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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 across a wide range of disciplines..

Tabla de contenidos provista por Syndetics

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

Reseñas proporcionadas por Syndetics

CHOICE Review

Samarskii (founder, Institute of Mathematical Modeling, Russian Academy of Science) and Mikhailov (department head, Institute of Mathematical Modeling) introduce mathematical modeling using ordinary and partial differential equations. Examples from many different applications are used to present the key ideas and practices of mathematical modeling. The major themes of the book include deriving models from conservation laws, variational principles, dimensional analysis, and the universality of mathematical models. A major weakness is that the book is oriented strongly toward deterministic models with no discussion of stochastic models. This book is a translation of an earlier Russian edition, and in many places the translation is unclear. Furthermore, the references cited by the authors are almost entirely in Russian and thus inaccessible to most readers. Other books with broader coverage of the same material include Frank R. Giordano, Maurice D. Weir, and William P. Fox's A First Course in Mathematical Modeling (3rd ed., 2002) and Michael Mesterton-Gibbons's A Concrete Approach to Mathematical Modelling (2nd ed., 1995). ^BSumming Up: Optional. Upper-division undergraduates through professionals. B. Borchers New Mexico Institute of Mining and Technology

Notas de autor provistas por Syndetics

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