This is a discursive introductory text in partial differential equations, that has good applications drawn from a number of fields including finance. It is aimed at undergraduates, is generally concrete in its approach, and has modest prerequisites. The present volume is an updated Dover 1999 reprint of the 1993 Wiley third edition, with some corrections and a new introduction and a new epilogue about then-recent developments.

The book is a sampler and doesn’t go into great depth on any topic. The “Hilbert space methods” of the title are mostly eigenfunction expansions, with a little abstract Hilbert space mixed in (although it does make good use of the concepts of discrete and continuous spectra).

The book is very traditional in many ways. It makes heavy use of separation of variables (primarily through Fourier series and eigenfunction expansion), with the other methods covered being Green’s functions and variational methods (mostly Dirichlet’s principle). One way the book is not traditional (and this may be its greatest strength) is that it keeps circling back to take repeated looks at particular ideas and techniques, developing them in greater and greater detail. All the key ideas are introduced very early. I think this repetition would work well for learning, even though it makes the book harder to use as a reference. (The footnotes are very erudite and worth reading just for themselves.)

There are a modest number of well-selected exercises, and these have answers or solutions to various degrees of completeness in the back of the book. There’s no drill per se; some problems ask for a solution to a particular PDE, some ask for a proof, and some ask what can be concluded from a given set of information. Thus they all encourage thought rather than technique. This is carried even further in the six “pauses” scattered through the book. These reflect on what has gone before and take a look at a few more difficult problems in great depth.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.