# Applied Partial Differential Equations: A Visual Approach

## Peter A. Markowich

Language: English

Pages: 210

ISBN: 3662518120

Format: PDF / Kindle (mobi) / ePub

This book presents selected topics in science and engineering from an applied-mathematics point of view. The described natural, socioeconomic, and engineering phenomena are modeled by partial differential equations that relate state variables such as mass, velocity, and energy to their spatial and temporal variations. Typically, these equations are highly nonlinear; in many cases they are systems, and they represent challenges even for the most modern and sophisticated mathematical and numerical-analytic techniques. The selected topics reflect the longtime scientific interests of the author. They include flows of fluids and gases, granular-material flows, biological processes such as pattern formation on animal skins, kinetics of rarified gases, free boundaries, semiconductor devices, and socioeconomic processes. Each topic is briefly introduced in its scientific or engineering context, followed by a presentation of the mathematical models in the form of partial differential equations with a discussion of their basic mathematical properties. The author illustrates each chapter by a series of his own high-quality photographs, which demonstrate that partial differential equations are powerful tools for modeling a large variety of phenomena influencing our daily lives.

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formulated a free boundary problem, which is now known as the classical Stefan problem and which has given rise to the modern research area of phase transition modeling by free boundary problems. As a basic reference we refer to [10]. Some of the photographs associated to this chapter show icebergs in lakes of Patagonia. The evolution of their water-ice phase transition free boundary is modeled by the 3-dimensional Stefan problem formulated below. Therefore, consider a domain G ∈ Rd (of course d

among prey and predators resp. This friction, caused for example by competition for nutrients, stabilizes the prey population even without predators. If both species are present initially, then their respective rates of change are supposed to be influenced by the number of their encounters, typically ending badly for the prey, i.e. b > 0, and good for the predator, i.e. c > 0. Note that due to the quadratic nature of the Lotka–Volterra system, only two-body interactions prey-prey,

1967 [11] L.V. Kantorovich, On the transfer of masses, Dokl. Akad. Nauk. SSSR 37, pp. 227–229, 1942 (in Russian) [12] L.V. Kantorovich, On a problem of Monge, Uspekhi Mat. Nauk. 3, pp. 225– 226, 1948 [13] G. Monge, M´emoire sur la Th´eorie des Deblais et des Remblais, Histoire de l’Acad. des Sciences de Paris, 1781 [14] C. Villani, Topics in Optimal Transportation, American Mathematical Society, in: Graduate Studies in Mathematics Series, vol. 58, 2003 [15] Q. Xia, Optimal paths related to

share the property that the steady (or, more generally, the self-similar asymptotic) states are different from the classical Maxwell distribution of the Boltzmann equation of gas dynamics presented in Chap. 1. Another analogy becomes evident when looking at the non-conservative properties of the economic and granular Boltzmann equations, resulting from inelastic binary collision models. Conservative exchange dynamics between individuals redistribute the wealth among people. Without conservation,

gas dynamics systems is the microscopic Boltzmann equation, but the involved numerical effort, due to the high dimension of the phase space (three velocity directions plus three spatial directions, and time!), does usually not justify its application in industrial aircraft design. Recently however, lattice Boltzmann equation simulations (for more details see below) have been employed for airfoil simulations and turbulence modeling in different applications, with striking success [6]. Comments on