Principles of Mathematical Modelling: Ideas, Methods, Examples

360 Pages·2001·21.374 MB·\360

by  Alexander A. Samarskii (Author), Alexander P. Mikhailov (Author)

Principles of Mathematical Modeling
Numerical Insights
Series Editor
A. Sydow, GMD-FIRST, Berlin, Germany
Editorial Board
P. Borne, Ecole de Lille, France; G. Carmichael, University of Iowa, USA;
L. Dekker, Delft University of Technology, The Netherlands; A. Iserles,
University of Cambridge, UK; A. Jakeman, Australian National University,
Australia; G. Korn, Industrial Consultants (Tucson), USA; G.P. Rao,
Indian Institute of Technology, India; J.R. Rice, Purdue University, USA;
A. A. Samarskii, Russian Academy of Science, Russia;
Y. Takahara, Tokyo Institute of Technology, Japan
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Modelling, Simulation and Control of Non-Linear Dynamical Systems: An Intelligent
Approach using Soft Computing and Fractal Theory
Patricia Melin and Oscar Castillo
Volume 3
Principles of Mathematical Modeling: Ideas, Methods, Examples
A,A. Samarskii andA.P. Mikhailov
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Principles of Mathematical
Modeling
Ideas, Methods, Examples
A.A. Samarskii
Founder of the Institute of Mathematical Modeling, Moscow,
Russia
and
A.P. Mikhailov
Head of Department, Institute of Mathematical Modeling,
Moscow, Russia
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
A TAYLOR & FRANCIS BOOK
Originally published in Russian in 1997 as MAT EMAT ICHESKOE MODELIROV ANIE:
IDEI. METOID. PRIMER! by Physical and Mathematical Literature Publishing
Company, Russian Academy of Sciences, Moscow.
First published 2002 by Taylor & Francis
Published 2018 by CRC Press
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© 2002 by Taylor & Francis Group, LLC
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ISBN 13: 978-0-415-27281-0 (pbk)
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Contents
INTRODUCTION 1
I THE ELEMENTARY MATHEMATICAL MODELS AND
BASIC CONCEPTS OF MATHEMATICAL MODELING 6
1 Elementary Mathematical Models 6
1.Fundamental laws of nature 6
2.Variational principles 13
3.Use of analogies in the construction of models 15
4.Hierarchical approach to the construction of models 17
5.On the nonlinearity of mathematical models 19
6.Preliminary conclusions 21
Exercises 22
2 Examples of Models Following from the Fundamental Laws of
Nature 23
1.The trajectory of a floating submarine 23
2.Deviation of a charged particle in an electron-beam tube 25
3.Oscillations of the rings of Saturn 27
4.Motion of a ball attached to a spring 29
5.Conclusion 31
Exercises 32
3 Variational Principles and Mathematical Models 32
1.The general scheme of the Hamiltonian principle 32
2.The third way of deriving the model of the system "ball-spring" 33
3.Oscillations of a pendulum in a gravity field 35
4.Conclusion 37
Exercises 38
4 Example of the Hierarchy of Models 38
1.Various modes of action of the given external force 38
2.Motion of an attaching point, the spring on a rotating axis 39
3.Accounting for the forces of friction 41
4.Two types of nonlinear models of the system "ball-spring" 43
5.Conclusion 46
Exercises 47
5 The Universality of Mathematical Models 47
1.Fluid in a U-shaped flask 47
2.An oscillatory electrical circuit 49
vi Contents
3. Small oscillations at the interaction of two biological
populations 50
4.Elementary model of variation of salary and employment 51
5.Conclusion 52
Exercises 52
6 Several Models of Elementary Nonlinear Objects 53
1. On the origin of nonlinearity 53
2.Three regimes in a nonlinear model of population 53
3.Influence of strong nonlinearity on the process of oscillations 55
4.On numerical methods 56
Exercises 57
II DERIVATION OF MODELS FROM THE FUNDAMENTAL
LAWS OF NATURE 59
1 Conservation of the Mass of Substance 59
1. A flow of particles in a pipe 59
2.Basic assumptions on the gravitational nature of flows of
underground waters 62
3.Balance of mass in the element of soil 62
4.Closure of the law of conservation of mass 65
5.On some properties of the Bussinesque equation 66
Exercises 68
2 Conservation of Energy 69
1.Preliminary information on the processes of heat transfer 69
2.Derivation of Fourier law from molecular-kinetic concepts 70
3.The equation of heat balance 72
4.The statement of typical boundary conditions for the equation of
heat transfer 75
5.On the peculiarities of heat transfer models 77
Exercises 79
3 Conservation of the Number of Particles 79
1. Basic concepts of the theory of thermal radiation 79
2.Equation of balance of the number of photons in a medium 82
3. Some properties of the equation of radiative transfer 84
Exercises 85
4 Joint Application of Several Fundamental Laws 86
1.Preliminaiy concepts of gas dynamics 86
2.Equation of continuity for compressible gas 86
3.Equations of gas motion 88
4.The equation of energy 90
5.The equations of gas dynamics in Lagrangian coordinates 91
6.Boundary conditions for the equations of gas dynamics 93
7.Some peculiarities of models of gas dynamics 94
Exercises 97
Contents vii
in MODELS DEDUCED FROM VARIATIONAL PRINCIPLES,
HIERARCHIES OF MODELS 98
l Equations of Motion, Variational Principles and Conservation
Laws in Mechanics 98
1.Equation of motion of a mechanical system in Newtonian form 98
2.Equations of motion in Lagrangian form 101
3.Variational Hamiltonian principle 105
4.Conservation laws and space-time properties 107
Exercises 111
2 Models of Some Mechanical Systems 111
1.Pendulum on the free suspension 112
2.Non-potential oscillations 116
3.Small oscillations of a string 119
4.Electromechanical analogy 123
Exercises 125
3 The Boltzmann Equation and its Derivative Equations 125
1.The description of a set of particles with the help of the
distribution function 126
2.Boltzmann equation for distribution function 127
3.Maxwell distribution and the //-theorem 129
4.Equations for the moments of distribution function 133
5.Chain of hydrodynamical gas models 139
Exercises 144
IVMODELS OF SOME HARDLY FORMALIZABLE
OBJECTS 146
1 Universality of Mathematical Models 146
1.Dynamics of a cluster of amoebas 146
2.Random Markov process 151
3.Examples of analogies between mechanical, thermodynamic and
economic objects 158
Exercises 162
2 Some Models of Financial and Economic Processes 162
1.Organization of an advertising campaign 162
2.Mutual offset of debts of enterprises 166
3.Macromodel of equilibrium of a market economy 173
4.Macromodel of economic growth 180
Exercises 183
3 Some Rivalry Models 184
1.Mutual relations in the system "predator - victim" 184
2.Arms race between two countries 187
3.Military operations of two armies 190
Exercises 194
4 Dynamics of Distribution of Power in Hierarchy 195
1.General statement of problem and terminology 195
viii Contents
2.Mechanisms of redistributing power inside the hierarchical structure 201
3.Balance of power in a level, conditions on boundaries of
hierarchy and transition to a continuous model 204
4.The legal system "power-society". Stationary distributions and
exit of power from its legal scope 209
5.Role of basic characteristics of system in a phenomenon of
power excess (diminution) 213
6.Interpretation of results and conclusions 214
Exercises 216
V STUDY OF MATHEMATICAL MODELS 218
1 Application of Similarity Methods 218
1. Dimensional analysis and group analysis of models 218
2.Automodel (self-similar) processes 224
3.Various cases of propagation of perturbations in nonlinear media 231
Exercises 239
2 The Maximum Principle and Comparison Theorems 240
1. The formulation and some consequences 240
2.Classification of blow-up regimes 245
3.The extension of "a self-similar method” 248
Exercises 254
3 An Averaging Method 254
1.Localized structures in nonlinear media 254
2.Various ways of averaging 258
3.A classification of combustion regimes of a thermal conducting
medium 261
Exercises 267
4 On Transition to Discrete Models 267
1.Necessity of numerical modeling, elementary concepts of the
theory of difference schemes 268
2.Direct formal approximation 272
3.The integro-interpolational method 279
4.Principle of complete conservatism 282
5.Construction of difference schemes by means of variational
principles 285
6.Use of the hierarchical approach in derivation of discrete models 289
Exercises 292
VI MATHEMATICAL MODELING OF COMPLEX OBJECTS 294
1 Problems of Technology and Ecology 294
1. Physically "safe” nuclear reactor 294
2.A hydrological "barrier” against the contamination of
underground waters 299
3.Complex regimes of gas flow around body 302
Contents ix
4.Ecologically acceptable technologies for burning hydrocarbon
fuels 306
2 Fundamental Problems of Natural Science 309
1.Nonlinear effects in laser thermonuclear plasma 309
2.Mathematical restoration of the Tunguska phenomenon 315
3.Climatic consequences of a nuclear conflict 318
4.Magnetohydrodynamic "dynamo” of the Sun 323
3 Computing Experiment with Models of Hardly Formalizable
Objects 326
1.Dissipative biological structures 327
2.Processes in transition economy 330
3.Totalitarian and anarchic evolution of power distribution in
hierarchies 334
REFERENCES 342
INDEX 347