Partial Differential Equations: Analytical and Numerical Methods

In Gockenbach's text, analytic methods for the solution of partial differential equations are not isolated from numerical methods. By focusing on linear operators, it is possible to discuss Fourier Series, Green's functions and Finite Element Methods in the same context. The parallels between the methods for linear algebraic systems and linear differential equations are repeatedly pointed out. Fourier Series are primarily treated as a spectral method, with the more traditional separation of variables approach taking a back seat. Green's functions are presented as inverse operators. The Finite Element Method relies inherently on linear algebra and so is a natural progression in the text rather than an isolated method. All of this provides a refreshingly modern alternative to the standard introductory PDE texts.

In order to build this alternative approach, the author includes a fairly extensive review of linear algebra techniques in Chapter 3 as the basis for the methods used to solve partial differential equations. The spectral method is not often used to solve linear systems, but is covered here as a precursor to its use for the solution of partial differential equations. It is explained very well using mathematics that would be familiar to the average undergraduate PDE student.

Unlike some other texts, there is a detailed discussion of eigenfunctions before actually using Fourier Series to solve a boundary value problem. Similarly, there is a lengthy treatment of the Galerkin method before Finite Elements are introduced. One-dimensional boundary value problems, the heat equation and the wave equation are treated in Chapters 5-7, with a discussion of both Fourier Series and Finite Elements in each chapter. Classical approaches are presented when they lead to physical insight. I particularly liked the overview of classification in Chapter 1 and the discussion of d'Alembert's formula in Chapter 7.

Chapters 8-10 provide material for really strong students. I would probably not cover them in a one-semester class, but would assign them as independent study reading for good students. The author includes an excellent treatment of Fourier Series for higher dimensional problems in Chapter 8. There is also a discussion of the Finite Element Method in multiple dimensions. I like the fact that the theoretical aspects of both Fourier Series and Finte Element Methods are included. I was particularly impressed with the inclusion of FEM implementation.

The further reading sections are excellent — providing many references for students and instructors. Another excellent feature is the discussion of physical models. The derivations are very well presented in Chapter 2. The exercises include many questions posed with physical units, which forces students to consider the application in addition to the differential equation. The only drawback is the lack of models from biological or social sciences.

Another excellent feature is the CD that comes with the book. It contains tutorials for Matlab, Maple & Mathematica — written specially for the needs of the text. The fact that the author provides this for three different software packages allows the instructor flexibility in his/her software choice. The text also comes with a free student membership to SIAM, thereby giving students exposure to a wealth of applications. Perhaps all of our professional societies should be doing this!

I love this book and look forward to using it as a text in the future. Green's functions do take a back seat in order to make room for a discussion of the more modern technique — Finite Element Methods. I would prefer to see them treated in more depth, but the introduction of FEM to an undergraduate PDE text is wonderful. It's the first truly modern approach that I've seen in a PDE text.

Maeve L. McCarthy (maeve.mccarthy@murraystate.edu) is associate professor of mathematics at Murray State University. Her mathematical interests include computational methods for inverse spectral problems and parameter identification for partial differential equations.