The book presents a wide range of methods for mathematical modeling of different problems and teaches how to formulate, solve, and interpret the results of various techniques mostly of differential equation kind, applied to numerous examples in many areas of science and technology, biology and medicine, economics and other fields of human needs and interests. In comparison with the first edition of 2014, the material is extended more than by a half, with new problems and approaches, exercises and projects. The book is organized in six chapters, each divided to multiple sections and subsections.
Chapter 1 “About Mathematical Modeling” introduces the main concepts of mathematical description of reality and briefs on the history of mathematical thought starting from ancient civilizations and going through the medieval world to the modern developments. On examples from physics, it describes units of measurement, dimensional analysis and scaling, discusses the needed steps in a problem modeling and various mathematical functions, for instance, logistic and Gompertz equations used in growth modeling, and considers populational dynamics of prey and predator in Holling’s models used in ecology. Miscellaneous examples and multiple exercises are given, including the damped Lotka-Volterra predator-prey model.
Chapter 2 “Discrete Models using Difference Equations” describes the recurrent relations and sequences involving differences, the linear and nonlinear, homogeneous and nonhomogeneous equations and their systems, possible analytical solutions, stability and points of equilibrium defined by specifics of the systems’ eigenvalues. Discrete-time models are considered for the population growth, immigration, Newton’s law of cooling, saving in bank account, drag delivery, Harrod’s economic model, arms race, Lanchester’s combat models, and tigers-deer model. Nonlinear techniques are presented for density-dependent logistic and Richer’s growth models, learning model, dynamic of alcohol in a body, and a model of two-species competition for black and grizzly bears. Cycle points, 2- and 3-cycles stability, bifurcations and chaos, criteria of chaos and Lyapunov exponent are described, and various additional problems are given. Codes in Mathematica and Matlab include Lanchester’s combat model, Lyapunov exponent, two-species competition, saddle-node and Neimark-Sacker bifurcations, and more. Four dozen exercises and two projects are suggested, particularly, a discrete dynamic system of the spread of infectious disease.
Chapter 3 “Continuous Models Using Ordinary Differential Equations” (ODE) is devoted to the continuous in time modeling. Examples of rumor propagation, and predator–prey systems of linear equations are given. Questions of steady-state solution, local stability by Routh-Hurwitz and Lyapunov criteria, Lyapunov’s condition for global stability, phase-plane diagrams of linear systems with nodes, saddle and spiral points are explained for different features of the eigenvalues. Continuous models are formulated for carbon dating, drug spreading in the body, growth and decay of electricity current in the inductance-resistance circuit, mechanical damped and forced oscillations, dynamics of rowing, and arms race models. Epidemic models, including susceptible-infective (SI), susceptible-infective-susceptible (SIS), susceptible-infective-recovered (SIR), and susceptible-infective-removed-susceptible (SIRS) are described. Lanchester’s combat models, such as conventional, guerilla, and mixed models are presented. Strogatz’s love affair model is described in detail featuring the exponential and steady love, love–hate relationship and endless love cycle. Bifurcations in one and two dimensions are considered, with characteristics of saddle-nodes, transcritical bifurcation by change of fixed points, pitchfork bifurcation, Sotomayar’s theorem, and Hopf bifurcation. Chaos in continuous models is discussed using Lyapunov exponents and Rössler systems. Other examples include the generalized Verhulst population model, amount of glucose in veins, chemical reactions, mechanics systems, sugar and oil pricing, trees growth, species competing for food, Shroud of Turin carbon dating, and more. Mathematica and Matlab codes are given for estimating parameters for ODE models in the least squares approach, and finding characteristics of bifurcation. Ninety exercises and three projects are also supplied.
Chapter 4 “Spatial Models Using Partial Differential Equations,” (PDE) extends modeling to operating with derivatives of at least two variables, combining time and space changes in the processes going in a real world. The initial and boundary value problems are described, including Dirichlet, Newmann, Cauchy, and Robin boundary conditions for the second-order PDE. Problems of the heat flow through a thin rod with a solution in Fourier series, two-dimensional diffusion heat-equation, steady-heat flow in Laplace equation, one and two-dimensional wave equations, vibrating string, fluid flow through a porous medium, traffic flow, reaction-diffusion systems, crime spread, and more are described via PDE. Mathematica and Matlab codes are given for the problem solutions. Two dozen exercises and two projects are suggested.
Chapter 5 “Modeling with Delay Differential Equations” (DDE) are applied to the lagged models with the derivatives of a present time depending on the independent variables at previous times. Questions of linear stability analysis and criteria are discussed. The delay-induced models are presented on examples of a National Football League (NFL) team performance, human behavior in controlling the water temperature in shower, arterial carbon dioxide level in breathing process, housefly behavior in laboratory conditions, two-neuron system, immunotherapy with interleukin-2 in cancer treatment, growth of blood cell density, Cooke’s epidemic model, microbial growth, gene regulatory system, Lotka-Volterra competition DDE model, reduction of cargo pendulation at bridge cranes, and more, with the corresponding Mathematica and Matlab codes. There are twenty exercises (for instance, Hutchinson’s delayed logistic equation), and also two projects on zombie epidemic and love affair.
Chapter 6 “Modeling with Stochastic Differential Equations” (SDE) discusses the stochastic impact on the dynamics and numerical solutions. It describes the event space and axiomatic definition of probability, random variables and their measures, normal distribution and its characteristic function, convergence and limit theorems, Gaussian and Markov stochastic processes, Wiener process or Brownian motion, white noise and Ito integral, existence and uniqueness of SDE solution and stochastic stability. Stochastic logistic growth, electric circuit with randomness, Heston stochastic volatility price model for analyzing bond and currency options, and species stochastic competition models are presented. Cancer self-remission and tumor stability problem is described in deterministic and stochastic modeling, with equilibria and local stability, numerical results and biological implications. Mathematica and Matlab codes are given for stochastic logistic growth and competition models, and there is a dozen exercises as well.
Additional Chapter 7 presents hints and solutions for all exercises. Bibliography contains 167 valuable sources, and there is an extensive index. The book holds over 250 illustrations, and 300 examples and exercises help to understand the presented topics on applied mathematics in engineering and natural sciences. It is not an easy reading, and requires a knowledge in advance calculus at the level of ordinary and partial differential equations and their systems. This manual, or rather a compendium and handbook, can be very much educational to graduate students and extremely useful to researchers who employ methods of mathematical modeling for solving their problems. Access to support materials is given at the link Mathematical Modeling: Models, Analysis and Applications—2nd Edition (routledge.com). Access to the instructor resources can be received at the Instructor Resources Download Hub where a registration is required. Some additional sources on the considered topics can be found in the references (Lipovetsky, Citation2018, Citation2019, Citation2020, Citation2021).
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References
- Lipovetsky, S. (2018), “Math Tools: 500+ Applications in Science and Arts, by G. Glaeser,” Technometrics, 60, 263.
- Lipovetsky, S. (2019), “Dynamic Data Analysis: Modeling Data with Differential Equations, by J. Ramsay and G. Hooker,” Technometrics, 61, 277–278.
- Lipovetsky, S. (2020), “2ˆ5 Problems for STEM Education, by V. Ochkov,” Technometrics, 62, 557–558. DOI: 10.1080/00401706.2020.1825604.
- Lipovetsky, S. (2021), “Advanced Engineering Mathematics, by K.A. Stroud and D.J. Booth,” Technometrics, 63, 566–570.