Mathematical Modelling
- Length: 442 pages
- Edition: 1
- Language: English
- Publisher: Chapman and Hall/CRC
- Publication Date: 2021-11-24
- ISBN-10: 0367474301
- ISBN-13: 9780367474300
- Sales Rank: #0 (See Top 100 Books)
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 through this uniform lens.
The author describes different classes of models, including lumped and distributed parameter systems, deterministic and stochastic models, continuous and discrete models, static and dynamical systems, and more. From a mathematical point of view, the considered models can be understood as equations and systems of equations of different nature and variational principles. In addition to this, mathematical features of mathematical models, applied control and optimization problems based on mathematical models, and identification of mathematical models are also presented.
Features
- Each chapter includes four levels: a lecture (main chapter material), an appendix (additional information), notes (explanations, technical calculations, literature review) and tasks for independent work; this is suitable for undergraduates and graduate students and does not require the reader to take any prerequisite course, but may be useful for researchers as well
- Described mathematical models are grouped both by areas of application and by the types of obtained mathematical problems, which contributes to both the breadth of coverage of the material and the depth of its understanding
- Can be used as the main textbook on a mathematical modelling course, and is also recommended for special courses on mathematical models for physics, chemistry, biology, economics, etc.
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. 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. 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
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