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Mathematical models in applied mechanics. Reprint with corrections. (English) Zbl 0994.35001

Oxford Texts in Applied and Engineering Mathematics. 4. Oxford: Clarendon Press. viii, 280 p. (2001).
This is a reprint of the original (1986; Zbl 0588.35001).
Applications of mathematics are based on the mathematical modelling. Mathematical modelling is intuitively evident and, at the same time, a hardly formalized concept. Creating and investigating mathematical models is important for natural and applied sciences, and usually the specific mathematical models are studied in the framework of the corresponding teaching courses. But there are also some general ideas, very useful and helpful in practice. The key word here is asymptotics.
“In the eyes of many people, ‘applied mathematical modelling’ is a computationally-intensive activity, concerned with addressing problems in systems and configurations which, in their complexity, come much closer to real world problems than the highly-idealized models susceptible to analytical treatment. …Exact solutions of even apparently simple problems are very difficult to find. So where is the theoretical framework for thought to be found? The answer is that “asymptotics” – the rational study of limiting cases – provides exactly what is needed. …Asymptotics should be seen as the framework for modelling, and the framework in which the whole problem, its formulation, solution, interpretation and refinement, should be embedded” (D. G. Crighton).
The reviewed book is written from this standpoint. Using examples from the various chapters of Applied Mechanics (Fluid Mechanics, Mechanics of Solids, etc.), the author shows fruitfulness of the asymptotic approach for modelling of nature phenomena and technical systems. The author applies analytical approaches omitting the discussion of ways of numerical implementation. It seems adequate because the most general features of mathematical modelling are considered. As a rule, partial differential equations, linear and nonlinear, are choosen as mathematical examples. The author aimed and succeeds “to demonstrate the power of mathematics and the diversity of its applications”. This book is a reader-friendly textbook in the fields of mathematical modelling. It can be recommended for students as a useful approach to modelling in Applied Mechanics.

MSC:

35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35Qxx Partial differential equations of mathematical physics and other areas of application
74-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids
35E20 General theory of PDEs and systems of PDEs with constant coefficients

Citations:

Zbl 0588.35001
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