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Mathematical and numerical modelling in electrical engineering. Theory and practice. (English) Zbl 0859.65128
Mathematical Modelling: Theory and Applications. 1. Dordrecht: Kluwer Academic Publishers. xiii, 300 p. (1996).
The object of this book, according to the authors, is to introduce engineers to the concepts of functional analysis as providing tools for the solution of real-life technical problems. It opens with a discussion of the philosophy of mathematical modelling and then gives a discussion of basic mathematical concepts such as abstract spaces, and the idea of least squares. There then follow introductory surveys of two particular concepts of numerical analysis, namely finite elements and conjugate gradients. These ideas are illustrated by a short chapter on the magnetic potential in the window of an ideal transformer.
Following on this, the authors use the ideas of monotone operators to discuss a problem involving nonlinear magnetostatic fields. The next topics discussed are some problems of heat transfer. At this stage, ideas such as Gâteaux/Fréchet differentials, weak solution and variational curves are introduced. Topics which follow are stationary semiconductor problems and nonstationary heat conduction in stators. Here again, particular attention is paid to the use of finite element methods.
A general treatment of solutions of Maxwell’s equations with a harmonic time variation is considered next and there follows a discussion of a problem connected with transformer shielding. After this, it is indicated how time dependent problems and anisotropic and inhomogeneous media may be analyzed. The text closes with drawing attention to the necessity of designing devices to have an optimal shape, the ideas being illustrated by an iron-core electromagnet and an air filled capacitor. The book closes with a list of nearly 300 references, the latest being as recent as 1995.
The general tone of the book is sophisticated, with very much of the treatment being in terms of abstract spaces, and possibly some more concrete examples would have been helpful. In accordance with this outlook, there is a discussion of topics such as strong and weak convergence and error estimates, although for a proof of some of the results mentioned it is necessary to look at some of the references.
The book is international being written in English by the authors from the Czech Republic and Finland and printed in the Netherlands. It reads well, the printing is clear and no mistakes were noticed. (It is unfortunate that the usual practice of denoting vectors in Clarendon type has not been followed.) Non-mathematically-minded engineers will find the going difficult, but for those who have a sufficient mathematical background there will be much interest and the book can be recommended to them. It is a pity that the price will put it out of reach of individuals and many libraries.

MSC:
65Z05 Applications to the sciences
78-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to optics and electromagnetic theory
65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
00A71 General theory of mathematical modeling
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.)
35Q60 PDEs in connection with optics and electromagnetic theory
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