Elements of Partial Differential Equations

An excellent text, Elements of Partial Differential Equations, by Pavel Dràbek and Gabriela Holubová of the University of West Bohemia in the Czech Republic, addresses itself to beginners in PDE, including, besides mathematicians, fellow travelers from engineering and science. Dràbek and Holubová state (in the Preface) that “our text is on a very elementary level.” I think the book is an outstanding pedagogical achievement, worthy of serious consideration as a textbook for any course in methods of applied mathematics or even a course specifically in PDE. It’s among the best I’ve seen, and if the students taking the course are well-trained and well-motivated, Elements of PDE is nigh on perfect!

The compact (ca. 240 page) book starts with physical principles (e.g., conservation laws), classification of PDE (properly focusing on second order PDE, of course), and a discussion of boundary and initial conditions. It quickly proceeds to linear PDE of order one, the wave equation, the heat (and diffusion) equation, and Laplace and Poisson equations. The latter special second order PDE are clearly critical in the subject, given their ubiquity in nature and the educational value provided by their methods of solution. And Dràbek and Holubová do a superb job in their coverage of these topics. It’s my guess that this material would make up a very nice first semester course.

I want to draw special attention to Dràbek and Holubová’s effective treatment of the Gauss kernel (p. 61 ff.), set in the context of the suitable Cauchy problem (vis à vis the heat equation, of course, although they prefer to cite it as the diffusion equation). This object, in the general case being the exponential of the negative of a positive definite quadratic form, but classically just a normalized avatar of exp(-x²), is of huge importance in a number of mathematical fields, including, if I may be forgiven the suggestive phrase, the strange but passionate and fecund pair of bedfellows, number theory and physics.

The conduct of the Gauss kernel, as also the broader behavior of the heat equation, is a deep study in itself; see, for example, my recent review of Jorgenson, Lang, The Heat Kernel and Theta Inversion on \(\mathrm{SL}_2(\mathbb{C})\). Say Dràbek and Holubová: “The … heat (diffusion) kernel or the fundamental solution of the diffusion equation … [is also called] Green’s function, source function, Gaussian, or propagator …,” which immediately hints at the different locales the heat kernel calls home, in itself a suggestive point. The last term, by the way, “propagator,” points toward the methods chosen by quantum field theorists, after Feynman, in dealing with certain kinds of integrals involving the indicated kernel functions.

Dràbek and Holubová immediately go on to a nice discussion of Dirac’s delta function, properly downplaying the issue of distributions (it’s too early at this point), but nonetheless doing justice to everything else. Then, in the space of a few pages, they proceed to the solution of the heat equation (as a certain BVP) via the error function. This is a theme of absolutely central importance, of course, dealt with compactly and very elegantly. I’m reminded of Richard Bellman’s beautiful discussion of this material in his famous little book on theta functions.

After all this, and perhaps proper to a second semester, the authors go on to discuss general principles: causality, energy conservation, the maximum principle, and so on; subsequently they focus on Laplace and Poisson in higher dimensions, saving Dirichlet’s principle for this context (p. 162), as also their discussion of Green’s functions (properly so-called).

Elements of PDE finishes with a pair of discussions of the heat and wave equations in higher dimensions, and then there is an appendix on Sturm-Liouville theory and Bessel functions. There are loads of good exercises in the book, some with parenthesized answers, covering the spectrum from standard to mildly fruity (I didn’t see anything particularly nasty hiding in the shadows). Dràbek and Holubová are also scrupulous to give proper attribution to other sources whence they lifted a problem or two.

This is a very, very good book. Given that I get to teach PDE and methods of applied mathematics rather often, I am happy to have it in my collection. I’ll adopt the book for the indicated course myself in the near future, if I have a sufficiently mathematically mature class (I fear that Czech undergraduate fellow travelers’ mathematical backgrounds are considerably stronger than what I’ve seen at my university, where engineering majors tend to whine rather deafeningly when real mathematics appears on the board: c’est la guerre).

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.