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Partial Differential Equations

Marcelo Epstein
Publication Date: 
Number of Pages: 
Mathematical Engineering
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

This introduction to partial differential equations for engineers was published in Springer’s “Mathematical Engineering” series. The goal of the series is to present “new or heretofore little-known methods to support engineers in finding suitable answers to their questions, presenting those methods in such a manner as to make them ideally comprehensible and applicable in practice”.

The author was assigned to teach a graduate course called “Mathematical Techniques for Engineers”. He observed that such a course did not have a well-defined scope, and chose to focus the course on partial differential equations (PDEs) because he saw them as the backbone of several engineering disciplines.

The book was written as a set of class notes. The author says that he found it necessary to establish a common foundation while he let each student choose one of the many available options of standard PDE textbooks. He calls what he includes in this book “barely edited so as to preserve some of the freshness of a class environment”.

What results is a rather good textbook. Part I begins with preliminaries on vector fields and ordinary differential equations. Part II introduces quasi-linear PDEs, shockwaves and nonlinear PDEs. In Part III the author addresses classification of second order quasi-linear equations and discusses systems of PDEs. Finally, in Part IV he focuses on what he calls paradigmatic equations — types of PDEs of particular interest in engineering practice.

Although the author is attentive to mathematical details, he gives special attention to physical considerations. For example, he is particularly careful to show how PDEs arise as consequences of balance, conservation, or constitutive laws. Beginning early in the book and continuing throughout, he emphasizes the breadth of application of PDEs in engineering — from traffic flow to diffusion, from continuum mechanics to the theory of beams and structural engineering.

Although the major strengths of the book are examples and applications, the development and exposition of the underlying theory is also nicely done. One of the appealing features of the text is the way it integrates theory with examples. One instance of this is the author’s treatment of shock waves, where he develops the theory piece by piece and then illustrates its application using the inviscid Burgers equation that has shocks arising with both smooth and discontinuous initial conditions.

Each chapter has a modest number of exercises. Quite a few are straightforward and ask for verification of arguments in the text. A few are pretty challenging and several are both clever and novel.

One gets the impression — although the author says nothing explicitly — that he finds other PDE textbooks inadequate to help engineering students connect the basic elements of PDEs to actual engineering practice. He seems to understand how to help students understand more general questions by giving careful attention to specific applications and examples and encouraging physical and geometric intuition.

The book would be accessible to strong undergraduates with some multivariable calculus, basic linear algebra and ordinary differential equations. The author provides references at each stage to several of the standard texts, including those of Garabedian and John. This would be a good textbook for an introduction to PDEs or as a supplement to a more standard mathematical treatment.

Bill Satzer ( was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

See the table of contents in the publisher's webpage.