Samarskii A.A., Mikhailov A.P. Principles Of Mathematical Modeling: Ideas, Methods, Examples

Boca Raton: CRC Press, 2018. — 360 p.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.The elementary mathematical models and basic concepts of mathematical modeling
Elementary Mathematical Models
Fundamental laws of nature
Variational principles
Use of analogies in the construction of models
Hierarchical approach to the construction of models
On the nonlinearity of mathematical models
Preliminary conclusions
ExercisesExamples of Models Following from the Fundamental Laws of Nature
The trajectory of a floating submarine
Deviation of a charged particle in an electron-beam tube
Oscillations of the rings of Saturn
Motion of a ball attached to a springExercisesVariational Principles and Mathematical Models
The general scheme of the Hamiltonian principle
The third way of deriving the model of the system "ball-spring
Oscillations of a pendulum in a gravity fieldExercisesExample of the Hierarchy of Models
Various modes of action of the given external force
Motion of an attaching point, the spring on a rotating axis
Accounting for the forces of friction
Two types of nonlinear models of the system "ball-springExercisesThe Universality of Mathematical Models
Fluid in a U-shaped flask
An oscillatory electrical circuit
Small oscillations at the interaction of two biological populations
Elementary model of variation of salary and employmentExercisesSeveral Models of Elementary Nonlinear Objects
On the origin of nonlinearity
Three regimes in a nonlinear model of population
Influence of strong nonlinearity on the process of oscillations
On numerical methods
ExercisesDerivation of models from the fundamental laws of nature
Conservation of the Mass of Substance
A flow of particles in a pipe
Basic assumptions on the gravitational nature of flows of underground waters
Balance of mass in the element of soil
Closure of the law of conservation of mass
On some properties of the Bussinesque equation
ExercisesConservation of Energy
Preliminary information on the processes of heat transfer
Derivation of Fourier law from molecular-kinetic concepts
The equation of heat balance
The statement of typical boundary conditions for the equation of heat transfer
On the peculiarities of heat transfer models
ExercisesConservation of the Number of Particles
Basic concepts of the theory of thermal radiation
Equation of balance of the number of photons in a medium
Some properties of the equation of radiative transfer
ExercisesJoint Application of Several Fundamental Laws
Preliminaiy concepts of gas dynamics
Equation of continuity for compressible gas
Equations of gas motion
The equation of energy
The equations of gas dynamics in Lagrangian coordinates
Boundary conditions for the equations of gas dynamics
Some peculiarities of models of gas dynamics
ExercisesModels deduced from variational principles, hierarchies of models
Equations of Motion, Variational Principles and Conservation Laws in Mechanics
Equation of motion of a mechanical system in Newtonian form
Equations of motion in Lagrangian form
Variational Hamiltonian principle
Conservation laws and space-time properties
ExercisesModels of Some Mechanical Systems
Pendulum on the free suspension
Non-potential oscillations
Small oscillations of a string
Electromechanical analogy
ExercisesThe Boltzmann Equation and its Derivative Equations
The description of a set of particles with the help of the distribution function
Boltzmann equation for distribution function
Maxwell distribution and the //-theorem
Equations for the moments of distribution function
Chain of hydrodynamical gas models
ExercisesModels of some hardly formalizable objects
Universality of Mathematical Models
Dynamics of a cluster of amoebas
Random Markov process
Examples of analogies between mechanical, thermodynamic and economic objects
ExercisesSome Models of Financial and Economic Processes
Organization of an advertising campaign
Mutual offset of debts of enterprises
Macromodel of equilibrium of a market economy
Macromodel of economic growth
ExercisesSome Rivalry Models
Mutual relations in the system "predator - victim
Arms race between two countries
Military operations of two armies
ExercisesDynamics of Distribution of Power in Hierarchy
General statement of problem and terminology
Mechanisms of redistributing power inside the hierarchical structure
Balance of power in a level, conditions on boundaries of hierarchy and transition to a continuous model
The legal system "power-society". Stationary distributions and exit of power from its legal scope
Role of basic characteristics of system in a phenomenon of power excess (diminution)
Interpretation of results and conclusions
ExercisesStudy of mathematical models
Application of Similarity Methods
Dimensional analysis and group analysis of models
Automodel (self-similar) processes
Various cases of propagation of perturbations in nonlinear media
ExercisesThe Maximum Principle and Comparison Theorems
The formulation and some consequences
Classification of blow-up regimes
The extension of "a self-similar method
ExercisesAn Averaging Method
Localized structures in nonlinear media
Various ways of averaging
A classification of combustion regimes of a thermal conducting medium
ExercisesOn Transition to Discrete Models
Necessity of numerical modeling, elementary concepts of the theory of difference schemes
Direct formal approximation
The integro-interpolational method
Principle of complete conservatism
Construction of difference schemes by means of variational principles
Use of the hierarchical approach in derivation of discrete models
ExercisesMathematical modeling of complex objects
Problems of Technology and Ecology
Physically "safe” nuclear reactor
A hydrological "barrier” against the contamination of underground waters
Complex regimes of gas flow around body
Ecologically acceptable technologies for burning hydrocarbon fuelsFundamental Problems of Natural Science
Nonlinear effects in laser thermonuclear plasma
Mathematical restoration of the Tunguska phenomenon
Climatic consequences of a nuclear conflict
Magnetohydrodynamic "dynamo” of the SunComputing Experiment with Models of Hardly Formalizable Objects
Dissipative biological structures
Processes in transition economy
Totalitarian and anarchic evolution of power distribution in hierarchies