Introduction to Partial Differential Equations with Applications

One of our graduates this year — let’s call him Harry — was a triple major: mathematics and physics and computer science. Harry was really only interested in mathematics insofar as it could plausibly be applied to physics, but fortunately this takes in a lot. He did his senior seminar talk on \(\mathrm{SO}(3)\) and \(\mathrm{SU}(2)\).

Two summers ago Harry asked me for an independent study on partial differential equations. (I will sometimes use the standard abbreviation PDEs, and ODEs sometimes for ordinary differential equations.) I am far from being an expert on the subject, although I had the graduate course at Wisconsin from Paul Rabinowitz. Moreover, I had done an independent study on PDEs myself as an undergraduate, using an old book by Frederic H. Miller that my father had, so I was favorably disposed to the idea. Harry was primarily interested in learning solution techniques, as I had been at his age and indeed still am.

Our library’s PDE collection is pretty skimpy, although it includes a few e-books that we might have tried. Instead I acquired a handful of Dover books and had Harry look at them, and he chose the book under review (hereafter ZT). It was developed at Purdue in the late 60s and early 70s, published in 1976, and transferred to Dover in 1986. Although the two authors have retired, partial differential equations is currently the second largest research group in Purdue’s department with 11 faculty, trailing only computational and applied mathematics with 14.

ZT was written for a course taken by advanced undergraduate and beginning graduate students in mathematics, engineering, and the physical sciences. “A course in Advanced Calculus or a strong course in Calculus with extensive treatment of functions of several variables, and a very elementary introduction to Ordinary Differential Equations” are the stated prerequisites, but the authors add that “the basic results of advanced calculus are recalled whenever needed.” Harry had not yet taken our Analysis course when we used ZT, but this rarely if ever caused us trouble. He had a strong calculus background and had also taken Mathematical Methods of Physics with me.

Chapter 1 is a brief review of some ideas from multivariable calculus and first order ODEs. Section 2.1 makes the fundamental connection between vector fields, first order linear PDEs, and an associated system of ODEs. Section 2.2 has some advice for and examples of solving the latter. (This section does not have enough problems, but later sections are better in this respect.) Section 2.3 connects these solutions to the allied PDE, and section 2.4 develops the theory a little further. Section 2.5 should really be two sections, one that does a little vector analysis and the other giving an application of the earlier sections to plasma physics.

Chapter 3 continues to develop the first order linear theory. Sections 3.3 and 3.4 discuss existence and uniqueness of solutions, though not very extensively. Section 3.5 begins “Conservation laws are first order quasi-linear PDEs that arise in many physical applications (see Section [3.6] for examples).” ZT never gives a more adequate explanation of the term “conservation laws”, but this is an interesting section for several reasons. It mentions shock waves very briefly, referring the reader to Peter Lax’s paper The formation and decay of shock waves from The American Mathematical Monthly 79 (1972), 227–241.

The last two sections are on applications, including an example of a shock wave in a traffic flow problem.

Chapter 4 is quite short and is on series solutions. Section 4.2 has the Cauchy-Kovalevsky theorem on analytic solutions of an initial value problem. Together with the first several paragraphs of section 3.5, the last problem in this chapter gives a derivation of Lagrange’s theorem on the expansion of a certain kind of implicit function in a power series. The argument, though unaccredited, is essentially due to Laplace and in fact gives an extension of Lagrange’s theorem. The authors might have said much more here — they merely call the result of the problem Lagrange series, with neither an article nor the possessive, and give a reference to a then-recent application to gas flow — and the temptation to do so would be hard for me to resist if I were lecturing on this subject.

Chapter 5 transitions to higher order equations. It discusses characteristics, gives a very brief treatment of the general initial value problem, and then brings in canonical forms for first and higher order equations, leading to the classification of second order equations as either elliptic, parabolic, or hyperbolic.

Chapter 6 is very short. It begins with a little vector analysis and ends with a brief discussion of what it means for a problem to be well posed: existence and uniqueness of solutions together with continuous dependence on the initial conditions. A famous example of Hadamard is presented in problem 5.2.

The other three sections of chapter 6 are on the heat equation, Laplace’s equation, and the wave equation, each of which then gets its own chapter. At 90 pages, chapter 7 on Laplace’s equation and harmonic functions is much the longest in the book. Since one could easily devote a whole course to this material, it is hard to quibble with this decision. Separation of variables for PDEs appears for the first time near the beginning of this chapter. Fourier series get about 16 pages, and Green’s functions are used extensively.

Chapter 8 is on the wave equation and runs to 70 pages. It takes a surprisingly long time to come back to Fourier series, but in the last few sections we do, and they appear again in chapter 9, a mere 26 pages on the heat equation. Chapter 10 is on systems of PDEs and is also rather short.

My experience was that ZT is quite suitable for an advanced undergraduate PDE course. In the unlikely event that I were to teach such a course, familiarity and price would make it an attractive option. There is more than enough material for one semester; the authors estimate 25% more. I might put it a little higher: we were able to reach chapter 9, but we skipped quite a few things. It could also be used at the beginning graduate level, though I think it would work better with physics and engineering students than with budding mathematicians.

Warren Johnson (wpjoh@conncoll.edu) is Associate Professor of Mathematics at Connecticut College. He is trying to finish a book on \(q\)-analysis.