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Principles of mathematical modelling : ideas, methods, examples

Author: A A Samarskiĭ; Alexander P Mikhailov
Publisher: Boca Raton : CRC Press, 2018.
Series: Numerical insights.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:

This book demonstrates the versatility of the increasingly multi-disciplinary field of mathematical modelling and will be an insightful book of interest to specialists, teachers and students in areas  Read more...

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Genre/Form: Electronic books
Additional Physical Format: Print version
Samarskii, A. A., (Aleksandr Andreevich)
Principles of mathematical modeling : ideas, methods, examples.
Boca Raton ; London : New York : CRC Press, Taylor & Francis Group, 2018, c2002
ix, 349 pages
(DLC) 2005283115
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: A A Samarskiĭ; Alexander P Mikhailov
ISBN: 9781420024449 1420024442 9781482288131 1482288133
OCLC Number: 1042329410
Description: 1 online resource : illustrations
Contents: Cover --
Half Title --
Title Page --
Copyright Page --
Contents --
INTRODUCTION --
Chapter I: THE ELEMENTARY MATHEMATICAL MODELS AND BASIC CONCEPTS OF MATHEMATICAL MODELING --
1 Elementary Mathematical Models --
1. Fundamental laws of nature --
2. Variational principles --
3. Use of analogies in the construction of models --
4. Hierarchical approach to the construction of models --
5. On the nonlinearity of mathematical models --
6. Preliminary conclusions --
Exercises --
2 Examples of Models Following from the Fundamental Laws of Nature --
1. The trajectory of a floating submarine --
2. Deviation of a charged particle in an electron-beam tube --
3. Oscillations of the rings of Saturn --
4. Motion of a ball attached to a spring --
5. Conclusion --
Exercises --
3 Variational Principles and Mathematical Models --
1. The general scheme of the Hamiltonian principle --
2. The third way of deriving the model of the system "ball-spring" --
3. Oscillations of a pendulum in a gravity field --
4. Conclusion --
Exercises --
4 Example of the Hierarchy of Models --
1. Various modes of action of the given external force --
2. Motion of an attaching point, the spring on a rotating axis --
3. Accounting for the forces of friction --
4. Two types of nonlinear models of the system "ball-spring" --
5. Conclusion --
Exercises --
5 The Universality of Mathematical Models --
1. Fluid in a U-shaped flask --
2. An oscillatory electrical circuit --
3. Small oscillations at the interaction of two biological populations --
4. Elementary model of variation of salary and employment --
5. Conclusion --
Exercises --
6 Several Models of Elementary Nonlinear Objects --
1. On the origin of nonlinearity --
2. Three regimes in a nonlinear model of population --
3. Influence of strong nonlinearity on the process of oscillations --
4. On numerical methods --
Exercises. Chapter II: DERIVATION OF MODELS FROM THE FUNDAMENTAL LAWS OF NATURE --
1 Conservation of the Mass of Substance --
1. A flow of particles in a pipe --
2. Basic assumptions on the gravitational nature of flows of underground waters --
3. Balance of mass in the element of soil --
4. Closure of the law of conservation of mass --
5. On some properties of the Bussinesque equation --
Exercises --
2 Conservation of Energy --
1. Preliminary information on the processes of heat transfer --
2. Derivation of Fourier law from molecular-kinetic concepts --
3. The equation of heat balance --
4. The statement of typical boundary conditions for the equation of heat transfer --
5. On the peculiarities of heat transfer models --
Exercises --
3 Conservation of the Number of Particles --
1. Basic concepts of the theory of thermal radiation --
2. Equation of balance of the number of photons in a medium --
3. Some properties of the equation of radiative transfer --
Exercises --
4 Joint Application of Several Fundamental Laws --
1. Preliminary concepts of gas dynamics --
2. Equation of continuity for compressible gas --
3. Equations of gas motion --
4. The equation of energy --
5. The equations of gas dynamics in Lagrangian coordinates --
6. Boundary conditions for the equations of gas dynamics --
7. Some peculiarities of models of gas dynamics --
Exercises --
Chapter III: MODELS DEDUCED FROM VARIATIONAL PRINCIPLES, HIERARCHIES OF MODELS --
1 Equations of Motion, Variational Principles and Conservation Laws in Mechanics --
1. Equation of motion of a mechanical system in Newtonian form --
2. Equations of motion in Lagrangian form --
3. Variational Hamiltonian principle --
4. Conservation laws and space-time properties --
Exercises --
2 Models of Some Mechanical Systems --
1. Pendulum on the free suspension --
2. Non-potential oscillations. 3. Small oscillations of a string --
4. Electromechanical analogy --
Exercises --
3 The Boltzmann Equation and its Derivative Equations --
1. The description of a set of particles with the help of the distribution function --
2. Boltzmann equation for distribution function --
3. Maxwell distribution and the H-theorem --
4. Equations for the moments of distribution function --
5. Chain of hydrodynamical gas models --
Exercises --
Chapter IV: MODELS OF SOME HARDLY FORMALIZABLE OBJECTS --
1 Universality of Mathematical Models --
1. Dynamics of a cluster of amoebas --
2. Random Markov process --
3. Examples of analogies between mechanical, thermodynamic and economic objects --
Exercises --
2 Some Models of Financial and Economic Processes --
1. Organization of an advertising campaign --
2. Mutual offset of debts of enterprises --
3. Macromodel of equilibrium of a market economy --
4. Macromodel of economic growth --
Exercises --
3 Some Rivalry Models --
1. Mutual relations in the system "predator --
victim" --
2. Arms race between two countries --
3. Military operations of two armies --
Exercises --
4 Dynamics of Distribution of Power in Hierarchy --
1. General statement of problem and terminology --
2. Mechanisms of redistributing power inside the hierarchical structure --
3. Balance of power in a level, conditions on boundaries of hierarchy and transition to a continuous model --
4. The legal system "power-society". Stationary distributions and exit of power from its legal scope --
5. Role of basic characteristics of system in a phenomenon of power excess (diminution) --
6. Interpretation of results and conclusions --
Exercises --
Chapter V: STUDY OF MATHEMATICAL MODELS --
1 Application of Similarity Methods --
1. Dimensional analysis and group analysis of models --
2. Automodel (self-similar) processes. 3. Various cases of propagation of perturbations in nonlinear media --
Exercises --
2 The Maximum Principle and Comparison Theorems --
1. The formulation and some consequences --
2. Classification of blow-up regimes --
3. The extension of "a self-similar method" --
Exercises --
3 An Averaging Method --
1. Localized structures in nonlinear media --
2. Various ways of averaging --
3. A classification of combustion regimes of a thermal conducting medium --
Exercises --
4 On Transition to Discrete Models --
1. Necessity of numerical modeling, elementary concepts of the theory of difference schemes --
2. Direct formal approximation --
3. The integro-interpolational method --
4. Principle of complete conservatism --
5. Construction of difference schemes by means of variational principles --
6. Use of the hierarchical approach in derivation of discrete models --
Exercises --
Chapter VI: MATHEMATICAL MODELING OF COMPLEX OBJECTS --
1 Problems of Technology and Ecology --
1. Physically "safe" nuclear reactor --
2. A hydrological "barrier" against the contamination of underground waters --
3. Complex regimes of gas flow around body --
4. Ecologically acceptable technologies for burning hydrocarbon fuels --
2 Fundamental Problems of Natural Science --
1. Nonlinear effects in laser thermonuclear plasma --
2. Mathematical restoration of the Tunguska phenomenon --
3. Climatic consequences of a nuclear conflict --
4. Magnetohydrodynamic "dynamo" of the Sun --
3 Computing Experiment with Models of Hardly Formalizable Objects --
1. Dissipative biological structures --
2. Processes in transition economy --
3. Totalitarian and anarchic evolution of power distribution in hierarchies --
REFERENCES --
INDEX.
Series Title: Numerical insights.
Other Titles: Matematicheskoe modelirovanie.
Responsibility: A.A. Samarskii and A.P. Mikhailov.

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