Mathematical Modelling with Differential Equations

0. Preliminaries. 0.1. Introduction. 0.2. Mathematical Modeling. 0.3. Elementary Modeling Examples. 0.4. What is Science? 0.5. Scaling of Variables. 0.6. Dominant Balance and Approximations. 0.7. Use of Handbooks. 0.8. The Use OF Wikipedia. 0.9. Discussion. 0.10. Notes and References. 1. What is the ?N? 1.1. Introduction. 1.2. Interative Guessing. 1.3. Series Expansion Method. 1.4. Newton Method Algorithm. 1.5. Discussion. Problems. Notes and References. 2. Damping/Dissipative Forces. 2.1. Introduction. 2.2. Properties of DDF Functions. 2.3. Dimensional Analysis and DDF Functions. 2.4. One Term Power-law DDF Functions. 2.5. Two Term Power-law DDF Functions. 2.6. Discussion. Problems. Notes and References. 3. The Thomas-Fermi Equation. 3.1. Introduction. 3.2. Exact Results. 3.3. Dynamic Consistency. 3.4. Two Rational Approximations. 3.5. Discussion. Problems. Notes and References. 4. Single Population Growth Models. 4.1. Introduction. 4.2. Logistic Equation. 4.3. Gompertz Model. 4.4. Non-logistic Models. 4.5. Allee Effect. 4.6. Discussion. Problems. Notes and References. 5. 1 + 2 + 3 + 4 + 5 + . . . = ?(1/2). 5.1. Introduction. 5.2. Preliminaries. 5.3. Numerical Values of Divergent Series. 5.4. Elementary Function Defined by an Integral. 5.5. Gamma Function. 5.6. Riemann Zeta Function. 5.7. Discussion. Problems. Notes and References. 6. A Truly Nonlinear Oscillator. 6.1. Introduction. 6.2. General Properties of Exact Solutions. 6.3. Approximate Solutions. 6.4. Summary. Problems. Notes and References. 7. Discretization of Differential Equations. 7.1. Introduction. 7.2. Exact Schemes. 7.3. NSFD Methodology. 7.4. NSFD Schemes for One's. 7.5. Partial Differential Equation Applications. 7.6. Discussion. Problems. Notes and References. 8. SIR Models for Disease Spread. 8.1. Introduction. 8.2. SIR Methodology. 8.3. Standard SIR Model. 8.4. Flattening the Curve. 8.5. Solveable SIR Model. 8.6. Resume. Problems. Notes and References. 9. Dieting Model. 9.1 Introduction. 9.2. Mathematical Model. 9.3. Analysis of a Model. 9.4. Approximate Solutions. 9.5. Discussion. Problems. Notes and References. 10. Alternate Futures. 10.1. Introduction. 10.2. Two Systems Exhibiting Alternative Futures. 10.3. Resume of Concepts and Definitions. 10.4. Counterfactual Histories. 10.5. Summary and Discussion. Problems. Notes and References. 11. Toy Model of the Universe.11.1 Introduction. 11.2. In the Beginning: Let There Be Rules. 11.3. Some "Dull" Model Universes. 11.4. Nontrivial TMOU. 11.5. Fibonacci Equation. 11.6. Discussion. Problems. Notes and References. 12. Diffusion and Heat Equations. 12.1. Introduction. 12.2. Heat Equation Derivation. 12.3. Diffusion Equation and Random Walks. 12.4. Diffusion and Probability. 12.5. Derivation Difficulties. 12.6. Heated Rod Problem. 12.7. Comments. Problems. Notes and References. Appendix A. Algebraic Relations. B. Trigonometric Relations. C. Hyperbolic Functions. D. Relations from Calculus. E. Fourier Series. F. Even and Odd Functions. G. Some Nonstandard but Important Functions. H. Differential Equations. I. Linearization of Certain Types of Nonlinear Differential Equations. Bibliography. Index.