Modelling Mathematical Methods and Scientific Computation. by Nicola Bellomo; Luigi PreziosiReview by: J. David LoganSIAM Review, Vol. 39, No. 1 (Mar., 1997), pp. 154-156Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2133022 .

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154 BOOK REVIEWS

which such strong estimates are not known and current consensus is that these esti- mates fail to hold. At the very least, they are extremely difficult to prove. Amann's book provides a different approach to such problems. Using (comparatively) simple es- timates, he considers these PDE problems in a more abstract setting and analyzes the abstract problem. The model for this ap- proach is the semigroup method of Kato, Yoshida, etc., but the nature of the prob- lems studied here begs for a more detailed study of the abstract problem.

The book reviewed here is the first volume of a trilogy on parabolic sys- tems. The goal is to study systems of quasi-linear parabolic systems of arbitrary order with general nonlinear boundary conditions. This ambitious goal is realized in three steps (one per volume): first, study a suitable abstract linear evolution prob- lem; then, apply the abstract linear theory to concrete problems; and, finally, consider quasi-linear problems.

Amann has written a very useful book. He presents several important concepts in a clear and concise way, and anyone inter- ested in the modern theory of parabolic dif- ferential equations will need to know much of the material in this book. In addition to a study of the parts of semigroup the- ory which are relevant to parabolic PDEs, this book includes a discussion of maximal regularity which is crucial to a complete understanding of such problems. To illus- trate this concept, consider the simple case of the inhomogeneous heat equation with Cauchy-Dirichlet data:

-Ut + Au = f(x, t) in Q x (O, T),

u= 0 on 9Q x (0,T), u-= pon Q x {0}.

If f lies in a suitable Holder space, stan- dard semigroup theory would assert that u is a continuous function into a corre- sponding Holder space. Specifically, if f E Co([O, T] : Co(Q)), then stan- dard semigroup theory would give u E C([O,T] : C2o (?!)) and ut E C([O,T] : C (Q)), but the maximal regularity is that these functions are Ca rather than C. This improvement is crucial for the study of nonlinear problems. A fur- ther important element of Amann's ap- proach is the study of variable domains.

In this context, the major use of vari- able domains is in allowing the differ- ential operator to have time-dependent coefficients, whereas standard semigroup theory considers only time-independent operators.

These additional considerations lead to complications in the structure of the book; however, Amann keeps them from becom- ing impediments to understanding. His writing style is clear, and that is impor- tant for such a necessarily abstract tome. There is a lot to absorb, but the material is well presented, and this reviewer is looking forward to the next two volumes. My only complaint is the price, but it seems that all mathematical books are increasing rapidly in this direction.

GARY M. LIEBERMAN Iowa State University

Modelling Mathematical Methods and Scientific Computation. By Nicola Bellomo and Luigi Preziosi. CRC Press, Inc., Boca Raton, FL, 1995. $61.95. xiv + 497 pp., cloth. ISBN 0-8493-8331-5.

In recent years, because of the emergence of modelling courses in colleges and uni- versities, several excellent texts on math- ematical modelling have appeared. The revolution may have begun with the now classic text of Lin and Segel [1], originally published in 1974. They, perhaps more than others, brought out the notion that applied mathematics is more than mathe- matical methods and techniques. Applied mathematics is intimately intertwined with the natural and applied sciences, and there- fore the entire modelling process (includ- ing formulation of the equations, analysis, comparison with empirics, and validation) lies in its domain. Mathematical modelling has come of age, and now there are many courses in mathematics departments with that title. The trend is continuing with the recent emphasis on industrial mathe- matics programs. No longer do we think of applied mathematics as the study of the classical equations of mechanics and elec- trodynamics; rather, mathematical mod- elling is fundamental in all of the applied

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BOOK REVIEWS 155

sciences, including medical science, biology, economics, and so on.

The authors note that, in spite of its recognized relevance, mathematical mod- elling is not a fully well-established disci- pline. One of the purposes of this text is to present a systematic structure that gives definition to the science of mathematical modelling. There is some effort, for exam- ple, to classify mathematical models into discrete, continuous, and stochastic mod- els. Flow charts are presented to illustrate how the modelling process evolves in spe- cific problems.

Consequently, this book differs from other books on modelling. Most current books are organized around specific, often simple, scientific problems. That is, a prob- lem from science or some technological area is stated, the equations are formulated, techniques are thus motivated and devel- oped, and solutions, either exact or approx- imate, are analyzed. Bellomo and Preziosi's book takes a more general approach and develops a broad context in which to study specific models. For example, in Chapter 3, on continuous models, they begin by developing the general equations of continuum mechanics, followed by a general, concise treatment of the equations of electrodynamics. The material is very well organized and nicely presented, but this general approach may make the text inaccessible to beginners. However, the authors state explicitly that the book is designed as an advanced textbook for higher-level "master" courses in mathemat- ics. In the reviewer's opinion, this is a correct assessment of the prerequisites. The book would be a good choice for graduate students who have had basic courses in ordinary and partial differen- tial equations and who have been exposed to intermediate-level physics or engineer- ing courses, or at least physical concepts, in some depth. Students reading Chap- ter 3 would certainly benefit from a prior understanding of concepts like diffusion and convection.

The text is divided into four chapters, followed by appendices. Chapter 1 (Math- ematical Modelling) is a brief (28 pages) excursion into defining and classifying mathematical models. Chapter 2 (Discrete

Models, 166 pages) undertakes a general discussion of systems of ordinary differen- tial equations. After several examples from mechanics, population dynamics, circuit theory, and nerve-pulse propagation, there is a brief discussion of basic theoretical results for ordinary differential equations. Following that discussion is a treatment of stability and linearization, bifurcation, and chaos. There is a nice sequence of examples involving dynamics of the heart, Lienard's equation, the May model of predation, and the classic Lorenz model. Woven into the examples are some general theorems about limit cycles and Hopf bifurcation. The plan of the book is to formulate and state careful versions of theorems but not to give proofs. The chapter on discrete models ends with a long discussion of numerical methods for solving initial value problems. Chapter 3 (Continuous Models, 142 pages) takes a similar approach for partial differential equations. Here there are several examples, and theoretical results are minimal; there is a good discussion of well posedness and classification. A substantial portion of Chapter 3 deals with numerical methods, especially finite difference methods for par- tial differential equations. Chapter 4 (In- verse and Stochastic Problems, 78 pages) begins with a classification of inverse prob- lems and well posedness for such problems. There is a discussion of domain decompo- sition and minimization techniques. There follows a brief treatment of the stochastic aspects of inverse problems. The three appendices (61 pages) are Function Spaces, Interpolation and Approximation, and Random Variables; they are not meant to be tutorial but rather a useful summary of the basic definitions and concepts.

Each chapter ends with a few illustrative exercises, not all simple; many of these take the form of small projects. There are more than 100 references, mostly to other books that treat the same, or related, topics.

An IBM-compatible diskette is sup- plied with the text, and it contains BASIC programs designed to carry out numerical solutions of some of the sam- ple problems introduced in the text. Among the programs included are a Runge-Kutta-Fehlberg method and a fourth-order Adams-Bashforth method for solving systems of ordinary differential

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156 BOOK REVIEWS

equations, explicit and implicit finite difference methods for solving parabolic equations, an explicit method for solving the telegrapher's equation, a program for solving a parabolic inverse problem, a program simulating population dynamics with stochastic interaction, and Lagrange interpolation and splines. The programs given in the text are well documented with ample comments in the program listings. The reviewer found these programs easy to use, with minimal input. The programs are not limited merely to simple cases; for example, the parabolic equation solver can accommodate both nonlinear convection and diffusion. The authors emphasize the important role that numerics and scientific computation play in the modelling process. The graphics in the programs are a little crude but adequate; the reader is invited to improve and optimize the programs given.

In summary, the reviewer was impressed with the overall organization and layout of the book. On occasion there is an unusual word order or usage, but it is not distract- ing; in general, the book is well written and well conceived. Students who read the book (it could be covered in a one-semester course meeting three times per week) will have a good grasp of the systematics of mathematical modelling; it is not the art of developing mathematical models that is emphasized (i.e., physical reasoning and equation derivation) but rather the struc- ture and format of mathematical models. For graduate students who have been exposed to ordinary and partial differential equations, as well as some physical reason- ing, this book could be of value in codifying their ideas about the interdependence of mathematics and science.

REFERENCE

[1] C. C. LIN AND L. A. SEGEL, Mathematics Applied to Deterministic Problems in the Natural Sciences, Macmillan Publishing Co., New York, 1974; reprinted by SIAM, Philadelphia, PA, 1988.

J. DAVID LOGAN University of Nebraska-Lincoln

Water Waves: Relating Modern Theory to Advanced Engineering Practice. By Matiur Rahman. Oxford University Press, London, UK, 1994. $79.00. xii+343 pp., cloth. ISBN 0-19- 853478-7.

This book is well written and can be easily read. It is self-contained in that all of the mathematical tools and equations the reader needs can be found in the book. It is of interest to a wide audience in the science community: mathematicians, physicists, and engineers. Any reader would have to refer to numerous other books to get all the information found here, and this book is more detailed and focused than [1] when it comes to water waves.

In Chapter 2, the author gives a good summary of the main equations and results in fluid mechanics as well as important analysis techniques. In Chapter 3, he summarizes solution techniques for partial differential equations: the D'Alembert solution of the wave equation, the method of characteristics, and the method of separation of variables. In Chapter 4, he summarizes the properties of surface waves, and he introduces the reader to the important definitions and phenomena for waves; the simplification assumptions are well stated. In Chapter 5, he concentrates on finite amplitude waves, in particular the steady motion of such waves by considering the wave in a reference system moving at a constant velocity C. He presents Stokes's solution of the water-waves equation. This chapter might have gained in clarity if the author had given more details about some of his statements.

Chapter 6 is devoted to the dynamics of tidal waves in various geometries: in a canal, in estuaries, and in a river. Rahman also considers tidal waves in two- dimensional geometries, with or without friction, when the amplitude of the wave is no longer small compared with the water depth. In Chapter 7, he gives a brief review of the statistic and probablity results he will need. He also considers ocean waves as random phenomena; he looks at the distribution of wave heights

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