Differential Equations: A Modern Approach with Wavelets

The author describes this book as a mixture of differential equations, Fourier analysis, and applications with equal weight. Fourier analysis is construed broadly to include wavelets. The introduction to students tells them that the book’s purpose is to teach the interrelationships between these topics. The book is intended for a broad audience including mathematics, engineering and physics students. No background in undergraduate real analysis is expected.

 

This is an unusual book with an occasionally odd selection of topics. It begins with ordinary differential equations (ODEs), and devotes about half the book to that. In some respects this part looks like a text written a few decades ago.  For example, it includes sections on solution techniques for special equations such as exact equations, the use of integrating factors, and the like.  At the same time the book never considers systems of linear ODEs, and no linear algebra appears in this part of the book. Some higher order linear equations are discussed briefly with a solution method that essentially uses the characteristic polynomial. 

 

The discussion of ODEs also includes series solutions, numerical methods, and a treatment of Sturm-Liouville and boundary value problems for ODEs. Perhaps the best parts in the section on ODEs are the application examples. A particularly good one describes design of a dialysis machine.

 

The second half of the book begins with more or less standard treatments of Fourier series with a quick introduction to the Fourier transform and a full chapter on the Laplace transform. A short chapter on distributions follows these. The author says that he wants to make the book “timely and exciting” by introducing the basic concepts of wavelets with some description of their applications to signal and image processing. He describes advantages that Haar series and wavelet transforms have over Fourier series and transform methods in applications. The book concludes with an introduction to partial differential equations and boundary value problems including discussions of the wave and heat equations.

 

There is a disconnected feeling to the book. The parts that deal with ODEs form one part and operate on one level of sophistication consistent with use of the book at the sophomore undergraduate level. After that it appears that the author raises his expectations for the level of sophistication of his readers. Despite his insistence in the introduction that no background in analysis is required, some of the later chapters assume more. The chapter on distributions, in particular, seems to treat Schwartz distributions, function spaces, Borel measure, semi-norms and even the Paley-Weiner theorem as if the reader had seen them before. This level of sophistication continues into the chapter on wavelets. It isn’t even very clear why the topic of distributions is included, since we only see it later with the Dirac delta function.

 

This is an odd and uneven book. Some topics are handled well – basic differential equations, Fourier series and wavelets, but some things that seem important - systems of linear ODEs, for example - don’t appear at all. The connections and interrelationships between topics that the author describes in his introduction just don’t come through.

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films, material science and the odd bit of high performance computing. He did his PhD work in dynamical systems and celestial mechanics.