The subject of PDEs is classical and suffers from no shortage of good textbooks. The reviewer of a new textbook on this topic is therefore forced firstly to confront the question of whether it justifies an expansion of this literature. In my opinion, the book under review does make a significant contribution, for the following reasons.

• It is comprehensive, up-to-date and (more-or-less) completely self-contained. As such, it straddles the space between an advanced-level textbook and a practitioners' reference.
• It is primarily aimed at physicists and engineers, and maintains its focus on this target audience throughout. In particular, there is a consistent emphasis of technique and method with less time devoted to general theory. There are also many carefully worked-out examples and a large number of problems originating from engineering and scientific disciplines.
• It is clear that the authors have put a lot of thought into exposition, resulting in a logical and flowing structure as well as an easy and unencumbered style of prose.
• It contains a large number of carefully thought-out exercises, ranging in difficulty from fairly routine to quite challenging. Solutions and hints to selected exercises are provided after the final chapter, but these are generally parsimonious enough not to spoil all the fun.

The items above are what I consider to be the general strengths of the book. I shall now briefly delve into its structure. Since there are fifteen chapters in all, my comments will be brief.

Chapter 1 is an introduction, and contains (among other things) a detailed account of the history of PDEs—I enjoyed this very much! Chapter 2, which is also introductory by nature, provides a fairly standard account of first-order equations. In Chapter 3 the book reveals its bias towards physics by deriving the PDEs associated with a number of problems from classical and quantum physics. Chapter 4 introduces second-order equations, with an emphasis on their classification. Next, Chapter 5 presents the general Cauchy problem for hyperbolic PDEs before studying initial-value problems and initial-boundary-value problems for the wave equation as concrete examples.

In Chapter 6 we find a self-contained survey of Fourier series, which can be read independently of the other chapters. This material is immediately put to use in Chapter 7, which presents the method of separation of variables for solving the PDEs associated with the vibrating string problem and the heat conduction problem, in particular. Chapter 8 generalizes the method of separation of variables by introducing the Sturm-Liouville theory. This chapter also contains accounts of eigenvectors and eigenfunctions, eigenfunction expansions, an introduction to Bessel's equation and special functions, as well as an introduction to Green's functions for ODEs. The chapter ends by considering Schrödinger's equation and the linear harmonic oscillator, as a concrete example. Chapter 9 is devoted to a study of boundary-value problems for elliptic PDEs, and introduces the maximum and minimum principles, as well as presenting uniqueness and continuity results. The examples in this chapter include Dirichlet and Neumann problems for Laplace's equation, as well as a Dirichlet problem for Poisson's equation.

Chapter 10 presents examples of boundary-value problems where the underlying PDE has more than two spatial dimensions, while Chapter 11 examines the application of Green's functions to the solution of boundary value problems. Next, Chapter 12 presents a very thorough and self-contained introduction to integral transforms with examples of their use in solving boundary-value problems. Chapter 13 is somewhat of an anomaly since it is devoted to non-linear PDEs. This chapter, which is a relatively recent addition to the book (it first appeared in the third edition), spends most of its time on non-linear extensions of the wave equation. Finally, Chapter 14 offers a fairly comprehensive account of numerical methods for PDEs, while Chapter 15 presents tables of integral transforms.

I found the book a pleasure to read. It is suitable as a textbook for graduate and advanced undergraduate courses on PDEs, as well as being a handy reference on the subject. Given that it is heavily focused on examples, my only complaint is that there is no discussion of the Black-Scholes PDE, and other related parabolic PDEs from mathematical finance. Such examples could easily have been accommodated—in fact, the book sets itself up ideally for this.