Mathematical Modeling with Multidisciplinary Applications E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2013

Beschreibung

Features mathematical modeling techniques and real-world processes with applications in diverse fields

Mathematical Modeling with Multidisciplinary Applications details the interdisciplinary nature of mathematical modeling and numerical algorithms. The book combines a variety of applications from diverse fields to illustrate how the methods can be used to model physical processes, design new products, find solutions to challenging problems, and increase competitiveness in international markets.

Written by leading scholars and international experts in the field, the book presents new and emerging topics in areas including finance and economics, theoretical and applied mathematics, engineering and machine learning, physics, chemistry, ecology, and social science. In addition, the book thoroughly summarizes widely used mathematical and numerical methods in mathematical modeling and features:

• Diverse topics such as partial differential equations (PDEs), fractional calculus, inverse problems by ordinary differential equations (ODEs), semigroups, decision theory, risk analysis, Bayesian estimation, nonlinear PDEs in financial engineering, perturbation analysis, and dynamic system modeling
• Case studies and real-world applications that are widely used for current mathematical modeling courses, such as the green house effect and Stokes flow estimation
• Comprehensive coverage of a wide range of contemporary topics, such as game theory, statistical models, and analytical solutions to numerical methods
• Examples, exercises with select solutions, and detailed references to the latest literature to solidify comprehensive learning
• New techniques and applications with balanced coverage of PDEs, discrete models, statistics, fractional calculus, and more

Mathematical Modeling with Multidisciplinary Applications is an excellent book for courses on mathematical modeling and applied mathematics at the upper-undergraduate and graduate levels. The book also serves as a valuable reference for research scientists, mathematicians, and engineers who would like to develop further insights into essential mathematical tools.

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

List of Figures

Preface

Acknowledgments

Editor and Contributors

Part I: Introduction and Foundations

Chapter 1: Differential Equations

1.1 Ordinary Differential Equations

1.2 Partial Differential Equations

1.3 Classic Mathematical Models

1.4 Other Mathematical Models

1.5 Solution Techniques

Exercises

References

Chapter 2: Mathematical Modeling

2.1 Mathematical Modeling

2.2 Model Formulation

2.3 Parameter Estimation

2.4 Mathematical Models

2.5 Numerical Methods

Exercises

References

Chapter 3: Numerical Methods: An Introduction

3.1 Direct Integration

3.2 Finite Difference Methods

Exercises

References

Chapter 4: Teaching Mathematical Modeling in Teacher Education: Efforts and Results

4.1 Introduction

4.2 Theoretical Frameworks Connected to Mathematical Modeling

4.3 Mathematical Modeling Tasks

4.4 Conclusions

Exercises

References

Part II: Mathematical Modeling with Multidisciplinary Applications

Chapter 5: Industrial Mathematics with Applications

5.1 Industrial Mathematics

5.2 Numerical Simulation of Metallurgical Electrodes

5.3 Numerical Simulation of Pit Lake Water Quality

Exercises

References

Chapter 6: Binary and Ordinal Data Analysis in Economics: Modeling and Estimation

6.1 Introduction

6.2 Theoretical Foundations

6.3 Estimation

6.4 Applications

6.5 Conclusions

Exercises

References

Chapter 7: Inverse Problems in ODEs

7.1 Banach’s Fixed Point Theorem & The Collage Theorem

7.2 Existence-Uniqueness of Solutions to Initial Value Problems

7.3 Solving Inverse Problems for ODEs

Exercises

References

Chapter 8: Estimation of Model Parameters

8.1 Estimation is an Inverse Problem

8.2 The Multivariate Normal Distribution

8.3 Model of Observations

8.4 Estimation

8.5 Conclusion

Exercises

References

Chapter 9: Linear and Nonlinear Parabolic Partial Differential Equations in Financial Engineering

9.1 Financial Derivatives

9.2 Motivation for a Model for the Price of Stocks

9.3 Stock Prices Involving the Wiener Process

9.4 Connection Between the Wiener Process and PDEs

9.5 The Black-Scholes-Merton Equation

9.6 Solution of the Black-Scholes-Merton Equation

9.7 Free Boundary-Value Problems

9.8 The Hamilton-Jacobi-Bellman Equation

9.9 Numerical Methods

9.10 Conclusion

Exercises

References

Chapter 10: Decision Modeling in Supply Chain Management

10.1 Introduction to Decision Modeling

10.2 Mathematical Programming Models

10.3 Introduction of Supply Chain Management

10.4 Applications in Supply Chain Management

10.5 Summary

Exercises

References

Chapter 11: Modeling Temperature for Pricing Weather Derivatives

11.1 Introduction

11.2 Stochastic Temperature Modeling

11.3 Continuous-Time Autoregressive Processes

11.4 Pricing of Temperature Futures Contracts

Exercises

References

Chapter 12: Decision Theory under Risk and Applications in Social Sciences: I. Individual Decision Making

12.1 Introduction

12.2 The Fundamental Framework

12.3 A Brief Introduction to Theory of Choice

12.4 Collective Choice

12.5 Preferences Under Uncertainty

12.6 Decisions Over Time

12.7 The Problem of Aggregation

12.8 Conclusion

Exercises

References

Chapter 13: Fractals, with Applications to Signal and Image Modeling307

13.1 Iterated Function Systems

13.2 Fractal Dimension

13.3 More on the Definition of Iterated Function System

13.4 The Chaos Game

13.5 An Application to Image Analysis

References

Chapter 14: Efficient Numerical Methods for Singularly Perturbed Differential Equations

14.1 Introduction

14.2 Characterization of SPPs

14.3 Numerical Approximate Solution

14.4 SPPs Arising in Chemical Reactor Theory

14.5 Layer-Adapted Nonuniform Meshes

References

Part III: Advanced Modeling Topics

Chapter 15: Fractional Calculus and its Applications

15.1 Introduction

15.2 Fractional Calculus Fundamentals

15.3 Fractional-Order Systems and Controllers

15.4 Stability of Fractional-Order Systems

15.5 Applications of Fractional Calculus

Exercises

References

Chapter 16: The Goal Programming Model: Theory and Applications

16.1 Multi-Criteria Decision Aid

16.2 The Goal Programming Model

16.3 Scenario-based Goal Programming

16.4 Applications

Exercises

References

Chapter 17: Decision Theory under Risk and Applications in Social Sciences: II. Game Theory

17.1 Introduction

17.2 Best Replies and Nash Equilibria

17.3 Mixed Strategies and Minimax

17.4 Nash Equilibria and Conservative Strategies

17.5 Zero-Sum Games and the Minimax Theorem

17.6 Nash Equilibria for Mixed Strategies

17.7 Cooperative Games

17.8 Conclusion

Exercises

References

Chapter 18: Control Problems on Differential Equations

18.1 Introduction

18.2 Ordinary Differential Equations

18.3 Partial Differential Equations

Exercises

References

Chapter 19: Markov-Jump Stochastic Models for Tropical Convection

19.1 Introduction

19.2 Random Numbers: Theory and Simulations

19.3 Markov Chains and Birth-Death Processes

19.4 A Birth-Death Process for Convective Inhibition

19.5 A Birth-Death Process for Cloud-Cloud Interactions

19.6 Further Reading

Exercises

References

Problem Solutions

Index

Mathematical Modeling with Multidisciplinary Applications

Copyright © 2013 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Yang, Xin-She. Mathematical modeling with multidisciplinary applications / Xin-She Yang.pages cm Includes bibliographical references and index. ISBN 978-1-118-29441-3 1. Differential equations. 2. Mathematical models. I. Title. QA371.Y28 2013 510.1’1—dc23 2012020899

LIST OF FIGURES

PREFACE

Mathematical modeling is a multidisciplinary endeavor that applies mathematical techniques to study real-world phenomena such as physical, chemical, biological, and economical processes. The quantities of a process of interest are often expressed as variables, while their interactions are often expressed as mathematical relationships or model equations, based on fundamental physical laws such as mass and energy conservation. Such mathematical models can be partial differential equations (PDEs), statistical relationships, or rule-based descriptions, though PDEs are mostly widely used.

Once of the main objectives of mathematical modeling is to model the process and mechanism of interest accurately so as to gain insight and make reasonably accurate predictions. This is a challenging, multidisciplinary task. Often, modeling is an interactive, iterative, time-consuming process. It is rarely the case that a first simple mathematical model will work well; more often, a modeler has to construct a series of mathematical models based on further assumptions, simplifications, adjustments, and improvement so that the revised/improved model can provide better predictions than initial crude models.

Even with the right mathematical models or a right set of differential equations, the task could be even more challenging. First, most mathematical models are highly nonlinear, and their mathematical analysis is often intractable. Even with some simplified models, analysis is possible, but the mathematical techniques involved are still not straightforward. In most cases, no analytical solution or solutions of any closed form is possible. Secondly, some approximation techniques have to be employed to get some estimates to the true solutions. Approximation methods can be very diverse, though some common techniques such as asymptotic analysis, perturbation methods, and model reduction are often used. In most cases, mathematical analysis and approximations still do not provide sufficient information to construct the exact solutions of the mathematical model. Numerical methods are usually used to provide a fuller picture of the solution characteristics. Numerical methods are also a diverse subject. Solutions of PDEs can be achieved by finite difference methods, finite element methods, finite volume methods, boundary element methods, and spectral methods among others. These topics can fill several books in computational methods if they are described in detail. However, numerical methods are not the main focus of this book, though we will introduce the basics of the numerical methods in relevant chapters.

The aims of this book are twofold: model formulation and analysis, and multidisciplinary applications. We will mainly focus on how to formulate mathematical models for a given process or phenomenon. For a given problem of interest, we will show to build a workable mathematical model, then we will show to do mathematical analysis to obtain solutions (analytical, approximation, or numerical). Another emphasis will be on the diverse mathematical models arisen from multidisciplinary applications such as physics, chemistry, climate, environment, finance, and economics. Though applications are multidisciplinary, key mathematical equations can be the same for different processes. For example, a parabolic PDE can be used to model mass diffusion, heat transfer, reaction-diffusion, pattern formation, and many other phenomena. Similarly, a random walk model can also be used to describe diffusion, search optimization, option pricing, and random samplings in Monte Carlo methods. Therefore, we will demonstrate the above key features throughout this book.

The basic requirement for this book is the good knowledge of basic calculus and mathematical foundations at university level. However, we will briefly review the key concepts of calculus and partial differential equations as well as the fundamental nature of mathematical modelling in the first few chapters. This makes it possible for readers to read all the relevant chapters without much difficulty.

This book strives to provide diverse coverage of multidisciplinary applications with a major focus on mathematical modeling. It divides into three parts. Part I reviews the fundamental of mathematics required for this book. Part II provides the basics of mathematical modeling and numerical methods. Part III covers a diverse range of multidisciplinary applications. Due to the multidisciplinary nature of this book, contributed by multiple authors who are leading experts in their fields, we have strived to make all chapters self-contained with enough background information and further reading materials, and consequently some chapters are more suitable for advanced graduates. A major advantage of this diverse coverage is that readers can choose topics and chapters of their own interest, while skipping some chapters without interrupting the flow and main scheme of modeling and applications. We hope to provide a solid foundation for readers to pursue further studies and research in their chosen area. The other advantage of this book is that all chapters are provided with exercises and answers so that readers can consolidate what they have learned. Thus, this book serves well as a textbook or reference for mathematical modeling courses as well as for self-study.

XIN-SHE YANG

Cambridge and London, UKDecember, 2012

ACKNOWLEDGMENTS

I would like to thank all contributing authors for their enthusiastic support for this book. Without their professional contributions, this book would not be possible.

I also would like to thank my Editor, Susanne Steitz-Filler, Associate Editor, Jacqueline Palmieri, Production Editor, Melissa Yanuzzi, Copyeditor, Liz Belmont, and staff at Wiley for their help and professionalism. I also thank my students, Aman Atak, Osaseri O. I. Guobadia and Qichen Xu, at Cambridge University for their help in proofreading some chapters of this book.

Last but not least, I thank my wife and son for their support and help.

X. S. Y.

Editor and Contributors

Editor

Xin-She Yang School of Science and Technology, Middlesex University, United Kingdom. ([email protected])

Contributors

Belaid Aouni School of Commerce and Administration, Faculty of Management, Laurentian University, Sudbury, Ontario, Canada. ([email protected])

Fred Espen Benth Center of Mathematics for Applications, University of Oslo, Blindern, Oslo, Norway. ([email protected])

Alfredo Bermúdez Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, Spain. ([email protected])

L. A. Boukas Department of Information and Communication Systems Engineering, University of the Aegean, Greece.

Cinzia Colapinto Department of Management, Ca’ Foscari University of Venice, San Giobbe Cannaregio, Italy. ([email protected])

Luz M. García García Instituto Español de Oceanografía, Spain.

Ivan Jeliazkov Department of Economics, University of California, Irvine, 3175 Social Science Plaza A, Irvine, CA, USA. ([email protected])

Boualem Khouider Mathematics and Statistics University of Victoria, Victoria, B.C., Canada. ([email protected])

Herb Kunze Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada. ([email protected])

Davide La Torre Department of Economics, Business and Statistics, University of Milan, via Conservatorio, Milan, Italy. ([email protected])

Thomas Lingefjärd Department of pedagogical, curricular and professional studies, University of Gothenburg, Gothenburg, Sweden. ([email protected])

S. Natesan Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, India. ([email protected])

E. V. Petracou Department of Geography, University of the Aegean, Greece.

Ivo Petráš Technical University of Kosice, Faculty of BERG, URaIVP, Kosice, Slovak Republic. ([email protected])

Robert Piché Department of Mathematics, Tampere University of Technology, Tampere, Finland. ([email protected])

Mohammad Arshad Rahman Department of Economics, University of California, Irvine, CA, USA.

Huajun Tang Faculty of Management and Administration, Macau University of Science and Technology, Macau. ([email protected])

K. I. Vasileiadis Laboratory for Financial and Actuarial Mathematics, Department of Statistics and Actuarial − Financial Mathematics, University of the Aegean, Greece.

S. Z. Xanthopoulos Laboratory for Financial and Actuarial Mathematics, Department of Statistics and Actuarial − Financial Mathematics, University of the Aegean, Greece.

Xin-She Yang School of Science and Technology, Middlesex University, London, UK. ([email protected])

A. N. Yannacopoulos Department of Statistics, Athens University of Economics and Business, Greece. ([email protected])

Chuang Zheng School of Mathematical Science, Beijing Normal University, Beijing, China. ([email protected])

PART I

INTRODUCTION AND FOUNDATIONS

CHAPTER 1

DIFFERENTIAL EQUATIONS

XIN-SHE YANG

School of Science and Technology, Middlesex University, London, UK

Also Mathematics and Scientific Computing, National Physical Laboratory, UK

The main requirement for this book is the basic knowledge of calculus and statistics as covered by most undergraduate courses in engineering and science subjects. However, we will provide a brief review of mathematical foundations in the first few chapters so as to help readers to refresh some of the most important concepts.

Most mathematical models in physics, chemistry, biology and many other applications are formulated in terms of differential equations. If the variables or quantities (such as velocity, temperature, pressure) change with other independent variables such as spatial coordinates and time, their relationship can in general be written as a differential equation or even a set of differential equations.

1.1 ORDINARY DIFFERENTIAL EQUATIONS

An ordinary differential equation (ODE) is a relationship between a function y(x) of an independent variable x and its derivatives y′, y″, …, y(n). It can be written in a generic form

(1.1)

(1.2)

is a first-order ODE, and the following equation of Euler-type

(1.3)

is a second order. The degree of an equation is defined as the power to which the highest derivative occurs. Therefore, both the Riccati equation and the Euler equation are of the first degree.

An equation is called linear if it can be arranged into the form

(1.4)

To find a solution of an ordinary differential equation is not always easy, and it is usually very complicated for nonlinear equations. Even for linear equations, solutions can be found in a straightforward way for only a few simple cases. The solution of a differential equation generally falls into three types: closed form, series form and integral form. A closed form solution is the type of solution that can be expressed in terms of elementary functions and some arbitrary constants. Series solutions are the ones that can be expressed in terms of a series when a closed form is not possible for certain types of equations. The integral form of solutions or quadrature is sometimes the only form of solution that is possible. If all these forms are not possible, the alternatives are to use approximate and numerical solutions.