- Email: [email protected]
Perturbation methods in applied mathematics
Book Reviews mechanics research: (1) “Finite Element Methods in Continuum Mechanics”, by T. H. H. Pian and P. Tong; (2) “Motion of Bubbles and Drops through Liquids”, by J. F. Harper; (3) “Shock Waves, Jump Relations and Structure”, by P. Germain; and (4) “Interplanetary Gas Dynamics”, by V. C. Liu. Researchers working in the numerical solution of field problems in continuum mechanicsshouldbeparticularlyinterested in the extremely thorough article on finite element methods by Professors Pian and Tong. The authors give a very lucid presentation of the formulation of finite element methods from different variational principles (principle of minimum potential energy, principle of complementary energy, and Reissner’s principle), and their modifications, in linear elastic theory. A significant feature of the article is their presentation of the theory Finite Element Models”, of “Hybrid which employs variational principles with relaxed continuity requirements at the interelement boundaries-most of which were originally pioneered by the authors themselves during the past 9 years. The hybrid model theory should be of special interest to those involved in complicated structural analysis, fracture mechanics, etc. In addition to the linear elasticity problem, the article also treats other continuum field problems such as heat transfer, steady-state temperature distribution and two-dimensional Stokes flow. Included is a highly comprehensive extremely welcome bibliography-an addition to the literature at a time when the finite element method is finding rapid applications in various areas of continuum mechanics. The article by Harper contains a summary of the cases of a bubble with constant surface tension rising under gravity, of a drop whose interior viscosity and density are taken into account, and the surface activity treats as well phenomenon. The article by Germain is concerned with the “local study” of shockwaves, and contains a vivid mathematical account of “shock relations” for a physical phenomenon governed by conservation laws, classification of shocks,
224
problems related to shock structures, and presents an excellent summary of the asymptotic expansion methods for solution in the neighborhood of a shock. Finally, the article by Liu gives a very readable account of current methods of gas kinetics and continuum flows, collective particle behavior of interplanetary gas and the free-expansion phenomenon. The volume as a whole can be very profitably used by both researchers and graduate students as a source of authoritative state-of-the-art surveys in the fields mentioned. SATYANADHAM ATLURI Department of Aeronautics and Astronautics likersity of Washington Seattle, Washingto//
PERTURBATION METHODS IN APPLIED MATHEMATICS, by Julian D. Cole. 260 pages, diagrams, 6 x 9 in. Waltham, Mass., Ginn-Blaisdell 1968. Price $9.50 (approx. 52.75). Among the many recent books dealing with asymptotic solutions to differential equations, Cole’s work under review is one of the better, in many aspects. The title is a bit misleading, implying more generality than the book really has. It concentrates almost exclusively on singular perturbations of differential equations, and the approach is to teach by example. This the author does very well. He selects a wide range of model problems, describes the physical background of the problem briefly so that someone with a fair acquaintance in basic physics and basic fluid mechanics can visualize the situation, and then works out the solution in considerable detail. Some of the underlying theoretical principles are discussed, but no rigorous proofs are given, and very few theorems even specifically mentioned as theorems. There are problems for students scattered about, and a number of references to the actual use of these ideas in the literature. The book is broken down into the following topics. Chapter One discusses the definitions of asymptotic. Chapter
.Jo~r~~al of The Vrsnklin
Institute
Book Reviews Two introduces ordinary and singular perturbations and many of the standard ideas: in particular, boundary layers and matched asymptotic expansions. This inchIdes some verysophisticatedexamples, such as the van der Pol oscillator for very large damping. Chapter Three discusses the twovariable method, applied to oscillatjor problems in ordinary differential equations and to problems which are essentially \VKB problems. Chapter Four is devoted to part’ial differential equations, largely those equations appearing in gas dynamics. The presentation of boundary-layer phenomena in hyperbolic equat,ions appears in no other book t,hat this reviewer has seen. The last chapt,er is devoted to deriving various equations defining limiting cases of complicated physical processes. The author shows how one can syst,ematically obtain the equations for subsonic, transsonic and supersonic gas flow from the full equations of t,hin air foil tjheory. He also derives t.he
various approximations which can describe acoustic shock waves in a pipe and nonlinear shallow water waves. One need not be an expert to understand these derivations. There are certain perturbation methods the author omlts, and little mention is made of the WKB method, which is discussed in the litt,le book, JWKEI Approximation, by N. and P. 0. Froman. There is no discussion of the BogoliubovMitropolski method of averaging, which is in many books. There is no coverage of the Poincare-Lighthill method [which is discllssed in t,he reviewer’s article in MAX Rev. Vol. 14, pp. 433-446, 19721. But, for the classes of problems it does discuss, this book is a very good teaching tool for students and researchers who wish to learn how to use singular perturbation met,hods in differential equations. CRAIG COMSTOCK
Postgraduate s&001 JloHtPly/, CaliforHiu Xuval
225
224
problems related to shock structures, and presents an excellent summary of the asymptotic expansion methods for solution in the neighborhood of a shock. Finally, the article by Liu gives a very readable account of current methods of gas kinetics and continuum flows, collective particle behavior of interplanetary gas and the free-expansion phenomenon. The volume as a whole can be very profitably used by both researchers and graduate students as a source of authoritative state-of-the-art surveys in the fields mentioned. SATYANADHAM ATLURI Department of Aeronautics and Astronautics likersity of Washington Seattle, Washingto//
PERTURBATION METHODS IN APPLIED MATHEMATICS, by Julian D. Cole. 260 pages, diagrams, 6 x 9 in. Waltham, Mass., Ginn-Blaisdell 1968. Price $9.50 (approx. 52.75). Among the many recent books dealing with asymptotic solutions to differential equations, Cole’s work under review is one of the better, in many aspects. The title is a bit misleading, implying more generality than the book really has. It concentrates almost exclusively on singular perturbations of differential equations, and the approach is to teach by example. This the author does very well. He selects a wide range of model problems, describes the physical background of the problem briefly so that someone with a fair acquaintance in basic physics and basic fluid mechanics can visualize the situation, and then works out the solution in considerable detail. Some of the underlying theoretical principles are discussed, but no rigorous proofs are given, and very few theorems even specifically mentioned as theorems. There are problems for students scattered about, and a number of references to the actual use of these ideas in the literature. The book is broken down into the following topics. Chapter One discusses the definitions of asymptotic. Chapter
.Jo~r~~al of The Vrsnklin
Institute
Book Reviews Two introduces ordinary and singular perturbations and many of the standard ideas: in particular, boundary layers and matched asymptotic expansions. This inchIdes some verysophisticatedexamples, such as the van der Pol oscillator for very large damping. Chapter Three discusses the twovariable method, applied to oscillatjor problems in ordinary differential equations and to problems which are essentially \VKB problems. Chapter Four is devoted to part’ial differential equations, largely those equations appearing in gas dynamics. The presentation of boundary-layer phenomena in hyperbolic equat,ions appears in no other book t,hat this reviewer has seen. The last chapt,er is devoted to deriving various equations defining limiting cases of complicated physical processes. The author shows how one can syst,ematically obtain the equations for subsonic, transsonic and supersonic gas flow from the full equations of t,hin air foil tjheory. He also derives t.he
various approximations which can describe acoustic shock waves in a pipe and nonlinear shallow water waves. One need not be an expert to understand these derivations. There are certain perturbation methods the author omlts, and little mention is made of the WKB method, which is discussed in the litt,le book, JWKEI Approximation, by N. and P. 0. Froman. There is no discussion of the BogoliubovMitropolski method of averaging, which is in many books. There is no coverage of the Poincare-Lighthill method [which is discllssed in t,he reviewer’s article in MAX Rev. Vol. 14, pp. 433-446, 19721. But, for the classes of problems it does discuss, this book is a very good teaching tool for students and researchers who wish to learn how to use singular perturbation met,hods in differential equations. CRAIG COMSTOCK
Postgraduate s&001 JloHtPly/, CaliforHiu Xuval
225