Book Review

Handbook of Nonlinear Partial Differential Equations

Andrei D. Polyanin and Valentin F. Zaitsev
Chapman and Hall/CRC, Boca Raton, FL, 2004
814 pp., $99.95 hb
ISBN 1-58488-355-3
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Reviewed by Andrew Resnick

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This book initially appears to be quite formidable, for it contains almost 800 densely packed pages of nonlinear partial differential equations and their solutions. The stated goal of this text is impressive: to present closed-form solutions to more than 1,600 nonlinear partial differential equations. The equations chosen appear in multiple physical and biological sciences problems, including heat and mass transfer, hydrodynamics, control theory, and chemical engineering. The authors are to be congratulated for somehow making this book so approachable. From the well-ordered table of contents to the clear index, this book promises to be one that will be used regularly, rather than gather dust on a shelf.

The first six chapters are concerned with parabolic, hyperbolic, and elliptic equations. They cover the bulk of equations found in the physical sciences. The next two chapters address mixed derivatives and general cases of second-order differential equations, while the last three chapters are concerned with third-, fourth-, and higher-order differential equations. Some examples, chosen at random, that illustrate the breadth of this book are the following: the Schrödinger (quantum mechanics) and Fitzhugh-Nagumo (signal propagation in nerve cells) equations are parabolic equations, the telegraph equation (transport in a conductive and diffusive medium) is a hyperbolic equation, the Monge-Ampère equation (gas dynamics and meteorology) contains mixed derivatives, the hydrodynamic boundary-layer equations are third-order, and the Navier-Stokes equations are fourth-order in the stream function.

The 12 supplementary chapters contain a wealth of information on methods of classification, transformations, and constructing closed-form solutions of differential equations. It is difficult to summarize the supplements, but a few sections that are particularly interesting to this reviewer are those on similarity transforms, methods of functional and generalized separation of variables, and group analysis methods. I found these supplements to be clearly written, with the goal of providing a “recipe” that can be applied toward a solution. Although the topics are rather advanced, the notation is uniformly clear and easy to follow.

The authors clearly state in the preface that an overriding goal was to make the book useful to a wide audience with different mathematical backgrounds. Thus, they prefer schematic approaches to specialized topics rather than introducing detailed terminology. Without a doubt, this approach makes the book useful to those of us lacking advanced mathematical abilities. Books like this are presumably prone to errors, either typographical or otherwise. I checked approximately a dozen of the equations, chosen at random throughout the book, and found no errors.

Handbook of Nonlinear Partial Differential Equations is a total success from the standpoint of offering a complete, easy-to-use solution guide. The authors are to be congratulated on what surely must have been a monumental effort to bring this book to fruition.

Biography

Andrew Resnick is a postdoctoral fellow at the Case Western Reserve School of Medicine, Department of Physiology and Biophysics, in Cleveland, Ohio. His current research concerns how cells sense their fluid environment using a primary cilium.