Essential Partial Differential Equations

This introduction to partial differential equations is designed for upper level undergraduates in mathematics. The first nine chapters are mostly accessible to students with just first-year calculus. The final four chapters are more specialized and require greater sophistication; they focus on computational finite difference methods.

The book is a mixture of classical and modern, analytical and computational. It has a fresh feel to it. The writing is lively, the authors make appealing use of computational examples and visualization, and they are very successful at conveying and integrating physical intuition.

Students are introduced immediately to nine examples of partial differential equations (PDEs). These range from the one-dimensional wave equation to the Korteweg-de Vries and Black-Scholes equations. Immediately following is a discussion of what it means to be a solution of a PDE with very specific examples. In line with the authors’ more concrete approach, the next chapter describes how PDEs arise as tools for mathematical modeling with Newton’s laws or via conservation laws arising in physical systems.

The authors introduce the concept of characteristics in the next chapter. They consider how the direction of characteristics is related to the imposition of boundary conditions that lead to well-posed problems. They do this gently and with clarity, starting first with simple examples and gradually introducing greater generality. The classification of second order PDEs then follows fairly naturally.

By introducing boundary value problems in one dimension the authors have the opportunity to develop the tools (maximum and comparison principles, infinite series solutions) that look a good deal more intimidating in a general setting. Once again we see a development that begins with and stays close to examples while building a framework for more sophisticated questions about convergence of finite difference approximations later on.

The book continues with this well-paced and measured treatment through chapters on maximum principles and the energy method, separation of variables and the method of characteristics. Three chapters toward the end treat finite difference methods for hyperbolic, parabolic and elliptic PDEs separately. These seriously step up the level of sophistication and would probably not be suitable for inclusion in an ordinary first course.

This is probably the best introductory book on PDEs that I have seen in some time. It is well worth a look.

Bill Satzer (wjsatzer@mmm.com) 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.