Full Description
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.
Contents
1 Dynamical system.- 2 Fluid outflow from a vessel.- 3 Equilibrium and auto-oscillations of fluid level in the vessel with simultaneous inflow and outflow.- 4 Transitive processes, equilibrium states and auto-oscillations.- 5 Dynamics of the water surface level in a reservoired hydropower station.- 6 Energetic model of the heart.- 7 Soiling a water reservoir with a bay and the Caspian Sea puzzles.- 8 Exponential processes.- 9 Dynamics in coexistence of populations.- 10 Flow biological reactor.- 11 Mathematical model for the immune response of a living organism to an infectious invasion.- 12 Mathematical model for the community "Producers —Products — Managers".- 13 Linear oscillators.- 14 Electromechanical analogies. Lagrange-Maxwell equations.- 15 Galileo-Huygens clock.- 16 Generator of electric oscillations.- 17 Soft and hard regimes of exciting auto-oscillations.- 18 Stochastic oscillator (the "contrary clock").- 19 Instability and auto-oscillations caused by friction.- 20 Forced oscillations of a linear oscillator.- 21 Parametric excitation and stabilization.- 22 Normal oscillations and beatings.- 23 Stabilizing an inverted pendulum.- 24 Controllable pendulum and a two-legged pacing.- 25 Dynamical models for games, teaching and rational behaviour.- 26 Perception and pattern recognition.- 27 Kepler laws and the two-body problem solved by Newton.- 28 Distributed dynamical models in mechanics and physics.- 29 Fundamental solution of the thermal conductivity equation.- 30 Running waves and the dispersion equation.- 31 Faraday-Maxwell theory of electromagnetism and the Maxwell-Hertz electromagnetic waves.- 32 Wave reflection and refraction.- 33 Standing waves and oscillations of a bounded string.- 34 Microparticles.- 34.1 Mathematical formalism in quantummechanics.- 34.2 Free microparticle.- 34.3 Microparticle in a potential well.- 34.4 Diffusion of a microparticle through a potential barrier.- 34.5 Atom of hydrogen.- 34.6 Quantum linear oscillator.- 34.7 Newton quantum equation.- 35 Space and time.- 36 Speeding up relativistic microparticles in a cyclotron.- 37 Mathematics as a language and as an operating system and models.- 38 Geometrical, physical, analogous, mathematical and imitative types of modelling.- 38.1 Physical modelling.- 38.2 Imitative modelling.- 39 General scheme of mathematical modelling.- 40 Models of vibratory pile driving.- 41 The fundamental mathematical model of the modern science and the theory of oscillations.- 41.1 A dynamical system as a basic mathematical model of the contemporary science.- 41.2 A.A. Andronov and the theory of oscillations.- 42 Mathematical model as a fruitful idea of research. The D-partition.- 43 Idealization, mathematical correctness and reality.- 43.1 Frictional regulator of rotating velocity.- 43.2 Panleve paradox and auto-oscillations under Coulomb friction.- 44 Dynamical interpretation of the least-square method and global search optimization with use of an adaptive model.- 44.1 A universal recurrent form of the LSQ method.- 44.2 Searching global optimization with an adaptive model.- 45 Theoretical game model of the human society.- 45.1 Game-like perception of life and a theoretical game model of the society.- 45.2 Organizational and management principles of the society.- 45.3 An ideal public game.- 45.4 A problem of involving managers and authorities into a general playing interaction.- 45.5 Conclusion.
