Fundamentals of Partial Differential Equations

According to the authors, this new book on partial differential equations (PDEs) is primarily directed at upper-level undergraduate students in physics and engineering. It would also be appropriate for mathematics students either at that stage or in early graduate studies. The text primarily treats linear PDEs with two or three independent variables and their applications. The authors argue in their preface that differential equations give a unifying theme to the study of physical systems. Their approach throughout the book consistently follows that idea. 

 

The topics treated here are mostly standard for courses at this level. About the first one-third of the book covers background material in vector analysis and ordinary differential equations. The treatment of PDEs begins with a short discussion of mathematical modeling and then goes on to describe prototypical PDEs: wave, heat, and Poisson-Laplace equations as well as the PDEs that model transport and transfer of mass, momentum and energy.

 

A more detailed discussion of PDEs begins almost halfway through the book. It has a more formal introduction to the main concepts of PDEs and begins with classification, canonical forms, a discussion of classical solutions, and then a general treatment of standard solution methods.

 

From the authors’ perspective, the core of the book consists of concepts relevant to standard analytical solution methods: the method of characteristics, separation of variables, Fourier series, eigenfunction expansion, and the transform techniques of Fourier and Laplace. Each of these is described in a chapter of its own. Every chapter also has a collection of relevant exercises as well as a set of references (mostly to other textbooks).

 

Although the book is aimed at science and engineering students, the mathematical level seems a better fit for advanced undergraduates in mathematics. Typical science and engineering students might well struggle with the overall level of mathematical sophistication.

 

The text is sometimes troubled by grammatical errors, missing definite articles and occasional odd word choices. This never obscures the underlying meaning, but it can be distracting. For the most part, the text is clear and well written.

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.