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Publisher:

John Wiley

Publication Date:

2008

Number of Pages:

454

Format:

Hardcover

Edition:

2

Price:

122.95

ISBN:

9780470054567

Category:

Textbook

[Reviewed by , on ]

Allen Stenger

07/28/2008

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 MathNerds.com, 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

Appendix

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

References

Answers and Hints to Selected Exercises

Index

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