Principles of Mathematical Modeling,9780415272803

Principles of Mathematical Modeling

by ; ; ;
ISBN13: 9780415272803
ISBN10: 0415272807
Format: Hardcover
Pub. Date: 2002-06-01
Publisher(s): Taylor & Francis
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Summary

Mathematical modeling is becoming increasingly versatile and multi-disciplinary. This text demonstrates the broadness of this field, as the authors consider the principles of model construction and use common approaches to build models from a range of subject areas. The book reflects the interests and experiences of the authors, but it explores mathematical modeling across a wide range of applications, from mechanics to social science. A general approach is adopted where ideas and examples are favored over rigorous mathematical procedures. This insightful book will be of interest to specialists, teachers and students of this fascinating discipline.

Author Biography

Professor Alexander A. Samarskii is the founder of the Institute of Mathematical Modeling at the Russian Academy of Science and has been the leader of the Russian National Program for Mathematical Modeling in Science and Technology since 1985 Professor Alexander P. Mikhailov is a head of department at the Institute of Mathematical Modeling at the Russian Academy of Science, and is a Professor of M.V. Lomonosov Moscow State University

Table of Contents

Introduction 1(5)
The Elementary Mathematical Models and Basic Concepts of Mathematical Modeling
6(53)
Elementary Mathematical Models
6(17)
Fundamental laws of nature
6(7)
Variational principles
13(2)
Use of analogies in the construction of models
15(2)
Hierarchical approach to the construction of models
17(2)
On the nonlinearity of mathematical models
19(2)
Preliminary conclusions
21(1)
Exercises
22(1)
Examples of Models Following from the Fundamental Laws of Nature
23(9)
The trajectory of a floating submarine
23(2)
Deviation of a charged particle in an electron-beam tube
25(2)
Oscillations of the rings of Saturn
27(2)
Motion of a ball attached to a spring
29(2)
Conclusion
31(1)
Exercises
32(1)
Variational Principles and Mathematical Models
32(6)
The general scheme of the Hamiltonian principle
32(1)
The third way of deriving the model of the system ``ball-spring''
33(2)
Oscillations of a pendulum in a gravity field
35(2)
Conclusion
37(1)
Exercises
38(1)
Example of the Hierarchy of Models
38(9)
Various modes of action of the given external force
38(1)
Motion of an attaching point, the spring on a rotating axis
39(2)
Accounting for the forces of friction
41(2)
Two types of nonlinear models of the system ``ball-spring''
43(3)
Conclusion
46(1)
Exercises
47(1)
The Universality of Mathematical Models
47(6)
Fluid in a U-shaped flask
47(2)
An oscillatory electrical circuit
49(1)
Small oscillations at the interaction of two biological populations
50(1)
Elementary model of variation of salary and employment
51(1)
Conclusion
52(1)
Exercises
52(1)
Several Models of Elementary Nonlinear Objects
53(6)
On the origin of nonlinearity
53(1)
Three regimes in a nonlinear model of population
53(2)
Influence of strong nonlinearity on the process of oscillations
55(1)
On numerical methods
56(1)
Exercises
57(2)
Derivation of Models from the Fundamental Laws of Nature
59(39)
Conservation of the Mass of Substance
59(10)
A flow of particles in a pipe
59(3)
Basic assumptions on the gravitational nature of flows of underground waters
62(1)
Balance of mass in the element of soil
62(3)
Closure of the law of conservation of mass
65(1)
On some properties of the Bussinesque equation
66(2)
Exercises
68(1)
Conservation of Energy
69(10)
Preliminary information on the processes of heat transfer
69(1)
Derivation of Fourier law from molecular-kinetic concepts
70(2)
The equation of heat balance
72(3)
The statement of typical boundary conditions for the equation of heat transfer
75(2)
On the peculiarities of heat transfer models
77(2)
Exercises
79(1)
Conservation of the Number of Particles
79(7)
Basic concepts of the theory of thermal radiation
79(3)
Equation of balance of the number of photons in a medium
82(2)
Some properties of the equation of radiative transfer
84(1)
Exercises
85(1)
Joint Application of Several Fundamental Laws
86(12)
Preliminary concepts of gas dynamics
86(1)
Equation of continuity for compressible gas
86(2)
Equations of gas motion
88(2)
The equation of energy
90(1)
The equations of gas dynamics in Lagrangian coordinates
91(2)
Boundary conditions for the equations of gas dynamics
93(1)
Some peculiarities of models of gas dynamics
94(3)
Exercises
97(1)
Models Deduced from Variational Principles, Hierarchies of Models
98(48)
Equations of Motion, Variational Principles and Conservation Laws in Mechanics
98(13)
Equation of motion of a mechanical system in Newtonian form
98(3)
Equations of motion in Lagrangian form
101(4)
Variational Hamiltonian principle
105(2)
Conservation laws and space-time properties
107(4)
Exercises
111(1)
Models of Some Mechanical Systems
111(14)
Pendulum on the free suspension
112(4)
Non-potential oscillations
116(3)
Small oscillations of a string
119(4)
Electromechanical analogy
123(2)
Exercises
125(1)
The Boltzmann Equation and its Derivative Equations
125(21)
The description of a set of particles with the help of the distribution function
126(1)
Boltzmann equation for distribution function
127(2)
Maxwell distribution and the H theorem
129(4)
Equations for the moments of distribution function
133(6)
Chain of hydrodynamical gas models
139(5)
Exercises
144(2)
Models of Some Hardly Formalizable Objects
146(72)
Universality of Mathematical Models
146(16)
Dynamics of a cluster of amoebas
146(5)
Random Markov process
151(7)
Examples of analogies between mechanical, thermodynamic and economic objects
158(4)
Exercises
162(1)
Some Models of Financial and Economic Processes
162(22)
Organization of an advertising campaign
162(4)
Mutual offset of debts of enterprises
166(7)
Macromodel of equilibrium of a market economy
173(7)
Macromodel of economic growth
180(3)
Exercises
183(1)
Some Rivalry Models
184(11)
Mutual relations in the system ``predator victim''
184(3)
Arms race between two countries
187(3)
Military operations of two armies
190(4)
Exercises
194(1)
Dynamics of Distribution of Power in Hierarchy
195(23)
General statement of problem and terminology
195(6)
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(5)
The legal system ``power-society''. Stationary distributions and exit of power from its legal scope
209(4)
Role of basic characteristics of system in a phenomenon of power excess (diminution)
213(1)
Interpretation of results and conclusions
214(2)
Exercises
216(2)
Study of Mathematical Models
218(76)
Application of Similarity Methods
218(22)
Dimensional analysis and group analysis of models
218(6)
Automodel (self similar) processes
224(7)
Various cases of propagation of perturbations in nonlinear media
231(8)
Exercises
239(1)
The Maximum Principle and Comparison Theorems
240(14)
The formulation and some consequences
240(5)
Classification of blow-up regimes
245(3)
The extension of ``a self similar method''
248(6)
Exercises
254(1)
An Averaging Method
254(13)
Localized structures in nonlinear media
254(4)
Various ways of averaging
258(3)
A classification of combustion regimes of a thermal conducting medium
261(6)
Exercises
267(1)
On Transition to Discrete Models
267(27)
Necessity of numerical modeling, elementary concepts of the theory of difference schemes
268(4)
Direct formal approximation
272(7)
The integro-interpolational method
279(3)
Principle of complete conservatism
282(3)
Construction of difference schemes by means of variational principles
285(4)
Use of the hierarchical approach in derivation of discrete models
289(3)
Exercises
292(2)
Mathematical Modeling of Complex Objects
294(48)
Problems of Technology and Ecology
294(15)
Physically ``safe'' nuclear reactor
294(5)
A hydrological ``barrier'' against the contamination of underground waters
299(3)
Complex regimes of gas flow around body
302(4)
Ecologically acceptable technologies for burning hydrocarbon fuels
306(3)
Fundamental Problems of Natural Science
309(17)
Nonlinear effects in laser thermonuclear plasma
309(6)
Mathematical restoration of the Tunguska phenomenon
315(3)
Climatic consequences of a nuclear conflict
318(5)
Magnetohydrodynamic ``dynamo'' of the Sun
323(3)
Computing Experiment with Models of Hardly Formalizable Objects
326(16)
Dissipative biological structures
327(3)
Processes in transition economy
330(4)
Totalitarian and anarchic evolution of power distribution in hierarchies
334(8)
References 342(5)
Index 347

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