Principles of Mathematical Modeling

, by Samarskii, Aleksandr Andreevich; Mikhailov, Alexander P.; Samarskii, A. A.; Mikhailov, A. P.

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ISBN: 9780415272803 | 0415272807
Cover: Hardcover
Copyright: 6/1/2002

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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.

Introduction

(5)

The Elementary Mathematical Models and Basic Concepts of Mathematical Modeling

(53)

Elementary Mathematical Models

(17)

Fundamental laws of nature

(7)

Variational principles

(2)

Use of analogies in the construction of models

(2)

Hierarchical approach to the construction of models

(2)

On the nonlinearity of mathematical models

(2)

Preliminary conclusions

(1)

Exercises

(1)

Examples of Models Following from the Fundamental Laws of Nature

(9)

The trajectory of a floating submarine

(2)

Deviation of a charged particle in an electron-beam tube

(2)

Oscillations of the rings of Saturn

(2)

Motion of a ball attached to a spring

(2)

Conclusion

(1)

Exercises

(1)

Variational Principles and Mathematical Models

(6)

The general scheme of the Hamiltonian principle

(1)

The third way of deriving the model of the system ``ball-spring''

(2)

Oscillations of a pendulum in a gravity field

(2)

Conclusion

(1)

Exercises

(1)

Example of the Hierarchy of Models

(9)

Various modes of action of the given external force

(1)

Motion of an attaching point, the spring on a rotating axis

(2)

Accounting for the forces of friction

(2)

Two types of nonlinear models of the system ``ball-spring''

(3)

Conclusion

(1)

Exercises

(1)

The Universality of Mathematical Models

(6)

Fluid in a U-shaped flask

(2)

An oscillatory electrical circuit

(1)

Small oscillations at the interaction of two biological populations

(1)

Elementary model of variation of salary and employment

(1)

Conclusion

(1)

Exercises

(1)

Several Models of Elementary Nonlinear Objects

(6)

On the origin of nonlinearity

(1)

Three regimes in a nonlinear model of population

(2)

Influence of strong nonlinearity on the process of oscillations

(1)

On numerical methods

(1)

Exercises

(2)

Derivation of Models from the Fundamental Laws of Nature

(39)

Conservation of the Mass of Substance

(10)

A flow of particles in a pipe

(3)

Basic assumptions on the gravitational nature of flows of underground waters

(1)

Balance of mass in the element of soil

(3)

Closure of the law of conservation of mass

(1)

On some properties of the Bussinesque equation

(2)

Exercises

(1)

Conservation of Energy

(10)

Preliminary information on the processes of heat transfer

(1)

Derivation of Fourier law from molecular-kinetic concepts

(2)

The equation of heat balance

(3)

The statement of typical boundary conditions for the equation of heat transfer

(2)

On the peculiarities of heat transfer models

(2)

Exercises

(1)

Conservation of the Number of Particles

(7)

Basic concepts of the theory of thermal radiation

(3)

Equation of balance of the number of photons in a medium

(2)

Some properties of the equation of radiative transfer

(1)

Exercises

(1)

Joint Application of Several Fundamental Laws

(12)

Preliminary concepts of gas dynamics

(1)

Equation of continuity for compressible gas

(2)

Equations of gas motion

(2)

The equation of energy

(1)

The equations of gas dynamics in Lagrangian coordinates

(2)

Boundary conditions for the equations of gas dynamics

(1)

Some peculiarities of models of gas dynamics

(3)

Exercises

(1)

Models Deduced from Variational Principles, Hierarchies of Models

(48)

Equations of Motion, Variational Principles and Conservation Laws in Mechanics

(13)

Equation of motion of a mechanical system in Newtonian form

(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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Principles of Mathematical Modeling

Summary

Author Biography

Table of Contents

Supplemental Materials

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Principles of Mathematical Modeling

Summary

Author Biography

Table of Contents

Supplemental Materials

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