Principles of Mathematical Modelling: Ideas, Methods, Examples

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

ISBN: 9780415272810 | 0415272815
Cover: Nonspecific Binding
Copyright: 12/20/2001

Rent

(Recommended)

$99.08

Term

Due

Price

Semester

Dec 12

$104.29

Quarter

Aug 13

$99.08

Short Term

Jul 18

$93.86

*This item is part of an exclusive publisher rental program and requires an additional convenience fee. This fee will be reflected in the shopping cart.
Buy New

Usually Ships in 3-5 Business Days

$133.52
eBook

Available Instantly

Online: 180 Days

Downloadable: 180 Days

$112.20

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	p. 1
The Elementary Mathematical Models and Basic Concepts of Mathematical Modeling	p. 6
Elementary Mathematical Models	p. 6
Fundamental laws of nature	p. 6
Variational principles	p. 13
Use of analogies in the construction of models	p. 15
Hierarchical approach to the construction of models	p. 17
On the nonlinearity of mathematical models	p. 19
Preliminary conclusions	p. 21
Exercises	p. 22
Examples of Models Following from the Fundamental Laws of Nature	p. 23
The trajectory of a floating submarine	p. 23
Deviation of a charged particle in an electron-beam tube	p. 25
Oscillations of the rings of Saturn	p. 27
Motion of a ball attached to a spring	p. 29
Conclusion	p. 31
Exercises	p. 32
Variational Principles and Mathematical Models	p. 32
The general scheme of the Hamiltonian principle	p. 32
The third way of deriving the model of the system "ball-spring"	p. 33
Oscillations of a pendulum in a gravity field	p. 35
Conclusion	p. 37
Exercises	p. 38
Example of the Hierarchy of Models	p. 38
Various modes of action of the given external force	p. 38
Motion of an attaching point, the spring on a rotating axis	p. 39
Accounting for the forces of friction	p. 41
Two types of nonlinear models of the system "ball-spring"	p. 43
Conclusion	p. 46
Exercises	p. 47
The Universality of Mathematical Models	p. 47
Fluid in a U-shaped flask	p. 47
An oscillatory electrical circuit	p. 49
Small oscillations at the interaction of two biological populations	p. 50
Elementary model of variation of salary and employment	p. 51
Conclusion	p. 52
Exercises	p. 52
Several Models of Elementary Nonlinear Objects	p. 53
On the origin of nonlinearity	p. 53
Three regimes in a nonlinear model of population	p. 53
Influence of strong nonlinearity on the process of oscillations	p. 55
On numerical methods	p. 56
Exercises	p. 57
Derivation of Models from the Fundamental Laws of Nature	p. 59
Conservation of the Mass of Substance	p. 59
A flow of particles in a pipe	p. 59
Basic assumptions on the gravitational nature of flows of underground waters	p. 62
Balance of mass in the element of soil	p. 62
Closure of the law of conservation of mass	p. 65
On some properties of the Bussinesque equation	p. 66
Exercises	p. 68
Conservation of Energy	p. 69
Preliminary information on the processes of heat transfer	p. 69
Derivation of Fourier law from molecular-kinetic concepts	p. 70
The equation of heat balance	p. 72
The statement of typical boundary conditions for the equation of heat transfer	p. 75
On the peculiarities of heat transfer models	p. 77
Exercises	p. 79
Conservation of the Number of Particles	p. 79
Basic concepts of the theory of thermal radiation	p. 79
Equation of balance of the number of photons in a medium	p. 82
Some properties of the equation of radiative transfer	p. 84
Exercises	p. 85
Joint Application of Several Fundamental Laws	p. 86
Preliminary concepts of gas dynamics	p. 86
Equation of continuity for compressible gas	p. 86
Equations of gas motion	p. 88
The equation of energy	p. 90
The equations of gas dynamics in Lagrangian coordinates	p. 91
Boundary conditions for the equations of gas dynamics	p. 93
Some peculiarities of models of gas dynamics	p. 94
Exercises	p. 97
Models Deduced from Variational Principles, Hierarchies of Models	p. 98
Equations of Motion, Variational Principles and Conservation Laws in Mechanics	p. 98
Equation of motion of a mechanical system in Newtonian form	p. 98
Equations of motion in Lagrangian form	p. 101
Variational Hamiltonian principle	p. 105
Conservation laws and space-time properties	p. 107
Exercises	p. 111
Models of Some Mechanical Systems	p. 111
Pendulum on the free suspension	p. 112
Non-potential oscillations	p. 116
Small oscillations of a string	p. 119
Electromechanical analogy	p. 123
Exercises	p. 125
The Boltzmann Equation and its Derivative Equations	p. 125
The description of a set of particles with the help of the distribution function	p. 126
Boltzmann equation for distribution function	p. 127
Maxwell distribution and the H-theorem	p. 129
Equations for the moments of distribution function	p. 133
Chain of hydrodynamical gas models	p. 139
Exercises	p. 144
Models of Some Hardly Formalizable Objects	p. 146
Universality of Mathematical Models	p. 146
Dynamics of a cluster of amoebas	p. 146
Random Markov process	p. 151
Examples of analogies between mechanical, thermodynamic and economic objects	p. 158
Exercises	p. 162
Some Models of Financial and Economic Processes	p. 162
Organization of an advertising campaign	p. 162
Mutual offset of debts of enterprises	p. 166
Macromodel of equilibrium of a market economy	p. 173
Macromodel of economic growth	p. 180
Exercises	p. 183
Some Rivalry Models	p. 184
Mutual relations in the system "predator - victim"	p. 184
Arms race between two countries	p. 187
Military operations of two armies	p. 190
Exercises	p. 194
Dynamics of Distribution of Power in Hierarchy	p. 195
General statement of problem and terminology	p. 195
Mechanisms of redistributing power inside the hierarchical structure	p. 201
Balance of power in a level, conditions on boundaries of hierarchy and transition to a continuous model	p. 204
The legal system "power-society". Stationary distributions and exit of power from its legal scope	p. 209
Role of basic characteristics of system in a phenomenon of power excess (diminution)	p. 213
Interpretation of results and conclusions	p. 214
Exercises	p. 216
Study of Mathematical Models	p. 218
Application of Similarity Methods	p. 218
Dimensional analysis and group analysis of models	p. 218
Automodel (self-similar) processes	p. 224
Various cases of propagation of perturbations in nonlinear media	p. 231
Exercises	p. 239
The Maximum Principle and Comparison Theorems	p. 240
The formulation and some consequences	p. 240
Classification of blow-up regimes	p. 245
The extension of "a self-similar method"	p. 248
Exercises	p. 254
An Averaging Method	p. 254
Localized structures in nonlinear media	p. 254
Various ways of averaging	p. 258
A classification of combustion regimes of a thermal conducting medium	p. 261
Exercises	p. 267
On Transition to Discrete Models	p. 267
Necessity of numerical modeling, elementary concepts of the theory of difference schemes	p. 268
Direct formal approximation	p. 272
The integro-interpolational method	p. 279
Principle of complete conservatism	p. 282
Construction of difference schemes by means of variational principles	p. 285
Use of the hierarchical approach in derivation of discrete models	p. 289
Exercises	p. 292
Mathematical Modeling of Complex Objects	p. 294
Problems of Technology and Ecology	p. 294
Physically "safe" nuclear reactor	p. 294
A hydrological "barrier" against the contamination of underground waters	p. 299
Complex regimes of gas flow around body	p. 302
Ecologically acceptable technologies for burning hydrocarbon fuels	p. 306
Fundamental Problems of Natural Science	p. 309
Nonlinear effects in laser thermonuclear plasma	p. 309
Mathematical restoration of the Tunguska phenomenon	p. 315
Climatic consequences of a nuclear conflict	p. 318
Magnetohydrodynamic "dynamo" of the Sun	p. 323
Computing Experiment with Models of Hardly Formalizable Objects	p. 326
Dissipative biological structures	p. 327
Processes in transition economy	p. 330
Totalitarian and anarchic evolution of power distribution in hierarchies	p. 334
References	p. 342
Index	p. 347
Table of Contents provided by Syndetics. All Rights Reserved.

Please wait while the item is added to your bag...

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Principles of Mathematical Modelling: Ideas, Methods, Examples

Summary

Author Biography

Table of Contents

Supplemental Materials

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Principles of Mathematical Modelling: Ideas, Methods, Examples

Summary

Author Biography

Table of Contents

Supplemental Materials

Digital License