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Methods of mathematical modelling. Continuous systems and differential equations. (English)
Springer Undergraduate Mathematics Series. Cham: Springer (ISBN 978-3-319-23041-2/pbk; 978-3-319-23042-9/ebook). xviii, 305~p. (2015).
The purpose of this text is to introduce the reader to the art of mathematical modelling which the authors view as having only two stages. β€œIn the {\it formulation phase}, the problem is described using basic principles or governing laws and assumed relations taken from some branches of knowledge, such as physics, biology, chemistry, economics, geometry, probability or others. Then, all side-conditions that are needed to completely define the problem must be identified: geometric constraints, initial conditions, material properties, boundary conditions and design parameter values. Finally, the properties of interest, how they are to be measured, relevant variables, coordinate systems and a system of units must be decided on. Then, in the {\it solution phase}, mathematical modelling provides approaches to reformulating the original problem into a more convenient structure from which it can be reduced into solvable parts that can ultimately be re-assembled to address the main questions of interest for the problem.” The structure of the book corresponds to the above vision of the modelling process. All material is organized in three parts. Part I, Formulation of models, consists of four chapters dealing with rate equations (models with time evolution), transport equations (models with structural changes), variational principles (models involving optimization and constraints) and dimensional scaling. Part II, Solution techniques, comprises seven chapters. The opening Chapter 5 presents self-similar solutions and similarity methods. In Chapters 6β€’10, attention of the authors is focused on perturbation methods. Both regular and singular cases are considered along with applications of asymptotic matching technique and boundary layer theory to partial differential equations, presentation of multiscale methods and brief discussion of the dynamics of systems with slow and fast variables. Chapter 11 describes the method of moments which is used for reducing complex models to simpler ones. Part III, titled Case studies, has only one chapter where several models from applied fluid dynamics are considered. However, a fair number of case studies including, for instance, shallow water equations, porous medium equation or van der Pol equation can be found in other chapters. The final part of the book contains an appendix with trigonometric formulas and basic facts about Fourier series, solutions to selected problems, a list of references containing 109 titles and a subject index. The book provides an account of a number of useful for mathematical modelling techniques which are illustrated with examples and complemented with problems for self study. The authors skip substantiation of methods concentrating more on procedural part which may appeal to readers interested in practical applications of mathematical modelling.
Reviewer: Yuriy V. Rogovchenko (Kristiansand)
Classification: M15
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