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Non-local partial differential equations for engineering and biology. Mathematical modeling and analysis. (English) Zbl 1387.00004

Mathematics for Industry 31. Cham: Springer (ISBN 978-3-319-67942-6/hbk; 978-3-319-67944-0/ebook). xix, 300 p. (2018).
This book presents some new results concerning non-local models arising in mathematical analysis. Basic concepts in mechanics, thermodynamics, game theory, and theoretical biology are examined in detail. It starts with a review and summary of the basic ideas of mathematical modeling frequently employed in sciences and engineering. The first part of this monograph is devoted to the investigation of some non-local models linked with applications from engineering.
Chapter 1 focuses on the study of non-local models associated with electrostatic micro-electro-mechanical-systems (MEMS) control. The authors describe the two main physical problems which build up the operation of an idealized MEMS device: the elastic and the electric problem. Next, the authors are concerned with the structure of the set of radially symmetric steady-state solutions, which are investigated along with their stability. Then, they study the circumstances under which finite-time quenching occurs.
Chapter 2 discusses some non-local models describing Ohmic heat production in various industrial processes. In the first part of the chapter, the process of food sterilization through Ohmic heating is considered on the basis of a one-dimensional non-local model. Next, and under different circumstances, a hyperbolic approach with a non-local convection velocity is built up. The second part of this chapter deals with another application of Ohmic heating process in a thermistor device.
Chapter 3 deals with an application arising in the process of linear friction welding applied in metallurgy. Next, a similar non-local model is derived for the hard-material case where the exponential nonlinearity is replaced by \(f(u)=(-u)^p\), for \(p=1/\alpha\).
Chapter 4 discusses a degenerate non-local model which is associated with the industrial process of resistance spot welding and the unknown \(u\) represents the temperature in the welding area.
Part II of this monograph deals with some applications of non-local models coming from biology. The authors are concerned with the Gierer-Meinhardt system, an application arising in evolutionary game dynamics, biological phenomena arising in chemotaxis, and a mathematical model in cell biology that describes the evolution of protein dimers within human cells.
The models developed in this book are based on various laws of physics such as mechanics of continuum, electromagnetic theory, and thermodynamics. For these reasons, the arguments come from many areas of mathematics such as calculus of variations, dynamical systems, integrable systems, blow-up analysis, and energy methods. The book under review is mainly addressed to researchers and upper grade students in mathematics, engineering, physics, economics, and biology.

MSC:

00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.)
00A69 General applied mathematics
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
35Q79 PDEs in connection with classical thermodynamics and heat transfer
35Q93 PDEs in connection with control and optimization
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