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Introduction to Partial Differential Equations and Hilbert Space Methods

Karl E. Gustafson
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

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 His mathematical interests are number theory and classical analysis.

1 The Usual Trinities
1.1 The usual Three Operators and Classes of Equations
1.2 The Usual Three Types of Problems
1.3 The Usual Three Questions (and the Other Three)
1.4 The Usual Three Types of "Boundary Conditions"
1.5 The Usual Three Solution Methods (with Historical Remarks)
First Pause: Examples, Explanations, Exercises
1.6 Three Important Mathematical Tools
Second Pause: Examples, Explanations, Exercises
1.7 Some Physical Considerations (and Examples)
1.8 Elements of Bifurcation Theory
1.9 Supplementary Discussions and Problems

2 Fourier Series and Hilbert Space
2.1 Lots of Separation of Variables
2.2 Mathematical Justifications of the Method
2.3 Fourier Series and Hilbert Space
2.4 Fourier Series and Sturm--Liouville Equations
2.5 Fourier Series and Green's Functions
Third Pause: Examples, Explanations, Exercises
2.6 Fourier Series and Variational and Numerical Methods
Fourth Pause: Examples, Explanations, Exercises
2.7 Some Unbounded Domain Considerations (and Continuous Spectra)
2.8 Elements of Scattering Theory
2.9 Supplementary Discussions and Problems

3 Appendices
A. First-Order Equations
Fifth Pause: Examples, Explanations, Exercises
B. Computational Methods
Sixth Pause: Examples, Explanations, Exercises
C. Advanced Fluid Dynamics

Selected Answers, Hints, and Solutions

E.1 Distribution Equations
E.2 Semiconductor Equations
E.3 Finance Equations

Author Index

Subject Index