Mathematical Modelling, A Case Studies Approach, Illner R., McCollum S., 2005

Mathematical Modelling, A Case Studies Approach, Illner R., McCollum S., 2005.

   Mathematical modelling is a subject without boundaries in every conceivable sense. Wherever mathematics is applied to another science or sector of life, the modelling process enters in a conscious or subconscious way. Significantly, this is the process in which much of mathematics was originally started, even in the geometric problems encountered by the Greeks so many centuries ago.




Will t he Valve Hold?
The generation of hydroelectric power involves a potentially destructive environmental force that must be carefully managed. Should, for any reason, a proposed scheme of power generation be flawed by design, a great deal of damage could be inflicted on neighbouring areas. This is the responsibility faced by B.C. Hydro in 1961 when attempting to tap a water reservoir by drilling a tunnel through a mountainside. Figure 1 illustrates this situation.

This tunnel was drilled so that a rock plug between the tunnel and the lake was left intact. At the opposite end a valve had been installed for later control of the water fl.ow. These control measures are depicted in Figure 1. The control valve was shut so that when the rock plug, which had been charged with explosives, was blown clear, water would rush into the tunnel to fill it and, in the process, compress the air that had become trapped. The valve had been designed to withstand twice the hydrostatic pressure of the lake. However, the evening before the rock plug was to be removed, it was pointed out that the compressed air bubble would have to absorb all of the dynamic energy of the water rushing in and filling the tunnel, and that the pressure might temporarily exceed the safe threshold. B.C. Hydro was faced with a pressing issue: When the rock plug was blown, would the valve be able to withstand the maximum pressure?

Contents.
Preface.
Chapter 1. Crystallization Dynamics.
§1.1. Derivation of the К-A Model.
§1.2. Emergence of the Poisson Distribution from the Binomial Distribution.
§1.3. Testing the К-A Model.
§1.4. A New Model.
§1.5. The Averaging Process.
§1.6. Choosing the Probability Density f.
§1.7. Why Did the К-A Model Fail?.
Exercises.
Notes.
Chapter 2. Will the Valve Hold?.
§2.1. Terminology.
§2.2. The Relevant Forces.
§2.3. The Equation of Motion.
§2.4. Analysis: Is the Initial Value Problem Well-Posed?.
§2.5. Revising the Model.
§2.6. Revision 1: Adding a Reference Distance.
§2.7. Revision 2: Changing the Initial Conditions.
§2.8. Pmax: The Maximal Pressure.
Exercises.
Notes.
Chapter 3. How Much Will that Annuity Cost Me?.
§3.1. Interest Basics.
§3.2. Mortgages.
§3.3. Loan Repayment.
§3.4. Present Value.
§3.5. Annuities.
§3.6. Hazard Rate Functions.
§3.7. Expected Lifetime.
§3.8. An Annuity Problem.
§3.9.    V(y): How the Expected Value of the Annuity Varies.
Exercises.
Chapter 4. Dimensional Analysis.
§4.1. A Classical Example: The Pendulum.
§4.2. Dimensional Analysis: The General Procedure.
§4.3. The Energy Released by a Nuclear Bomb.
§4.4. Exploration: How to Cook a Turkey Exercises.
Chapter 5. Predator-Prey Systems.
§5.1. The Lotka-Volterra Model.
§5.2. The Effect of Interference on the System.
§5.3. Linearization: The General Procedure.
§5.4. Solving Linear Systems.
§5.5. Classification of the Equilibria.
§5.6. The Phase Paths.
§5.7. Multiple Species.
§5.8. Exploration A: Structural Stability.
§5.9. Exploration B: The Lorenz Attractor Exercises.
Chapter 6. A Control Problem in Fishery Management.
§6.1. Variables and Parameters.
§6.2. The Logistic Growth Model.
§6.3. Maximizing the Sustainable Catch.
§6.4. Maximizing the Profit Exercises.
Chapter 7. Formal Justice.
§7.1. The Basic Functional Equation.
§7.2. Formal Justice: A Generalized Approach.
§7.3. Multiple Qualifications.
§7.4. Exploration: Exotic Solutions of Cauchy’s Functional Equation Exercises.
Chapter 8. Traffic Dynamics: A Microscopic Model.
§8.1. The Braking Force.
§8.2. Density and Flux at Equilibrium.
§8.3. A Case Study: Propagation of a Perturbation.
§8.4. Exploration: Peano’s Existence Theorem Exercises.
Chapter 9. Traffic Dynamics: Macroscopic Modelling.
§9.1. Scalar Conservation Laws.
§9.2. Solving Initial Value Problems for First-Order PDEs.
§9.3. The Green Light Problem.
§9.4. Smooth Initial Data, and General Scalar Conservation Laws.
§9.5. Intersecting Characteristics Exercises.
Bibliography.



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Теги: учебник по математике :: математика :: Illner :: McCollum