11R3. Perturbation Methods for Differential Equations. - BK Shivamoggi (Dept of Math, Univ of Central Florida, Orlando FL 32816-1364). Birkhauser Boston, Cambridge MA. 2003. 354 pp. ISBN 0-8176-4189-0. $59.95.

Reviewed by J Awrejcewicz (Dept of Autom and Biomech, Tech Univ of Lodz, 1/15 Stefanowskiego St, Lodz, 90-924, Poland).

This book is focused on perturbation methods mainly applied to solve both ordinary and partial differential equations, as its title implies. As explained by the author, one of the unusual features of the treatment is motivated by his lecture notes devoted to a mix of students in applied mathematics, physics and engineering. Therefore, it is intended to serve as a textbook for both undergraduale students of the previously mentioned branches of science. However, I wonder if the students will be able to understand fully physical aspects of many various examples of completely separated fields such as solid mechanics, fluid dynamics and plasma physics. This aspect has been probably understood by the author, who added many appendices to the chapters. On the other hand, looking for the cited 26 references authored or co-authored by BK Shivamoggi, it is not surprising that his research covers the above-mentioned branches of science. This book can serve also as an example how an asymptotic analysis may easily move between various different disciplines.

The book is 354 pages long and has 130 references. It is divided into seven chapters.

Chapter 1 introduces a reader with asymptotic series and expansions of some arbitrarily chosen functions. It can be treated as a brief panoramic picture to the further problems dealt with the book.

In Chapter 2 regular perturbation methods are addressed. First algebraic equations are considered (four examples), then differential equations are analyzed (four examples), and finally partial differential equations are studied (1 example). The author originally introduced some of the outlined examples (for example, Section 2.5 is devoted to application to fluid dynamics published already by the author in 1998) and some were taken from other cited sources. Eight exercises are given at the end of this chapter to be solved by a reader or student.

In Chapter 3 the method of strained coordinates (parameters) is described. In Section 3.2, the Poincare´-Lindteadt-Lighthill method of perturbed eigenvalues is briefly stated with the supplement three examples. In addition, the eigenfunction expansion method (Section 3.3), Lighthill’s method of shifting singularities (Section 3.4), and the Pritulo’s method of renormalization (Section 3.5) are presented with supporting examples. It is worth noticing that the applications come from various fields including wave propagation in a homogeneous medium, nonlinear buckling of elastic columns, and a few examples within the field of fluid dynamics and plasma physics. The main limitation of the strained coordinates method, ie, an incapability of determining transient responses of dissipative systems, is illustrated and discussed. Nine exercises are added for the reader to solve.

Chapter 4 discusses the method of averaging. After a brief introduction, the Krylov-Bogoliubov method of averaging is described and two classical examples adopted from the Nayfeh work are given. Section 4.3 includes one sentence describing the so called generalized Krylov-Bogoliubov-Mitropolski method, and then two classical examples of the Duffing and van der Pol oscillators are considered. Witham’s average Lagrangian method is addressed in Section 4.4 using a nonlinear dispersive wave propagation problem. In the next section the Hamiltonian perturbation method is introduced followed by three examples. Then the averaged Lagrangian method is applied to study a nonlinear evolution of a modulated gravity wave packet on the surface of a fluid. At the end of the chapter, seven exercises are included.

The method of matched asymptotic expansions is described in Chapter 5. After a brief introduction and physical motivation the method of matched asymptotic expansion is explained through a simple example by computing inner, outer, and composite expansions. Applying Cole (1968) and Keviorkian and Cole (1996) results, the linear hyperbolic partial differential equation is analyzed in Section 5.4, the elliptic equations are described in section 5.5, and the parabolic equations are analyzed in Section 5.6. The interior layers are illustrated in Section 5.7 using an example introduced earlier by Lagerstrom (1988). In Section 5.8 Latta’s (1951) method of composite expansions are illustrated via three examples (two of them are borrowed form Nayfeh (1973) and Keller (1968)).

Section 5.9 titled Turning-point problems, includes a description of the JWKB approximation [with two examples borrowed from Holmes (1995)], the solution near the turning point and the Langer’s method. An application of the matched asymptotic expansion is taken from the field of fluid dynamics. Namely, a boundary layer flow past a flat plate is studied. Next, ten exercises to be solved follow. A method of multiple scales is illustrated in Chapter 6. After a brief introduction to the method, the differential equations with constant coefficients are addressed in Section 6.2, where eight examples are included (six of them are borrowed form other references). Struble’s method is described in Section 6.3, where two examples are given. In Section 6.4 differential equations with slowly varying coefficients are considered. Two supplemented examples illustrate application of the multiple scale method. The generalized multiple scale method, following Nayfeh (1964), is presented via two boundary-value problems. The considered applications include dynamic buckling of a thin elastic plate (solid mechanics) and a few examples taken from fields of fluid dynamics and plasma physics. The chapter finishes with eleven examples to be solved.

The last chapter, 7, is devoted to miscellaneous perturbation method. The main purpose of this chapter is to describe some special perturbation techniques that are very useful in some applications. The series of discussed methods include a quantum-field-theoretic perturbative procedure and a perturbation method for linear stochastic differential equations. Four exercises to be solved are given at the end of this chapter.

Since Perturbation Methods for Differential Equations covers a great deal of material, it is recommended to students and researchers, already familiar with solid and fluid mechanics, as well as with plasma physics. In general the figures and tables are fine, and the index is adequate, hence I recommend the book to be purchased by both individuals and libraries.