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Mathematical modelling /

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"Mathematical Modelling sets out the general principles of mathematical modelling as a means comprehending the world. Within the book, the problems of physics, engineering, chemistry, biology, medicine, economics, ecology, sociology, psychology, political science, etc. are all considered throug...

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Bibliographic Details
Main Author: Serovajsky, Simon (Author)
Corporate Author: Taylor & Francis
Format: eBook
Language:English
Published: Boca Raton, FL : Chapman & Hall/CRC Press, 2022.
Edition:First edition.
Subjects:
Mathematical models.
Models, Theoretical
Modèles mathématiques.
mathematical models.
MATHEMATICS / General
MATHEMATICS / Applied
MATHEMATICS / Arithmetic
Mathematical models
Electronic books.
Online Access:Connect to the full text of this electronic book
Table of Contents:
  • Cover
  • Half Title
  • Title Page
  • Copyright Page
  • Dedication
  • Contents
  • Preface
  • List of Figures
  • List of Tables
  • 1. Foundations of mathematical modeling
  • Lecture
  • 1. Cognition and modeling
  • 2. Natural Sciences and Mathematics
  • 3. Content or form?
  • 4. Copernicus or Ptolemy?
  • 5. Mathematical model of a body falling
  • 6. Principles for determining mathematical models
  • 7. Classification of mathematical models
  • Appendix
  • 1. Probe movement
  • 2. Missile flight
  • 3. Glider flight
  • Notes
  • I. Systems with lumped parameters
  • 2. Approximate solving of differential equations
  • Lecture
  • 1. Conception of approximate solution
  • 2. Euler method
  • 3. Probe movement
  • 4. Missile flight
  • 5. Glider flight
  • Appendix
  • 1. Runge-Kutta method
  • 2. Two-body problem
  • 3. Predator-pray model
  • Notes
  • 3. Mechanical oscillations
  • Lecture
  • 1. Determination of the pendulum oscillation equation
  • 2. Solving of the pendulum oscillation equation
  • 3. Pendulum oscillation energy
  • 4. Oscillation of a pendulum with friction
  • 5. Equilibrium position of the pendulum
  • 6. Forced oscillations of the pendulum
  • Appendix
  • 1. Spring oscillation
  • 2. Large pendulum oscillations
  • 3. Problems of nonlinear oscillation theory
  • Notes
  • 4. Electrical oscillations
  • Lecture
  • 1. Electrical circuit
  • 2. Energy of circuit
  • 3. Circuit with resistance
  • 4. Forced circuit oscillations
  • Appendix
  • 1. Forced oscillations of spring
  • 2. Circuit with nonlinear capacity
  • 3. Van der Pol circuit
  • Notes
  • 5. Elements of dynamical system theory
  • Lecture
  • 1. Evolutionary processes and differential equations
  • 2. General notions of dynamic systems theory
  • 3. Change in species number with excess food
  • 4. Oscillations of pendulum
  • 5. Stability of the equilibrium position
  • 6. Limit cycle
  • Appendix.
  • 1. Exponential growth systems
  • 2. Brussellator
  • 3. System with two limit cycle
  • Notes
  • 6. Mathematical models in chemistry
  • Lecture
  • 1. Chemical kinetics equations
  • 2. Monomolecular reaction
  • 3. Bimolecular reaction
  • 4. Lotka reaction system
  • Appendix
  • 1. Brusselator
  • 2. Oregonator
  • 3. Chemical niche
  • 4. Laser healing model
  • Notes
  • 7. Mathematical model in biology
  • Lecture
  • 1. One species evolution
  • 2. Biological competition model
  • 3. Predator-prey model
  • 4. Symbiosis model
  • Appendix
  • 1. Models of chemical and physical competition
  • 2. Fluctuations in yield and fertility
  • 3. Ecological niche model
  • 4. SIR model for spread of disease
  • 5. Antibiotic resistance model
  • Notes
  • 8. Mathematical model of economics
  • Lecture
  • 1. One company evolution
  • 2. Economic competition model
  • 3. Economic niche model
  • 4. Free market model
  • 5. Monopolized market model
  • Appendix
  • 1. Ecological niche model
  • 2. Inflation model
  • 3. Model of economic cooperation
  • 4. Racketeer-entrepreneur model
  • 5. Solow model of economic growth
  • Notes
  • 9. Mathematical models in social sciences
  • Lecture
  • 1. Political competition
  • 2. Political niche
  • 3. Bipartisan system
  • 4. Trade union activity
  • 5. Allied relations
  • Appendix
  • 1. Competition models
  • 2. Niche models
  • 3. Predator-prey models
  • Notes
  • II. Systems with distributed parameters
  • 10. Mathematical models of transfer processes. 1
  • Lecture
  • 1. Heat equation
  • 2. First boundary value problem for the homogeneous heat equation
  • 3. Non-homogeneous heat equation
  • Appendix
  • 1. Generalizations of the heat equation
  • 2. Second boundary value problem for the heat equation
  • 3. Diffusion equation
  • Notes
  • 11. Mathematical models of transfer processes. 2
  • Lecture
  • 1. Heat equation and similarity theory.
  • 2. Goods transfer equation
  • 3. Finite difference method for the heat equation
  • 4. Diffusion of chemical reactants
  • 5. Stefan problem for the heat equation
  • Appendix
  • 1. Overview of transfer processes
  • 2. Finite difference method: Implicit scheme
  • 3. Competitive species migration
  • 4. Hormone treatment of the tumor with hormone resistance
  • Notes
  • 12. Wave processes
  • Lecture
  • 1. Vibration of string
  • 2. Vibrations of string with fixed ends
  • 3. Infinitely long string
  • 4. Electrical vibrations in wires
  • Appendix
  • 1. Energy of vibrating string
  • 2. Mathematical models of wave processes
  • 3. Beam vibrations
  • 4. Maxwell equations
  • 5. Finite difference method for the vibrating string equation
  • Notes
  • 13. Mathematical models of stationary systems
  • Lecture
  • 1. Stationary heat transfer
  • 2. Spherical and cylindrical coordinates
  • 3. Vector fields
  • 4. Electrostatic field
  • 5. Gravity field
  • Appendix
  • 1. Stationary fluid flow
  • 2. Steady oscillations
  • 3. Bending a thin elastic plate
  • 4. Variable separation method for the Laplace equation in a circle
  • 5. Establishment method
  • Notes
  • 14. Mathematical models of fluid and gas mechanics
  • Lecture
  • 1. Material balance in a moving fluid
  • 2. Ideal fluid movement
  • 3. Ideal fluid under the gravity field
  • 4. Viscous fluid movement
  • Appendix
  • 1. Burgers equation
  • 2. Surface wave movement
  • 3. Boundary layer model
  • 4. Acoustic problem
  • 5. Thermal convection
  • 6. Problems of magnetohydrodynamics
  • Notes
  • 15. Mathematical models of quantum mechanical systems
  • Lecture
  • 1. Quantum mechanics problems
  • 2. Wave function
  • 3. Schrödinger equation
  • 4. Particle movement under an external field
  • 5. Potential barrier
  • Appendix
  • 1. Wave function normalization
  • 2. Particle movement in a well with infinitely high walls
  • Notes.
  • III. Other mathematical models
  • 16. Variational principles
  • Lecture
  • 1. Brachistochrone problem
  • 2. Lagrange problem
  • 3. Shortest curve
  • 4. Body falling problem and the concept of action
  • 5. Principle of least action
  • 6. Vibrations of string
  • Appendix
  • 1. Law of conservation of energy
  • 2. Fermat's principle and light refraction
  • 3. River crossing problem
  • 4. Pendulum oscillations
  • 5. Approximate solution of minimization problems
  • Notes
  • 17. Discrete models
  • Lecture
  • 1. Discrete population dynamics models
  • 2. Discrete heat transfer model
  • 3. Transportation problem
  • 4. Traveling salesman problem
  • 5. Prisoner's dilemma
  • Appendix
  • 1. Discrete model of epidemic propagation
  • 2. Potential method for solving a transportation problem
  • 3. Production planning
  • 4. Concepts of game theory
  • Notes
  • 18. Stochastic models
  • Lecture
  • 1. Stochastic model of pure birth
  • 2. Monte Carlo method
  • 3. Stochastic model of population death
  • 4. Stochastic Malthus model
  • Appendix
  • 1. Malthus model with random population growth
  • 2. Models with random parameters
  • 3. Discrete model of selling goods
  • 4. Passage of a neutron through a plate
  • Notes
  • IV. Additions
  • 19. Mathematical problems of mathematical models
  • Lecture
  • 1. Cauchy problem properties for differential equations
  • 2. Properties of boundary value problems
  • 3. Boundary value problems for the heat equation
  • 4. Hadamard's example and well-posedness of problems
  • 5. Classical and generalized solution of problems
  • Appendix
  • 1. Nonlinear boundary value problems
  • 2. Euler's elastic problem
  • 3. Bénard problem
  • 4. Generalized model of stationary heat transfer
  • 5. Sequential model of stationary heat transfer
  • Notes
  • 20. Optimal control problems
  • Lecture
  • 1. Maximizing the shell flight range.
  • 2. Maximizing the missile flight range
  • 3. General optimal control problem
  • 4. Solving of the maximization problem of the missile flight range
  • 5. Time-optimal control problem
  • Appendix
  • 1. Maximizing the probe's ascent height
  • 2. Approximate methods for solving optimality conditions
  • 3. Gradient methods
  • Notes
  • 21. Identification of mathematical models
  • Lecture
  • 1. Problem of determining the system parameters
  • 2. Inverse problems and their solving
  • 3. Heat equation with data at the final time
  • 4. Differentiation of functionals and gradient methods
  • 5. Solving of the heat equation with reversed time
  • Appendix
  • 1. Boundary inverse problem for the heat equation
  • 2. Inverse problem for the falling of body
  • 3. Inverse gravimetry problem
  • 4. Well-posedness of optimal control problems
  • Notes
  • Epilogue
  • Bibliography
  • Index.