9R2. Beyond Perturbation: Introduction to the Homotopy Analysis Method. - Edited by Shijun Liao (Shanghai Jiao Tong University, Shanghai, China). Chapman and Hall/CRC Press, Boca Raton FL. 2004. 322 pp. ISBN 1-58488-407-X.

Reviewed by SA Sherif (Dept of Mech and Aerospace Eng, Univ of Florida, 232 MAE Bldg B, PO Box 116300, Gainesville FL 32611-6300).

This book deals with a very interesting mathematical technique that is rather powerful. While perturbation methods work nicely for slightly nonlinear problems, the homotopy analysis technique addresses nonlinear problems in a more general manner. Through this method, the author demonstrates that a nonlinear problem that normally has a unique solution can have an infinite number of different solution expressions whose convergence region and rate are dependent on an auxiliary parameter. The method provides for ways to control and adjust the convergence region. This makes the method particularly suited for problems with strong nonlinearity.

The book is comprised of two parts. Part I contains Chapters 1 through 5, while Part II contains Chapters 6 through 18. The first part covers the basic ideas and concepts of the method, while the second part focuses on applications of the method to different situations. In addition to introducing the method in Part I, the author discusses the relation of the method to other analytical methods as well as the advantages and limitations of the method. Applications discussed in Part II are varied in scope covering areas such as simple bifurcation of nonlinear problems, nonlinear eigenvalue problems, the Thomas-Fermi atom model, free oscillation systems with both odd and quadratic nonlinearities, Blasius’ viscous flow, boundary layer flows with exponential and algebraic properties, von Karman’s swirling viscous flow, and nonlinear progressive waves in deep water.

The book should serve as an excellent reference to researchers, engineers, and interested individuals in helping them tackle nonlinear problems in an analytical fashion. It has a good subject index and an outstanding list of bibliography with 136 references cited. The book is very well written and is relatively easy to follow to the mathematically literate person. I highly recommend that it be acquired by interested individuals and libraries throughout.