, and science students. However, I have also intentionally written the material as a monograph and/or information source book. After the first six chapters of standard classical material, each chapter is written as a self-contained unit, except for an occasional reference to elementary definitions, theorems, and lemmas in previous chapters. Consequently, I believe that material can be drawn from the book as needed for a variety of courses, such as a standard course in partial differential equations, a course in initial-boundary-value problems for the heat equation, a course in not-well-posed problems and their numerical solution, a course in free-boundary-value problems, a course in parameter identification, and several others.

The treatment begins with a chapter of preliminary material in order to reduce the need for other reference material. This is followed by six chapters containing the standard basic material for the heat equation, such as the weak maximum principle, elementary solutions, and fundamental solution, and the usual initial- and/or boundary-value problems. One exception to the usual material is the treatment of the noncharacteristic Cauchy problem in Chapter 2. Utilizing the solution representations in Chapters 3 through 6, a fair number of initial-boundary-value problems are reduced to an equivalent system of integral equations in Chapter 7.