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Partial Differential Equations: An Introduction to Theory and Applications

Michael Shearer and Rachel Levy
Princeton University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on

This new textbook on partial differential equations is an exciting addition to the current textbook literature on the subject. It is (for the most part, anyway) accessible to undergraduates and would serve as an interesting text for an introductory undergraduate course, yet at the same time leads students to some of the more theoretical aspects of the subject.

The first eight chapters cover topics that are typically taught in an introductory course on PDEs: first order equations and characteristic curves, the “big three” second order linear equations (wave, heat, and Laplace), and Fourier series (in connection with the method of separation of variables). However, the overall approach is at a somewhat, but not unreasonably, more sophisticated level than is found in many other undergraduate books in this area. The intended audience is mathematics majors, specifically majors who have some prior background in the basics of analysis and linear algebra. For these students, the book, by virtue of very clear and engaging writing, should prove quite attractive; students from other disciplines who are just looking for a “plug-and-chug” approach to the subject would be advised to look elsewhere, however. Although applications are not ignored in this book, the authors clearly view the study of PDEs as a genuine area of mathematics, not just as a tool for solving physical problems.

In more detail: The book begins at the beginning with the definition of a partial differential equation, but little or no time is spent with truly trivial examples such as \(u_x=0\). Instead the book begins with a quick survey of some important PDEs, both linear (e.g., the wave and heat equations) and nonlinear (such as the Korteweg-deVries and Navier-Stokes equations), giving brief paragraph-long descriptions of each. Chapter 2 (“Beginnings”) opens with a survey of some of the theoretical issues that the subject entails: questions of existence, uniqueness, continuous dependence on data, and regularity of solutions. No theorems are proved at this point, but the reader is at least given an understanding and appreciation of the kind of questions that motivate research in this area. The chapter then discusses the classification of second-order PDEs; unlike many undergraduate textbooks, the authors make use of matrix terminology (orthogonal diagonalizability) to explain what is going on here, and also pay more attention to the case of more than two variables than is perhaps standard. This chapter also states (without proof) the existence and uniqueness theorems of Cauchy-Kovalevskaya and Holmgren.

Chapter 3 is on linear and quasilinear first order equations, which are studied via the method of characteristic curves. The case of two independent variables is studied first, and then generalized to an arbitrary number. The chapter ends with a look at a particular first-order nonlinear equation, the inviscid Burgers equation, the characteristic curves of which are straight lines that intersect, thus creating interesting complications. The treatment of nonlinear equations so early in the text distinguishes this book from many others at this level.

Chapters 4 and 5 are about, respectively, the wave and heat equations; the basic properties of both are set out. When these equations are studied on bounded domains, a useful method of solution is the method of separation of variables, which in turn leads to Fourier series. These ideas are introduced in chapter 6 and discussed further in chapter 7 (including a heuristic introduction to the L2 spaces in chapter 7). Separation of variables is one method that is then used, in chapter 8, to discuss Laplace’s equation (as well as the related Poisson equation).

The preceding material comprises a reasonable first undergraduate course in partial differential equations, but is done here in only about 130 pages of text. (As the quick survey of topics discussed above indicates, numerical methods are not covered.) The reader will not find scads of worked-out problems done in tedious detail here, but will find good, clear exposition, putting the underlying mathematical ideas front and center. Personally, I preferred this, but, as previously noted, a young undergraduate who just wants to know “how to do it” may find other, more “cookbooky”, texts to be more suitable.

The remainder of the book is devoted to more advanced topics, including an introduction to the methods of functional analysis in PDE. Chapters 9–11, for example, introduce distributions and (after a brief summary of function spaces) Sobolev spaces. The authors take care to introduce topics gradually and motivate them well with reference to specific problems that show the utility of these ideas. Other chapters address additional topics in nonlinear theory, including traveling wave solutions of nonlinear PDE, more on the Burgers equation, and systems of first-order hyperbolic equations. The book ends with a chapter on some of the PDEs associated with fluid dynamics. The material in the second half of the book will likely prove beyond the ken of most undergraduates, but there is plenty in the first half of the book from which to fashion an undergraduate course.

Because of this more sophisticated material, the text might also be suitable for a beginning graduate course in PDE, though it is less demanding and encyclopedic than is, say, Evans’ Partial Differential Equations, which is also about 400 pages longer. I would say, in fact, that this text lies somewhere between the standard undergraduate texts (by, say, Haberman, Strauss or Coleman) and the book by Evans. As such, the book may have an audience problem — too demanding for many undergraduate courses, not sophisticated enough for graduate ones. I think, however, that, as I mentioned earlier, that an undergraduate course based on this book would be interesting and rewarding, and the authors have done their very best to make this text suitable for such a course.

Each chapter ends with an assortment of problems, averaging around 11 or 12 per chapter. A reasonable number are of the “solve this PDE” variety, but some are more conceptual in nature. The conceptual questions may call for proofs, but may also just call for the student to explain some idea or interpret some term in a mathematical model. A solutions manual is available, to instructors only, from the Princeton University Press. (The publisher was good enough to send me a pre-publication version of it, and, even though it is only about twenty pages long, it looks pretty useful; solutions are given, in detail adequate enough for instructors but not adequate enough for student responses, to most of the problems in the text.)

Another nice feature that may prove useful for instructors using this text is the inclusion, on the publisher’s webpage, of all the illustrations that are used in the book, making them easily usable (in slides, for example) in lecture.

Let me close this review with a disclaimer that perhaps should have come at the beginning: I am an algebraist by training and inclination, and therefore am by no means an expert on the subject of partial differential equations. My knowledge of the subject is largely limited to what I learned from a couple of courses (one as an undergraduate, one as a graduate student) that I sat in on a long time ago, and from several books that I have looked at over the years. What I learned from these sources, however, always interested me, at least once I got past some of the lengthier and more tedious calculations. I liked seeing how second-order PDEs could be divided into three groups, each with a prototype example, with each group illustrating different properties. I liked seeing how linear algebra played more of a role in the subject than one might have first imagined it would. I liked seeing some of the applications. And now, having seen this book, I like the subject even more.

Mark Hunacek ( teaches mathematics at Iowa State University.