MATHEMATICAL MODELING A Comprehensive Introduction

MATHEMATICAL MODELING A Comprehensive Introduction

Gerhard Dangelmayr and Michael Kirby Department of Mathematics Colorado State University Fort Collins, Colorado, 80523

PRENTICE HALL, Upper Saddle River, New Jersey 07458

Contents

Preface

5

1 Mathematical Modeling

7

1.1 Examples of Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Modeling with Difference Equations . . . . . . . . . . . . . . 7

1.1.2 Modeling with Ordinary Differential Equations . . . . . . . . 7

1.1.3 Modeling with Partial Differential Equation . . . . . . . . . . 8

1.1.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.5 Modeling with Simulations . . . . . . . . . . . . . . . . . . . 9

1.1.6 Function Fitting: Data Modeling . . . . . . . . . . . . . . . . 9

1.2 The Modeling Process . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.1 An Algorithm for Modeling? . . . . . . . . . . . . . . . . . . 10

1.3 The Delicate Science of Errors . . . . . . . . . . . . . . . . . . . . . 10

1.4 Purpose of this Course . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Qualitative Modeling with Functions

13

2.1 Modeling Species Propagation . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Supply and Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2 Market Adjustment . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.3 Taxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Modeling with Proportion and Scale . . . . . . . . . . . . . . . . . . 18

2.3.1 Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.2 Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.1 Dimensional homogeneity . . . . . . . . . . . . . . . . . . . . 29

2.4.2 Discovering Joint Proportions . . . . . . . . . . . . . . . . . . 30

2.4.3 Procedure for Nondimensionalization . . . . . . . . . . . . . . 31

2.4.4 Modeling with Dimensional Analysis . . . . . . . . . . . . . . 32

Bibliography

38

3 Linear Programming

39

3.1 Examples of Linear Programs . . . . . . . . . . . . . . . . . . . . . . 40

3.1.1 Red or White? . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.2 How Many Fish? . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Geometric Solution of a 2D Linear Program . . . . . . . . . . . . . . 41

3.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3.1 Price Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3.2 Resource Sensitivity . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.3 Constraint Coefficient Sensitivity . . . . . . . . . . . . . . . . 45

3.4 Linear Programs with Equality Constraints . . . . . . . . . . . . . . 45

3.4.1 A Task Scheduling Problem . . . . . . . . . . . . . . . . . . . 46

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3.4.2 Transportation Problems . . . . . . . . . . . . . . . . . . . . 47 3.5 A Targeting Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5.1 Discretization and Solution of the Equations of Motion . . . 48 3.5.2 Formulation as Linear Program . . . . . . . . . . . . . . . . . 49 3.5.3 Targeting Problem with Air Resistance . . . . . . . . . . . . 53 3.5.4 Additional Constraints . . . . . . . . . . . . . . . . . . . . . . 55 3.6 Analysis of the Targeting Problem . . . . . . . . . . . . . . . . . . . 55 3.6.1 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . 57 3.6.2 Dimensionless Variables . . . . . . . . . . . . . . . . . . . . . 58 3.6.3 Maximum Altitude . . . . . . . . . . . . . . . . . . . . . . . . 59

4 Modeling with Nonlinear Programming

64

4.1 Unconstrained Optimization in One Dimension . . . . . . . . . . . . 65

4.1.1 Bisection Algorithm . . . . . . . . . . . . . . . . . . . . . . . 65

4.1.2 Newton's Method . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2 Unconstrained Optimization in Higher Dimensions . . . . . . . . . . 67

4.2.1 Taylor Series in Higher Dimensions . . . . . . . . . . . . . . . 67

4.2.2 Roots of a Nonlinear System . . . . . . . . . . . . . . . . . . 67

4.2.3 Newton's Method . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2.4 Steepest Descent . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3 Constrained Optimization and Lagrange Multipliers . . . . . . . . . 69

4.4 Geometry of Constrained Optimization . . . . . . . . . . . . . . . . . 70

4.4.1 One Equality Constraint . . . . . . . . . . . . . . . . . . . . . 70

4.4.2 Several Equality Constraints . . . . . . . . . . . . . . . . . . 74

4.4.3 Inequality Constraints . . . . . . . . . . . . . . . . . . . . . . 77

4.5 Modeling Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 Empirical Modeling with Data Fitting

88

5.1 Linear Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.1.1 The Mammalian Heart Revisited . . . . . . . . . . . . . . . . 89

5.1.2 General Formulation . . . . . . . . . . . . . . . . . . . . . . . 89

5.1.3 Exponential Fits . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.1.4 Fitting Data with Polynomials . . . . . . . . . . . . . . . . . 92

5.1.5 Interpolation versus Least Squares . . . . . . . . . . . . . . . 95

5.2 Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2.1 Linear Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2.2 Cubic Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3 Data Fitting and the Uniform Approximation . . . . . . . . . . . . . 99

5.3.1 Error Model Selection? . . . . . . . . . . . . . . . . . . . . . . 102

6 Modeling with Discrete Dynamical Systems

106

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.2 Linear First Order Difference Equations . . . . . . . . . . . . . . . . 110

6.2.1 Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . 110

6.2.2 Modeling Examples . . . . . . . . . . . . . . . . . . . . . . . 115

6.3 Linear Second Order Equations . . . . . . . . . . . . . . . . . . . . . 119

6.3.1 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . 119

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6.3.2 The Cobweb Model Revisited . . . . . . . . . . . . . . . . . . 122 6.4 Nonlinear Difference Equations and Systems in Population Modeling 124

6.4.1 Systems of Equations and Competing Species . . . . . . . . . 125 6.5 Empirical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.5.1 Non-Newtonian Fish? . . . . . . . . . . . . . . . . . . . . . . 128 6.5.2 Predator or Prey? . . . . . . . . . . . . . . . . . . . . . . . . 131

7 Simulation Modeling

140

7.1 The Tire Distributor Problem . . . . . . . . . . . . . . . . . . . . . . 140

7.2 Blackjack Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

APPENDICES

A Matlab Code for Data Fitting

149

A.1 Mammalian Heart Rate Problem . . . . . . . . . . . . . . . . . . . . 149

A.2 Least Squares with Normal Equations . . . . . . . . . . . . . . . . . 151

A.3 Least Squares with Overdetermined System . . . . . . . . . . . . . . 153

A.4 Non-Newtonian Fish . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

A.5 Preditor or Prey? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

A.6 Tire Distributor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

A.7 Blackjack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Preface

These materials are being developed with support from National Science Foundation Award no. 0126650 entitled A Mathematical Modeling Program for Undergraduates in Science, Mathematics, Engineering and Technology.

The objective of this project is the development of innovative educational materials that incorporate a novel educational approach and perspective to enhance the teaching and learning of mathematics for the purposes of knowledge discovery. The general undergraduate educated with these materials will possess a readily applicable toolbox of mathematical ideas for quantifying real world problems as well as problem solving skills, and possibly the most importantly, the ability to interpret results and further understanding.

Our pedagogical perspective consists of the observation that mathematical modeling is often taught backwards. An application of interest is presented and then appropriate mathematical tools are subsequently invoked. The beginner is left with the obvious concern. How does one decide which method to use on a new problem? Our proposed solution to this dilemma is to teach mathematics first and then show why a given mathematical methodology can be applied to the modeling problem. We will be successful if the student completes their modeling course based on these materials with a good sense of what makes various mathematical methods inherently different. Furthermore, students that are aware of the fundamental distinguishing characteristics of the array of methodologies should now be equipped to address this question of central importance in modeling, i.e., which method when!

This text is the first of two planned works to establish "proof of concept" of a new approach to teaching mathematical modeling. The scope of the text is the basic theory of modeling from a mathematical perspective. A second applications focussed text will build on the basic material of the first volume.

It is typical that students in a mathematical modeling class come from a wide variety of disciplines. In addition, their preparation and mathematical sophistication can vary as widely as their areas of interest. This heterogeneity makes the teaching and learning of mathematical modeling a significant challenge. One of the main student prototypes is a intelligent although possibly mathematically naive student that must learn mathematically modeling to make progress in an area of research. If a course or textbook does not provide the necessary information for these good students to bridge educational gaps students everyone suffers. Indeed, most textbooks fail to be accessible to such audiences.

With enhancing accessibility as our motivation, we propose to implement a simple pedagogical device to facilitate the use of the text by students of widely varying backgrounds. This device consists of graded levels of presentation denoted by (E) for elementary, (I) for intermediate and (A) for advanced.

? (E) Mathematical beginners will find much of interest in the elementary sections as well as foundation material for further study. The diligent student can use this self-contained treatment to pave the way to reading of more advanced sections. The basic properties of mathematical techniques will be presented with an emphasis on how methods lead to specific applications.

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