7R2. Handbook of Green’s Functions and Matrices. - VD Seremet (State Agrarian Univ of Moldova, Rep of Moldova). WIT Press, Southampton, UK. Distributed in USA by Comput Mech, Billerica MA. 2003. 295 pp. CD-Rom included. ISBN 1-85312-933-X. $229.00.
Reviewed by GC Gaunaurd (Code AMSRL-SE-RU, Army Res Lab, 2800 Powder Mill Rd, Adelphi MD 20783-1197).
This handbook is intended for the specialist interested in the solution of Poisson’s equation and the Navier equations of elastostatics (here called the Lame´ equations) in a number of particular situations. The handbook comprises a book and a compact disc (CD). The book has two parts each with two chapters. Part I contains the theoretical part. Part II gives a list of ‘Green’s functions’ for Poisson’s equation and a list of ‘dyadic Green’s functions’ (here called Green’s matrices) for the Navier equations of elastostatics. (These are not to be confused with the Navier-Stokes equations of Fluid Mechanics, which are time-dependent and nonlinear, since they are expressed in Eulerian coordinate frames.) Part III is a CD that contains six appendices to the book, with numerous practical examples. Appendix I shows how to construct Green’s functions for Poisson’s equation, and Appendices III and V, the ones for the Navier equations. The other three appendices give examples of cases in which these Green’s functions appear in applied problems of mathematical physics.
It is difficult to summarize this book in this brief review and do it justice. The author has done an excellent job. It covers material not well known in the West. As far as this reviewer knows, this book is unique in this respect. These Green’s functions are the basic tools needed to solve boundary value problems in elastostatics. Part I introduces the method of “incompressible influence elements.” This method leads to a new theory developed by the author for the construction of influence functions for bulk-dilatation and dyadic Green’s functions, whose components are the displacement components for boundary-value problems of elastostatics.
Part II implements the theory presented in Part I and constructs the Green’s functions of some specific boundary-value problems (BVP). Here is a practical collection of hundreds of problems on Green’s functions for these equations. These problems and their answers are presented to the reader for use in practical cases or to practice his own mathematical skills. The derivations are not shown; only the answers are listed. As mentioned above, Part III is the CD. Here many typical 2D and 3D BVPs for Poisson’s and Navier equations are solved. Some explanations on how to construct the Green’s Function in question are given here.
The book is a collection of Green’s functions for two elliptic second-order partial differential equations, and as such, is quite heavy on the math. There are no graphs, and there are also some shortcomings. For example, only the first basic problem of elasticity is addressed, only domains describable in Cartesian coordinate systems are presented, and there are many grammatical errors due to perhaps the fact that the author is not very familiar with the technical terminology in English.
Handbook of Green’s Functions and Matrices is quite excellent, and in many respects, unique. Its intended audience is the advanced graduate student, and since there are no derivations, maybe it is intended for the research specialist, who already knows how to derive these results, but wishes to have quick access to them. In other words, it is like a Table of Integrals. Its steep price will undoubtedly restrict its circulation. It is certainly a reference of high value for institutional libraries. This reviewer believes that it would be advantageous if the author would some day extend this work to the elastodynamic (ie, time-dependent) cases governed by the various types of wave-equations and/or the Navier-equations of elastodynamics that govern elastic vibrations. These dynamic situations are more likely to be encountered in practical instances, rather than their static counterparts.