11R39. Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems. Monographs and Surveys in Pure and Applied Mathematics, Vol 113. - PL Sachdev (Indian Inst of Sci, Bangalore, India). Chapman and Hall/CRC, Boca Raton FL. 2000. 319 pp. ISBN 1-58488-211-5. $94.95.
Reviewed by WS Janna (Herff Col of Eng, Univ of Memphis, 201E Eng Admin, Memphis TN 38152).
In his own words, the author of this text says he has “enjoyed finding exact solutions of nonlinear problems for several decades.” These decades of searching have afforded him the pleasure of associating with a large number of students, postdoctoral fellows, and other colleagues in his pursuit. It is clear from the Preface and from the text itself that the author greatly enjoys and is fascinated by the subject. The writing style is such that it seems as if he is introducing the reader to an old friend. The author refers to his 319-page text as a “monograph.”
Chapter 1 gives an introduction to the text. The author describes research work in the area and describes the types of problems where nonlinear differential equations are found. The search for exact solutions is motivated by the desire to understand the mathematical structure of the solutions themselves and to obtain a deeper understanding of the physical phenomena being described. The idea is to illustrate and bring out main points of the problems included. Most examples are drawn from real physical situations, mainly from fluid mechanics and nonlinear diffusion.
Chapter 2 gives an interesting discussion of first order, linear partial differential equations. Quasilinear partial differential equations of first order are described as are initial value problems. Fourteen example problems are solved. Chapter 3 is on exact similarity solutions of nonlinear partial differential equations, presented in two parts. Part A is on the reduction of partial differential equations by infinitesimal transformations. Systems of partial differential equations are described. Part B discusses a nonlinear heat equation in three dimensions. Also discussed is the similarity solution of Burgers equation (elementary nonlinear diffusive traveling wave equation) by the direct method. Exact solutions of equations for free surface flows for shallow water are solved by the direct similarity approach. An example from gas dynamics is solved by the multipronged approach to finding exact solutions of nonlinear partial differential equations.
Chapter 4 is about traveling wave problems. The chapter begins with a description of the problem and the general wave equation. The one-dimensional wave equation in gas dynamics is given and a solution is provided. The elementary nonlinear diffusive traveling wave equation is described next, followed by equations for traveling waves for higher-order diffusive systems. Multidimensional systems of homogeneous partial differential equations written to describe simple wave flows are discussed and solved. Traveling wave equations for nonhomogeneous hyperbolic or dispersive systems are written and solved, as are equations for hydromagnetic traveling waves. Simple waves on shear flows are also discussed.
Chapter 5 is on the exact linearization of nonlinear partial differential equations. Burgers equation in one and higher dimensions is written and analyzed. The nonlinear degenerate diffusion equation is also written and a numerical solution is described. An equation for the one-dimensional motion of an ideal compressible isentropic gas and the Born-Infeld equation are discussed. Another interesting example involves the mathematical description of water waves on a uniformly sloping beach. The chapter concludes with a description of simple waves on shear flows.
Chapter 6 is about the nonlinearization and embedding of special solutions. The objective here is to first linearize a given nonlinear partial differential equation, solve the resulting simpler equation, and use it to build up nonlinear effects either approximately or exactly. The method is applied to obtain exact solutions for the generalized Burgers equation. The method is then described for Burgers equation in cylindrical coordinates, as well as for nonplanar and other modified forms of Burgers equation.
Chapter 7 is on asymptotic solutions by balancing arguments. The method here would be applied specifically to nonlinear problems for which an exact solution is not obtainable. The search is for an asymptotic solution that will exist for large distances or for very long times. The chapter illustrates the method by using examples from ordinary differential equations. The method is also applied to the nonplanar Burgers equation with wave initial conditions and to equations for one-dimensional contaminant transport through porous media.
Chapter 8 describes a method involving the derivation of series solutions of nonlinear partial differential equations. The first example is expansion of a gas sphere into a vacuum, followed by the description of the collapse of a spherical or cylindrical cavity. The chapter concludes with a discussion of a converging shock wave from a spherical or cylindrical piston.
The text contains the Anglican spelling of a number words (eg, travelling), but the English is very good and the text is very well written. There are few figures in the text, but those included are nicely done. The text contains an extensive bibliography, citing over 140 titles, however, 22 of them list the author of this text as first author.
The index appears to be adequate and contains no entries that refer the reader to another entry. The text does not contain tables of fluid properties, and in fact, there are no appendices. There are no end-of-chapter practice problems.
The text can be used in a second or third course in differential equations in either an engineering or mathematics curriculum. The text contains an emphasis on applications, and there is a notable absence of theorems and corollaries; so would have an engineering slant. It is not known whether the author is an engineer, physicist, or a mathematician.
Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems is extremely interesting and is quite readable. It would make a great addition to any academic or personal library.