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Partial Differential Equations: An Introduction

Walter A. Strauss
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This undergraduate text spreads itself too thin — despite being 454 pages long, it covers so many topics that you don't really get an in-depth treatment of anything.

The book gets off to a good start looking at several traditional PDE problems, such as the vibrating string, the drumhead, and heat flow, and derives the PDE for each one. Then it gradually builds more and more complete solutions to each problem, gradually bringing in boundary conditions, initial conditions, and eigenvalues. This takes the first quarter of the book.

Then it veers off in a pure-math direction for the second quarter of the book, giving fairly complete accounts of the theory of Fourier series (including convergence theorems, completeness, and the Gibbs phenomenon), harmonic functions, and Green's functions. There are no applications, though, so students who were excited by the good discussion of applications in the first quarter may lose interest.

Now the book starts being spread too thin. The last half of the book deals again with physical problems: very good ones such as the vibrating drum head (Bessel functions), Schrödinger's wave equation, the hydrogen atom, spherical harmonics, solitons, Maxwell's equations, shock waves, water waves, and more! But it only gives 3 or 4 pages to each subject. There's also some more pure-math stuff, such as a long chapter on eigenfunctions that is not tied very well to applications.

The book is unsatisfying because it is trying to do too many things. It's very clearly written but there's not enough depth. A book I like that is very old but covers many of the same physical problems in much more detail, and develops the needed mathematical background, and is one-tenth the price, is Donald H. Menzel's Mathematical Physics (Dover reprint, original publication 1947). Another book that I like, and which will give you a much better skill at solving PDEs in about 140 pages (although it skimps on the applications), is Erwin Kreyszig's Advanced Engineering Mathematics.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

Chapter 1 / Where PDEs Come From
1.1 What is a Partial Differential Equation?
1.2 First-Order Linear Equations
1.3 Flows, Vibrations, and Diffusions
1.4 Initial and Boundary Conditions
1.5 Well-Posed Problems
1.6 Types of Second-Order Equations

Chapter 2 / Waves and Diffusions
2.1 The Wave Equation
2.2 Causality and Energy
2.3 The Diffusion Equation
2.4 Diffusion on the Whole Line
2.5 Comparison of Waves and Diffusions

Chapter 3 / Reflections and Sources
3.1 Diffusion on the Half-Line
3.2 Reflections of Waves
3.3 Diffusion with a Source
3.4 Waves with a Source
3.5 Diffusion Revisited

Chapter 4 / Boundary Problems
4.1 Separation of Variables, The Dirichlet Condition
4.2 The Neumann Condition
4.3 The Robin Condition

Chapter 5 / Fourier Series
5.1 The Coefficients
5.2 Even, Odd, Periodic, and Complex Functions
5.3 Orthogonality and General Fourier Series
5.4 Completeness
5.5 Completeness and the Gibbs Phenomenon
5.6 Inhomogeneous Boundary Conditions

Chapter 6 / Harmonic Functions
6.1 Laplace's Equation
6.2 Rectangles and Cubes
6.3 Poisson's Formula
6.4 Circles, Wedges, and Annuli

Chapter 7 / Green's Identities and Green's Functions
7.1 Green's First Identity
7.2 Green's Second Identity
7.3 Green's Functions
7.4 Half-Space and Sphere

Chapter 8 / Computation of Solutions
8.1 Opportunities and Dangers
8.2 Approximations of Diffusions
8.3 Approximations of Waves
8.4 Approximations of Laplace's Equation
8.5 Finite Element Method

Chapter 9 / Waves in Space
9.1 Energy and Causality
9.2 The Wave Equation in Space-Time
9.3 Rays, Singularities, and Sources
9.4 The Diffusion and Schršdinger Equations
9.5 The Hydrogen Atom

Chapter 10 / Boundaries in the Plane and in Space
10.1 Fourier's Method, Revisited
10.2 Vibrations of a Drumhead
10.3 Solid Vibrations in a Ball
10.4 Nodes
10.5 Bessel Functions
10.6 Legendre Functions
10.7 Angular Momentum in Quantum Mechanics

Chapter 11 / General Eigenvalue Problems
11.1 The Eigenvalues Are Minima of the Potential Energy
11.2 Computation of Eigenvalues
11.3 Completeness
11.4 Symmetric Differential Operators
11.5 Completeness and Separation of Variables
11.6 Asymptotics of the Eigenvalues

Chapter 12 / Distributions and Transforms
12.1 Distributions
12.2 Green's Functions, Revisited
12.3 Fourier Transforms
12.4 Source Functions
12.5 Laplace Transform Techniques

Chapter 13 / PDE Problems from Physics
13.1 Electromagnetism
13.2 Fluids and Acoustics
13.3 Scattering
13.4 Continuous Spectrum
13.5 Equations of Elementary Particles

Chapter 14 / Nonlinear PDEs
14.1 Shock Waves
14.2 Solitons
14.3 Calculus of Variations
14.4 Bifurcation Theory
14.5 Water Waves

A.1 Continuous and Differentiable Functions
A.2 Infinite Series of Functions
A.3 Differentiation and Integration
A.4 Differential Equations
A.5 The Gamma Function


Answers and Hints to Selected Exercises