There is no particular scarcity of textbooks on partial differential equations for undergraduates, and any new entry into that market must compete with, among others, Haberman’s *Applied Partial Differential Equations*, Strauss’s *Partial Differential Equations: An Introduction*, Coleman’s *An Introduction to Partial Differential Equations with MATLAB*, Asmar’s *Partial Differential Equations with Fourier Series and Boundary Value Problems*, and Farlow’s *Partial Differential Equations for Scientists and Engineers*. It follows, I think, that any new book on the subject should distinguish itself in some way from the rest of the available textbook literature. This junior/senior level text on PDE has several such distinguishing features, which, depending on your particular tastes, may elevate this book to the top of your short list of possible texts, or remove it altogether.

The first such feature is the organization of material. I commented in my review of Coleman’s book that elementary texts on this subject vary considerably in the way in which the material is organized and presented, and this book certainly illustrates that fact. The first three chapters cover, respectively, Fourier series, integral transforms (Laplace and Fourier), and Sturm-Liouville ODE problems. All of this material plays an important role in the solutions of PDEs, and presumably the authors wanted to get it discussed up front so that it would be available later on, when needed. The downside to this, of course, is that the student, in a course on partial differential equations, has to go through a full 200 pages of text before even finding out what a partial differential equation *is*. While starting a PDE book with Fourier series is by no means unheard of, there don’t seem to be all that many texts that delay the entrance of PDEs themselves quite so dramatically. Of course it may be possible here to simply omit some of the earlier material, but since there is no section dependency chart, an instructor will have to think hard about just what is safe to delete.

Once PDEs make their appearance in chapter 4, the textual development of them is quite logical and standard. After the basic terminology and examples are discussed, first-order linear PDEs are examined and the use of characteristic curves to solve them is explained. Then second-order linear PDEs are introduced, and the basic classification (hyperbolic, parabolic, elliptic) established. The prototypical examples of each of these types (wave equation, heat equation and Laplace’s equation, respectively) then each receive a chapter in which they are studied in some depth. A final chapter then discusses finite difference numerical methods.

Another aspect of this text that sets it apart from the rest of the pack is the extensive use of *Mathematica*. However, as was also the case with Coleman’s book (which made use of *MATLAB*) the computer software here enhances the discussion but is not essential to it. Every chapter except the last ends with a section entitled “Projects Using *Mathematica*”, in which problems (with solutions) are presented using *Mathematica* to gain insight into the material, but these sections can be easily skipped by people not invested in this software; likewise, there are *Mathematica*-generated pictures throughout the text, but these can be studied and enjoyed by people without computers of their own. An Appendix (“Basics of *Mathematica*”) provides a quick (roughly 15 page) introduction, but, as is usually the case in situations like this, the Appendix is not really a substitute for prior experience with the software.

The exposition throughout is reasonably clear but seemed to me to be pitched at a somewhat higher level than the books cited in the first paragraph above. The intended audience of scientists and engineers is kept in mind, and applications are certainly discussed, but proofs are often provided as well (though on occasion the authors omit a proof as being beyond the scope of the text).

In addition, the prerequisites for this text sometimes exceed the standard courses in ordinary differential equations and multivariable calculus; on occasion, for example, results from complex function theory are used (another Appendix summarizes the necessary background).

Numerous worked examples do appear throughout the text, and there are also quite a lot of exercises. Solutions to most of these are provided, but these solutions often consist, for example, of just a function, with the details of how that function was arrived at left unspecified (which is, I think, a definite pedagogical plus).

Verdict: I suspect this book would give the junior-level PDE students at Iowa State University some difficulty, but if you’re teaching a fairly sophisticated course in the subject and agree with the authors’ decision to delay the introduction of PDEs until after a lot of preliminary material has been discussed, this book is certainly worth a look.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.