Mathematical Models in Natural Science and Engineering

This book helps the reader become familiar with various mathematical models for mechanical, electrical, physical, atronomical, chemical, biological, ecological, cybernetical and other systems and processes. The models examined are evolutionary models, i.e. the models of time-varying processes known as dynamic systems, such as fluid flow, biological processes, oscillations in mechanical and electromagnetic systems, and social systems. The book shows readers how to identify appropriate situations for modelling, how to address difficulties in creating models, and how to learn what mathematics teaches us about the modelling of dynamical phenomena in our surrounding world.It is interesting for a wide spectrum of readers, students of natural science and engineering, and also for researchers in these fields.

Klappentext

This book has come into being as a result ofthe author's lectures on mathematical modelling rendered to the students, BS and MS degree holders specializing in applied mathematics and computer science and to post-graduate students in exact sciences of the Nizhny Novgorod State University after N.!. Lobatchevsky. These lectures are adapted and presented as a single whole ab out mathematical models and modelling. This new course of lectures appeared because the contemporary Russian educational system in applied mathematics rested upon a combination of fundamental and applied mathematics training; this way of training oriented students upon solving only the exactly stated mathematical problems, and thus there was created a certain estrangement to the most essential stages and sides of real solutions for applied problems, such as thinking over and deeply piercing the essence of a specific problem and its mathematical statement. This statement embraces simplifications, adopted idealizations and creating a mathematical model, its correction and matching the results obtained against a real system. There also existed another main objective, namely to orient university graduates in their future research not only upon purely mathematical issues but also upon comprehending and widely applying mathematics as a universal language of contemporary exact science, and mathematical modelling as a powerful me ans for studying nature, engineering and human society.

Inhalt

Dynamic system.- Fluid flowing-out from a vessel.- Equilibrium and auto-oscillations of fluid level in a vessel with simultaneous inflow and outflow.- Transitive process, equilibrium state and auto-oscillations.- Dynamics of water level in a reservoired hydropower station.- Energetic model of a heart.- Soiling a water reservoir with a bay and the Caspian Sea puzzles.- Exponential processes.- Populations coexistence dynamics.- Flow biological reactor.- Mathematical model of an immune response of a living organism to an infectious invastion.- Mathematic model for the community 'Manufacturers-Products-Managers'.- Linear oscillator.- Electromechanical analogies. Lagrange-Maxwell equations.- Galileo-Huygens watch.- Generator of electric oscillations.- Soft and hard regimes of exciting auto-oscillations.- Stochastic oscillator.- Instability and auto-oscillations caused by friction.- Forced oscillations of a linear oscillator.- Parametric excitation and stabilization.- Normal oscillations and beatings.- Stabilizing an inverted pendulum.- Controllable pendulum and a two-legged pacing.- Dynamic models for games, teaching and rational behaviour.- Perceptron and pattern recognition.- Kepler laws and the two-body problem solved by Newton.- Distributed dynamic models in mechanics and physics.- Fundamental solution of the thermal conductivity equation.- Travelling waves and the dispersion equation.- Faraday-Maxwell theory of electromagnetism and Maxwell-Hertz electromagnetic waves.- Wave reflection and refraction.- Standing waves and oscillations of a bounded string.- Microparticle.- Space and time.- Speeding up relativistic microparticles in a cyclotron.- Mathematics as a language and as an operating system and models.- Geometrical, physical, analogous, mathematical and imitative types of modelling.- General scheme of methematical modelling.- Models of vibratory pile driving.- The fundamental mathematical model of the modern science and the theory of oscillations.- Mathematicalmodel as a fruitful idea of research. D-partition.- Idealization, mathematical correctness and reality.- Dynamical interpretation of the least square method and global searching optimization with an adaptive model.- Theoretic game model of the human society.