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Solution Techniques for Elementary Partial Differential Equations

Christian Constanda
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

See our review of the second edition. This third edition is a modest update, with a small amount of additional material, examples, and exercises, and some rearrangement of the text.

This is a leisurely cookbook of PDE solution methods, aimed at engineering students. The approach is very traditional. There is no numerical work. There is a great deal of Mathematica code, although in my opinion this does not add much to the text. Mathematica knows all the cookbook methods too, and the code merely asks for the exact solution to the example problem. These are described as “verifications”, but they do raise the question: Why do we need to learn these techniques, when Mathematica already knows them?

There is a small amount of instructor support material on the publisher’s web site, consisting of a solutions manual a set of viewgraphs based on the text. I did not examine these.

This book is well-done, and has comprehensive coverage of explicit solution methods, and many students will benefit from the completely written-out solutions. On the other hand, it may be overkill for many students to have a whole book on this subject, when engineering math books (such as Kreyszig’s Advanced Engineering Mathematics) also have good coverage, although more concise and not as extensive as the present book.

On the third hand, there is more to PDEs today than explicit solutions, and most engineering students will also want to learn qualitative methods and numerical methods. Surprisingly, the Schaum’s Outline of Partial Differential Equations (Paul DuChateau & David W. Zachmann, McGraw-Hill, 3rd edition, 2011) is much more up-to-date than the present book, as well as being much cheaper at $20. It covers generally the same topics, although at less length, but also has a lot on difference equation methods, the finite element method, and qualitative behavior of elliptic, hyperbolic, and parabolic PDEs.

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.

Ordinary Differential Equations: Brief Revision
First-Order Equations
Homogeneous Linear Equations with Constant Coefficients
Nonhomogeneous Linear Equations with Constant Coefficients
Cauchy–Euler Equations
Functions and Operators

Fourier Series
The Full Fourier Series
Fourier Sine and Cosine Series
Convergence and Differentiation
Series Expansion of More General Functions

Sturm–Liouville Problems
Regular Sturm–Liouville Problems
Other Problems
Bessel Functions
Legendre Polynomials
Spherical Harmonics

Some Fundamental Equations of Mathematical Physics
The Heat Equation
The Laplace Equation
The Wave Equation
Other Equations

The Method of Separation of Variables
The Heat Equation
The Wave Equation
The Laplace Equation
Other Equations
Equations with More Than Two Variables

Linear Nonhomogeneous Problems
Equilibrium Solutions
Nonhomogeneous Problems

The Method of Eigenfunction Expansion
The Nonhomogeneous Heat Equation
The Nonhomogeneous Wave Equation
The Nonhomogeneous Laplace Equation
Other Nonhomogeneous Equations

The Fourier Transformations
The Full Fourier Transformation
The Fourier Sine and Cosine Transformations
Other Applications

The Laplace Transformation
Definition and Properties

The Method of Green’s Functions
The Heat Equation
The Laplace Equation
The Wave Equation

General Second-Order Linear Equations
The Canonical Form
Hyperbolic Equations
Parabolic Equations
Elliptic Equations
Other Problems

The Method of Characteristics
First-Order Linear Equations
First-Order Quasilinear Equations
The One-Dimensional Wave Equation
Other Hyperbolic Equations

Perturbation and Asymptotic Methods
Asymptotic Series
Regular Perturbation Problems
Singular Perturbation Problems

Complex Variable Methods
Elliptic Equations
Systems of Equations


Further Reading