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 1584883553
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Reviewed by Andrew Resnick
see all book reviews 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 closedform 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 wellordered 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
secondorder differential equations, while the last three chapters are concerned
with third, fourth, and higherorder differential equations. Some examples,
chosen at random, that illustrate the breadth of this book are the following:
the Schrödinger (quantum mechanics) and FitzhughNagumo (signal propagation
in nerve cells) equations are parabolic equations, the telegraph equation (transport
in a conductive and diffusive medium) is a hyperbolic equation, the MongeAmpère
equation (gas dynamics and meteorology) contains mixed derivatives, the hydrodynamic
boundarylayer equations are thirdorder, and the NavierStokes equations are
fourthorder in the stream function.
The 12 supplementary chapters contain a wealth of information on methods of
classification, transformations, and constructing closedform 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, easytouse 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.
