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Applied Partial Differential Equations

J. David Logan
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Charles Ashbacher
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Courses in partial differential equations (PDEs) tend to have very standard coverage. In general, the major ideas are: the wave equation, the diffusion equation, the heat equation, Laplace transforms, Fourier transforms, and Fourier series. Logan covers these areas and goes a bit beyond the basics. The book begins with a chapter on the physical problems that are the origins of PDEs. This is very important to setting an appropriate stage for the study of PDEs. By establishing the concept of the mathematical model, the student is immediately convinced that there are practical reasons for using PDEs.

The last chapter demonstrates how PDEs are used in the life sciences. This inclusion is in step with a major initiative called Meeting the Challenges: Education Across the Biological, Mathematical and Computer Sciences, a joint project of the Mathematical Association of America (MAA), the National Science Foundation Division of Undergraduate Education (NSF DUE), the National Institute of General Medical Sciences (NIGSM), the American Association for the Advancement of Science (AAAS) and the American Society for Microbiology (ASM). The overwhelming consensus of these groups is that the formerly distinct disciplines of mathematics and biology needed to generate more interdisciplinary courses if future biologists are to be properly trained. The Mathematical Association of America recently published the book, Math & Bio 2010: Linking Undergraduate Disciplines which is a description of the initiative. Clearly, movements like this demonstrate the wisdom of including such a chapter.

An occasional solution in the symbolic mathematics package Maple is also included, showing the reader how the problem can be coded and what the solution will look like. There are a large number of exercises, but no solutions are given. I consider this to be a major weakness. My opinion is that if you are going to include exercises in a math book, then solutions to at least some of them should be included. As a student I always appreciated it, and as a math instructor who has not worked with PDEs for many years, I would welcome the ability to immediately verify the veracity of my recollections.

The emphasis is on the explanations of the form of the solution strategy and how to implement it. While most of the techniques are presented in the form of a theorem, I was hard pressed to find any that were justified with a proof. The coverage is appropriate and the explanations are understandable and thorough. Therefore, I would not hesitate to use it as a textbook.

Charles Ashbacher ([email protected]) teaches at Mount Mercy College in Cedar Rapids, Iowa.

The Physical Origins of Partial Differential Equations * Partial Differential Equations on Unbounded Domains * Orthogonal Expansions * Partial Differential Equations on Bounded Domains * PDE Models in Biology * Appendix: Ordinary Differential Equations * Table of Laplace Transforms * References * Index