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9780821836507

Mathematical Modelling

by ; ; ;
  • ISBN13:

    9780821836507

  • ISBN10:

    0821836501

  • Format: Paperback
  • Copyright: 2005-01-01
  • Publisher: Amer Mathematical Society

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Summary

Mathematical modelling is a subject without boundaries. It is the means by which mathematics becomes useful to virtually any subject. Moreover, modelling has been and continues to be a driving force for the development of mathematics itself. This book explains the process of modelling real situations to obtain mathematical problems that can be analyzed, thus solving the original problem. The presentation is in the form of case studies, which are developed much as they would be in true applications. In many cases, an initial model is created, then modified along the way. Some cases are familiar, such as the evaluation of an annuity. Others are unique, such as the fascinating situation in which an engineer, armed only with a slide rule, had 24 hours to compute whether a valve would hold when a temporary rock plug was removed from a water tunnel. Each chapter ends with a set of exercises and some suggestions for class projects. Some projects are extensive, as with the explorations of the predator-prey model; others are more modest. The text was designed to be suitable for a one-term course for advanced undergraduates. The selection of topics and the style of exposition reflect this choice. The authors have also succeeded in demonstrating just how enjoyable the subject can be. This is an ideal text for classes on modelling. It can also be used in seminars or as preparation for mathematical modelling competitions.

Table of Contents

Preface xi
Chapter 1. Crystallization Dynamics 1(20)
§1.1. Derivation of the K-A Model
3(3)
§1.2. Emergence of the Poisson Distribution from the Binomial Distribution
6(1)
§1.3. Testing the K-A Model
7(1)
§1.4. A New Model
8(3)
§1.5. The Averaging Process
11(1)
§1.6. Choosing the Probability Density f
11(4)
§1.7. Why Did the K-A Model Fail?
15(1)
Exercises
15(5)
Notes
20(1)
Chapter 2. Will the Valve Hold? 21(20)
§2.1. Terminology
23(1)
§2.2. The Relevant Forces
24(2)
§2.3. The Equation of Motion
26(1)
§2.4. Analysis: Is the Initial Value Problem Well-Posed?
27(1)
§2.5. Revising the Model
28(3)
§2.6. Revision 1: Adding a Reference Distance
31(1)
§2.7. Revision 2: Changing the Initial Conditions
32(2)
§2.8. Pmax: The Maximal Pressure
34(3)
Exercises
37(2)
Notes
39(2)
Chapter 3. How Much Will that Annuity Cost Me? 41(22)
§3.1. Interest Basics
41(2)
§3.2. Mortgages
43(6)
§3.3. Loan Repayment
49(1)
§3.4. Present Value
50(1)
§3.5. Annuities
50(1)
§3.6. Hazard Rate Functions
51(5)
§3.7. Expected Lifetime
56(1)
§3.8. An Annuity Problem
57(2)
§3.9. V(Y): How the Expected Value of the Annuity Varies
59(1)
Exercises
60(3)
Chapter 4. Dimensional Analysis 63(20)
§4.1. A Classical Example: The Pendulum
63(4)
§4.2. Dimensional Analysis: The General Procedure
67(3)
§4.3. The Energy Released by a Nuclear Bomb
70(4)
§4.4. Exploration: How to Cook a Turkey
74(7)
Exercises
81(2)
Chapter 5. Predator-Prey Systems 83(36)
§5.1. The Lotka-Volterra Model
83(4)
§5.2. The Effect of Interference on the System
87(5)
§5.3. Linearization: The General Procedure
92(3)
§5.4. Solving Linear Systems
95(3)
§5.5. Classification of the Equilibria
98(3)
§5.6. The Phase Paths
101(3)
§5.7. Multiple Species
104(1)
§5.8. Exploration A: Structural Stability
105(5)
§5.9. Exploration B: The Lorenz Attractor
110(4)
Exercises
114(5)
Chapter 6. A Control Problem in Fishery Management 119(12)
§6.1. Variables and Parameters
120(1)
§6.2. The Logistic Growth Model
120(2)
§6.3. Maximizing the Sustainable Catch
122(3)
§6.4. Maximizing the Profit
125(4)
Exercises
129(2)
Chapter 7. Formal Justice 131(18)
§7.1. The Basic Functional Equation
131(5)
§7.2. Formal Justice: A Generalized Approach
136(2)
§7.3. Multiple Qualifications
138(4)
§7.4. Exploration: Exotic Solutions of Cauchy's Functional Equation
142(4)
Exercises
146(3)
Chapter 8. Traffic Dynamics: A Microscopic Model 149(20)
§8.1. The Braking Force
149(2)
§8.2. Density and Flux at Equilibrium
151(5)
§8.3. A Case Study: Propagation of a Perturbation
156(7)
§8.4. Exploration: Peano's Existence Theorem
163(4)
Exercises
167(2)
Chapter 9. Traffic Dynamics: Macroscopic Modelling 169(24)
§9.1. Scalar Conservation Laws
170(3)
§9.2. Solving Initial Value Problems for First-Order PDEs
173(5)
§9.3. The Green Light Problem
178(5)
§9.4. Smooth Initial Data, and General Scalar Conservation Laws
183(1)
§9.5. Intersecting Characteristics
184(9)
Exercises 193(2)
Bibliography 195

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