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Boundary Value Problems and Partial Differential Equations

David L. Powers
Publication Date: 
Number of Pages: 
Hardcover with CDROM
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Jeffrey A. Graham
, on

Boundary Values and Partial Differential Equations, by David Powers, is a no nonsense engineering approach to solving partial differential equations, primarily via separation of variables and Fourier series, although Laplace transforms and d'Alembert's method are included. I see two strengths in this book. The first strength is that the basic equations of mathematical physics (the heat, wave and Laplace equations) are each given a nice physical derivation. The second strength is that the author has incorporated many interesting applied problems taken from the scientific literature into the problem sets. These strengths make this book worthy of consideration if you are teaching a class to a room filled with engineers or very applied mathematicians.

On the other hand, I do have some problems with this book. The main problem is that this book is definitely intended for teaching engineers. This fact is clearly on display in the footnote on page 60 of the text which reads "The word orthogonality should not be thought of in the geometric sense." When I was an undergraduate, I took a course out of this book. I was very puzzled as to how I should be thinking of the word orthogonality. Should I read the word to mean that a certain computation comes out to be zero? Certainly one can take this approach. What harm is there in relating this to a geometric concept? There is no harm in thinking about orhogonality geometrically. In fact, it is inherently a geometric concept and it means the same thing in this setting as it does in good old Rn. What good comes from thinking of it geometrically? The answer to this last question is obvious to many mathematicians; it's why we invented the inner product space in the first place. That footnote did no permanent harm to my brain; in fact, the act of puzzling over it may have done me a lot of good. If that was the author's intention, then more power to him.

Finally, books of this type often include a chapter on numerical methods. These chapters are inevitably very superficial and more than likely violate the Hypocratic Oath (First, Do No Harm) and should probably be left off.

Overall, this is a decent introduction to using Fourier series to solve basic partial differential equations. It is very suitable for use with engineers and applied mathematicians although Applied Partial Differential Equations by Richard Haberman is, for my money, a better choice. If you do choose to use this book, be sure you have the bookstore clip the footnote off of page 60 before you let the students have it.

Jeff Graham received his PhD from Rensselaer under the direction of Margaret Cheney. He is currently employed as an Assistant Professor at Susquehanna University in Selinsgrove, Pennsylvania. His interests include differential equations, modeling, and the mathematical aspects of music theory.


Chapter 0 -- Differential Equations
Chapter 1 -- Fourier Series and Integrals
Chapter 2-- The Heat Equation
Chapter 3 -- The Wave Equation
Chapter 4 -- The Potential Equation
Chapter 5 -- Higher Dimensions & Other Coordinates
Chapter 6 -- Laplace Transform
Chapter 7 -- Numerical Methods
Answers to Odd Numbered Exercises