This introduction to partial differential equations is addressed to advanced undergraduates or graduate students in mathematics, the sciences, and engineering. Prerequisites include advanced calculus, the basics of ordinary differential equations, and linear algebra. It is an imposing book that includes plenty of material for two semesters even at the graduate level. The author succeeds at integrating basic theory, solution methods, qualitative properties of solutions and methods for numerical approximation of solutions into a single comprehensive — if sometimes overwhelming — textbook.
No background in partial differential equations is presumed, so the author starts with very basic stuff about classical solutions, initial conditions and boundary conditions, and linear versus nonlinear equations. After that he moves quickly into a discussion of linear and nonlinear waves, stationary and traveling waves, and the basic wave equation. Some introductory discussion of nonlinear equations and shocks follows, but could be deferred until it gets a full treatment in a later chapter or omitted. For the most part the author focuses on partial differential equations with two independent variables. The book’s final two chapters concentrate on equations with two or three spatial dimensions. This material constitutes almost one-fourth of the book and includes a lot of good mathematical physics.
Two primary solution methods get extensive treatment: Fourier series methods and Green’s function techniques. Fourier methods (essentials of Fourier series, Fourier transforms and solution methods) are examined in great detail. The author shows how separation of variables in conjunction with Fourier techniques can be applied to construct solutions to the heat, wave, and Laplace equations. With the introduction of Green’s function methods students also see an informal treatment of generalized functions and, in particular, the delta function.
The discussion of numerical methods is divided into two parts. A first part treats finite difference methods, and describes applications to the heat equation, wave equation and transport equations. Later on (after a discussion of the minimization principle) the author describes a general construction of the finite element method and then focuses on a practical implementation. The treatment of numerical methods is intended to introduce students to the basic ideas and present some examples. The author succeeds at conveying the basic concepts while wisely staying away from thorny issues of computer implementation.
Exercises are not in short supply. Many are of the form “Find the solution…”; others require derivations or explanations, some are computational, and a few ask for proofs. They range in complexity from the relatively straightforward to fairly extensive projects. One novel feature of the book is a notation that associates a static drawing in the text with a movie available on the author’s website. It is a clever way to introduce animation at points where the dynamical effects can give critical insight.
The author succeeds at maintaining a good balance between solution methods, mathematical rigor, and applications. With appropriate selection of topics this could serve for a one semester introductory course for undergraduates or a full year course for graduate students. It is a large book, with a certain intensity, but the author has clearly taken pains to make it readable and accessible.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.