Principles of mathematical modeling : ideas, methods, examples /
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| Author / Creator: | Samarskiĭ, A. A. (Aleksandr Andreevich) |
|---|---|
| Imprint: | London ; New York : Taylor & Francis, 2002. |
| Description: | ix, 349 p. : ill. ; 26 cm. |
| Language: | English |
| Series: | Numerical insights ; v. 3 |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/4818615 |
Table of Contents:
- Introduction
- 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
- 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
- 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
- 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
- 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
- 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
