Mathematical Modeling 4th Mark M. Meerschaert 2013.pdf

发布时间：2022-06-08 发布人：admin 分类：说明书资料大小：3.74M 资料格式：pdf 举报版权申诉

anyway_hi-9415021-4744302542863811129.pdf-第1页.png

第1页 / 共368页

anyway_hi-9415021-4744302542863811129.pdf-第2页.png

第2页 / 共368页

anyway_hi-9415021-4744302542863811129.pdf-第3页.png

第3页 / 共368页

anyway_hi-9415021-4744302542863811129.pdf-第4页.png

第4页 / 共368页

anyway_hi-9415021-4744302542863811129.pdf-第5页.png

第5页 / 共368页

anyway_hi-9415021-4744302542863811129.pdf-第6页.png

第6页 / 共368页

anyway_hi-9415021-4744302542863811129.pdf-第7页.png

第7页 / 共368页

anyway_hi-9415021-4744302542863811129.pdf-第8页.png

第8页 / 共368页

Cover

Title Page

Preface

Part I: Optimization Models

1 One Variable Optimization

2 Multivariable Optimization

3 Computational Methods for Optimization

Part II: Dynamic Models

4 Introduction to Dynamic Models

5 Analysis of Dynamic Models

6 Simulation of Dynamic Models

Part III: Probability Models

7 Introduction to Probability Models

8 Stochastic Models

9 Simulation of Probability Models

Afterword

Index

Mathematical Modeling Fourth Edition

Mathematical Modeling Fourth Edition Mark M. Meerschaert Department of Statistics and Probability Michigan State University A430 Wells Hall East Lansing, MI 48824-1027 USA AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an Imprint of Elsevier

Academic Press is an imprint of Elsevier 225 Wyman Street, Waltham, MA 02451, USA 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands F ourth edition 2013 Copyright © 2013, Elsevier Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or other- wise without the prior written permission of the publisher. Permissions may be sought directly from Elsevier Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: permissions@elsevier.com. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material. ’s Notices No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data Meerschaert, Mark M., 1955- Mathematical modeling / Mark M. Meerschaert. – 4th ed. p. cm. Includes index. ISBN 978-0-12-386912-8 1. Mathematical models. I. Title. QA401.M48 2013 511'.8– dc23 2012037454 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-386912-8 For information on all Academic Press publications visit our web site at store.elsevier.com Printed and bound in USA 13 14 15 16 17 10 9 8 7 6 5 4 3 2 1

Preface Mathematical modeling is the link between mathematics and the rest of the world. You ask a question. You think a bit, and then you reﬁne the question, phrasing it in precise mathematical terms. Once the question becomes a math- ematics question, you use mathematics to ﬁnd an answer. Then ﬁnally (and this is the part that too many people forget), you have to reverse the process, translating the mathematical solution back into a comprehensible, no–nonsense answer to the original question. Some people are ﬂuent in English, and some people are ﬂuent in calculus. We have plenty of each. We need more people who are ﬂuent in both languages and are willing and able to translate. These are the people who will be inﬂuential in solving the problems of the future. This text, which is intended to serve as a general introduction to the area of mathematical modeling, is aimed at advanced undergraduate or beginning graduate students in mathematics and closely related ﬁelds. Formal prerequi- sites consist of the usual freshman–sophomore sequence in mathematics, includ- ing one–variable calculus, multivariable calculus, linear algebra, and diﬀerential equations. Prior exposure to computing and probability and statistics is useful, but is not required. Unlike some textbooks that focus on one kind of mathematical model, this book covers the broad spectrum of modeling problems, from optimization to dynamical systems to stochastic processes. Unlike some other textbooks that assume knowledge of only a semester of calculus, this book challenges students to use all of the mathematics they know (because that is what it takes to solve real problems). The overwhelming majority of mathematical models fall into one of three categories: optimization models; dynamic models; and probability models. The type of model used in a real application might be dictated by the problem at hand, but more often, it is a matter of choice. In many instances, more than one type of model will be used. For example, a large Monte Carlo simulation model may be used in conjunction with a smaller, more tractable deterministic dynamic model based on expected values. This book is organized into three parts, corresponding to the three main categories of mathematical models. We begin with optimization models. A ﬁve- step method for mathematical modeling is introduced in Section 1 of Chapter 1, in the context of one–variable optimization problems. The remainder of the ﬁrst chapter is an introduction to sensitivity analysis and robustness. These vii

viii PREFACE fundamentals of mathematical modeling are used in a consistent way throughout the rest of the book. Exercises at the end of each chapter require students to master them as well. Chapter 2, on multivariable optimization, introduces decision variables, feasible and optimal solutions, and constraints. A review of the method of Lagrange multipliers is provided for the beneﬁt of those students who were not exposed to this important technique in multivariable calculus. In the section on sensitivity analysis for problems with constraints, we learn that Lagrange multipliers represent shadow prices (some authors call them dual variables). This sets the stage for our discussion of linear programming later in Chapter 3. At the end of Chapter 3 is a section on discrete optimization that was added in the second edition. Here we give a practical introduction to integer programming using the branch–and–bound method. We also explore the connection between linear and integer programming problems, which allows an earlier introduction to the important issue of discrete versus continuous models. Chapter 3 covers some important computational techniques, including Newton’s method in one and several variables, and linear and integer programming. In the next part of the book, on dynamic models, students are introduced to the concepts of state and equilibrium. Later discussions of state space, state variables, and equilibrium for stochastic processes are intimately connected to what is done here. Nonlinear dynamical systems in both discrete and continuous time are covered. There is very little emphasis on exact analytical solutions in this part of the book, since most of these models admit no analytic solution. At the end of Chapter 6 is a section on chaos and fractals that was added in the second edition. We use both analytic and simulation methods to explore the behavior of discrete and continuous dynamic models, to understand how they can become chaotic under certain conditions. This section provides a practical and accessible introduction to the subject. Students gain experience with sensitive dependence to initial conditions, period doubling, and strange attractors that are fractal sets. Most important, these mathematical curiosities emerge from the study of real–world problems. Finally, in the last part of the book, we introduce probability models. No prior knowledge of probability is assumed. Instead we build upon the material in the ﬁrst two parts of the book, to introduce probability in a natural and intuitive way as it relates to real–world problems. Chapter 7 introduces the basic notions of random variables, probability distributions, the strong law of large numbers, and the central limit theorem. At the end of Chapter 7, Introduction to Prob- ability Models, is a section on diﬀusion, which was added in the third edition. Here we give a gentle introduction to partial diﬀerential equations by focusing on the diﬀusion equation. We provide a simple derivation of the point source solution to this partial diﬀerential equation, using Fourier transforms, to arrive at the normal density. Then we connect the diﬀusion model to the central limit theorem introduced in the previous Section 7.3, Introduction to Statistics. This new section on diﬀusion grew out of a class taught at the University of Nevada for beginning graduate students in the earth sciences. The applications are to contaminant migration in the atmosphere and ground water. Chapter 8 covers the basic models of stochastic processes, including Markov chains, Markov pro-

PREFACE ix cesses, and linear regression. At the end of Chapter 8, Stochastic Models, a new section on time series was added in the third edition. This section also serves as an introduction to multivariate regression models with more than one predic- tor. As a natural follow-up to the discussion in Section 8.3, Linear Regression, the new section on time series introduces the important idea of correlation. It also shows how to recognize correlated variables and include the dependence structure in a time series model. The discussion is focused on autoregressive models, since these are the most generally useful time series models. They are also the most convenient, in that they can be handled using widely available linear regression software. For the beneﬁt of students with access to a statistical package, this section illustrates the proper application and interpretation of ad- vanced methods including autocorrelation plots and sequential sums of squares. However, the entire section can also be covered using only a basic implemen- tation of regression that allows multiple predictors and outputs the two basic measures: R2 and the residual standard deviation s. This can all be done with a good spreadsheet or hand calculator. Chapter 9 treats simulation methods for stochastic models. The Monte Carlo method is introduced, and Markov property is applied to create eﬃcient simulation algorithms. Analytic simula- tion methods are also explored, and compared to the Monte Carlo method. In the fourth edition, two new sections were added to the end of Chapter 9. The ﬁrst new section covers particle tracking methods, for solving partial diﬀerential equations via Monte carlo simulation of the underlying stochastic process. The ﬁnal section of the book introduces fractional calculus in the context of anoma- lous diﬀusion. The fractional diﬀusion equation is solved by particle tracking, and applied to a problem in ground water pollution. This section ties together the concepts of fractals, fractional derivatives, and probability distributions with heavy tails. Each chapter in this book is followed by a set of challenging exercises. These exercises require signiﬁcant eﬀort, as well as a certain amount of creativity, on the part of the student. I did not invent the problems in this book. They are real problems. They were not designed to illustrate the use of any particular mathematical technique. Quite the opposite. We will occasionally go over some new mathematical techniques in this book because the problem demands it. I was determined that there would be no place in this book where a student could look up and ask, “What is all of this for?” Although typically oversim- pliﬁed or grossly unrealistic, story problems embody the fundamental challenge in applying mathematics to solve real problems. For most students, story prob- lems present plenty of challenge. This book teaches students how to solve story problems. There is a general method that can be applied successfully by any reasonably capable student to solve any story problem. It appears in Chapter 1, Section 1. This same general method is applied to problems of all kinds throughout the text. Following the exercises in each chapter is a list of suggestions for further reading. This list includes references to a number of UMAP modules in applied mathematics that are relevant to the material in the chapter. UMAP modules can provide interesting supplements to the material in the text, or extra credit

x PREFACE projects. All of the UMAP modules are available at a nominal cost from the Consortium for Mathematics and Its Applications (www.comap.com). One of the major themes of this book is the use of appropriate technology for solving mathematical problems. Computer algebra systems, graphics, and numerical methods all have their place in mathematics. Many students have not had an adequate introduction to these tools. In this course we introduce modern technology in context. Students are motivated to learn because the new technology provides a more convenient way to solve real–world problems. Computer algebra systems and 2–D graphics are useful throughout the course. Some 3–D graphics are used in Chapters 2 and 3 in the sections on multivari- able optimization. Students who have already been introduced to 3–D graphics should be encouraged to use what they know. Numerical methods covered in the text include, among others, Newton’s method, linear programming, the Euler method, linear regression, and Monte Carlo simulation. The text contains numerous computer–generated graphs, along with instruc- tion on the appropriate use of graphing utilities in mathematics. Computer al- gebra systems are used extensively in those chapters where signiﬁcant algebraic calculation is required. The text includes computer output from the computer algebra systems Maple and Mathematica in Chapters 2, 4, 5, and 8. The chap- ters on computational techniques (Chapters 3, 6, and 9) discuss the appropriate use of numerical algorithms to solve problems that admit no analytic solution. Sections 3.3 and 3.4 on linear-integer programming include computer output from the popular linear programming package LINDO. Sections 8.3 and 8.4 on linear regression and time series include output from the commonly used statistical package Minitab. Students need to be provided with access to appropriate technology in or- der to take full advantage of this textbook. We have tried to make it easy for instructors to use this textbook at their own institution, whatever their situa- tion. Some will have the means to provide students with access to sophisticated computing facilities, while others will have to make do with less. The bare ne- cessities include: (1) a software utility to draw 2–D graphs; and (2) a machine on which students can execute a few simple numerical algorithms. All of this can be done, for example, with a computer spreadsheet program or a programmable graphics calculator. The ideal situation would be to provide all students access to a good computer algebra system, a linear programming package, and a statis- tical computing package. The following is a partial list of appropriate software packages that can be used in conjunction with this textbook.

分享到：

赞收藏

资料库

Mathematical Modeling 4th Mark M. Meerschaert 2013.pdf

相关推荐

课程资源

热门标签

最新资料