數(shù)學(xué)建模

前言

　　To facilitate an early initiation of the modeling experience， the first edition of this text was designed to be taught concurrently or immediately after an introductory business or engineering calculus course. In the second edition, we added chapters treating discrete dynamical systems, linear programming and numerical search methods, and an introduction to probabilistic modeling. Additionally, we expanded our introduction of simulation. In the third edition we included solution methods for some simple dynamical systems to reveal their long-term behavior. We also added basic numerical solution methods to the chapters covering modeling with differential equations. In this edition, we have added a new chapter to address modeling using graph theory. Graph theory is an area of burgeoning interest tor modeling contemporary scenarios. Our chapter is intended to introduce graph theory from a modeling perspective and encourage students to pursue the subject in greater detail. We have also added two new sections to the chapter on modeling with a differential equation: discussions of separation of variables and linear equations. Many of our readers had expressed a desire that analytic solutions to first-order differential equations be included as part of their modeling course. The text has been reorganized into two parts: Part One,Discrete Modeling (Chapters 1-9), and Part Two, Continuous Modeling (Chapters 10-13).　　This organizational structure allows for teaching an entire modeling course that is based on Part One and does not require the calculus. Part Two then addresses continuous models based on optimization and differential equations that can be presented concurrently with freshman calculus. The text gives students an opportunity to cover all phases of the mathematical modeling process. The new CD-ROM accompanying the text contains software, additional modeling scenarios and projects, and a link to past problems from the Mathematical Contest in Modeling. We thank Sol Garfunkel and the COMAP staff for their support of modeling activities that we refer to under Resource Materials below. .　　Goals and Orientation　　The course continues to be a bridge between the study of mathematics and the applications of mathematics to various fields. The course affords the student an early opportunity to see how the pieces of an applied problem fit together. The student investigates meaningful and practical problems chosen from common experiences encompassing many academic disciplines, including the mathematical sciences, operations research, engineering, and the management and life sciences.　　This text provides an introduction to the entire modeling process. Students will have opportunities to practice the following facets of modeling and enhance their problem-solving capabilities:　　1. Creative and Empirical Model Construction: Given a real-world scenario, the student learns to identify a problem, make assumptions and collect data, propose a model, test the assumptions, refine the model as necessary, fit the model to data if appropriate, and analyze the underlying mathematical structure of the model to appraise the sensitivity of the conclusions when the assumptions are not precisely met.　　2. Model Analysis: Given a model, the student learns to work backward to uncover the implicit underlying assumptions, assess critically how well those assumptions fit the scenario at hand, and estimate the sensitivity of the conclusions when the assumptions are not precisely met.　　3. Model Research: The student investigates a specific area to gain a deeper understanding of some behavior and learns to use what has already been created or discovered.　　Student Background and Course Content　　Because our desire is to initiate the modeling experience as early as possible in the students program, the only prerequisite for Chapters 10, 11, and 12 is a basic understanding of single-variable differential and integral calculus. Although some unfamiliar mathematical ideas are taught as part of the modeling process, the emphasis is on using mathematics that the students already know after completing high school. This is especially true in Part One.The modeling course will then motivate students to study the more advanced courses such as linear algebra, differential equations, optimization and linear programming, numerical analysis, probability, and statistics. The power and utility of these subjects are intimated throughout the text.　　Further, the scenarios and problems in the text are not designed for the application of a particular mathematical technique. Instead, they demand thoughtful ingenuity in us-ing fundamental concepts to find reasonable solutions to "open-ended" problems. Certain mathematical techniques (such as Monte Carlo simulation, curve fitting, and dimensional analysis) are presented because often they are not formally covered at the undergraduate level. Instructors should find great flexibility in adapting the text to meet the particular needs of students through the problem assignments and student projects. We have used this material to teach courses to both undergraduate and graduate students——and even as a basis for faculty seminars.

內(nèi)容概要

數(shù)學(xué)建模是用數(shù)學(xué)方法解決各種實(shí)際問(wèn)題的橋梁。本書(shū)分離散建模和連續(xù)建模兩部分介紹了整個(gè)建模過(guò)程的原理，通過(guò)本書(shū)的學(xué)習(xí)，學(xué)生將有機(jī)會(huì)在創(chuàng)造性模型和經(jīng)驗(yàn)?zāi)Ｐ偷臉?gòu)建、模型分析以及模型研究方面進(jìn)行實(shí)踐，增強(qiáng)解決問(wèn)題的能力?！　”緯?shū)特點(diǎn)　　論證了離散動(dòng)力系統(tǒng)、離散優(yōu)化等技術(shù)對(duì)現(xiàn)代應(yīng)用數(shù)學(xué)發(fā)展的促進(jìn)作用。　　在創(chuàng)造性模型和經(jīng)驗(yàn)?zāi)Ｐ偷臉?gòu)建，模型分析以及模型研究中融入個(gè)人項(xiàng)目和小組項(xiàng)目，并且包含大量的例子和習(xí)題?！　”景嫘略隽岁P(guān)于圖論建模的新的一章，從數(shù)學(xué)建模的角度介紹圖論并鼓勵(lì)學(xué)生對(duì)圖論進(jìn)行更深入的學(xué)習(xí)?！　‰S書(shū)光盤(pán)中包含大學(xué)數(shù)學(xué)應(yīng)用教學(xué)單元(UMAP)，過(guò)去的建模競(jìng)賽試題，充滿(mǎn)活力的跨學(xué)科應(yīng)用研究課題，利用電子表格(Excel)、計(jì)算機(jī)代數(shù)系統(tǒng)(Maple、Mathematica、Matlab)以及圖形計(jì)算器(TI)等技術(shù)的廣泛的例子，在實(shí)驗(yàn)室環(huán)境下為學(xué)生設(shè)計(jì)的例子和習(xí)題。

作者簡(jiǎn)介

Frank R．Giordano畢業(yè)于美國(guó)西點(diǎn)軍校，曾任西點(diǎn)軍校數(shù)學(xué)系系主任，現(xiàn)為美國(guó)海軍研究生院教授，多年來(lái)一直是美國(guó)大學(xué)生數(shù)學(xué)建模競(jìng)賽的主要組織者，也是美國(guó)大學(xué)生數(shù)學(xué)建模競(jìng)賽組委會(huì)主任。

書(shū)籍目錄

1 Modeling Change 　Introduction 　1.1 Modeling Change with Difference Equations 　1.2 Approximating Change with DifferenceEquations 　1.3 Solutions to Dynamical Systems 　1.4 Systems of Difference Equations 2 The Modeling Process, Proportionality,and Geometric Similarity 　2.1 Mathematical Models 　2.2 Modeling Using Proportionality 　2.3 Modeling Using Geometric Similarity 　2.4 Automobile Gasoline Mileage 　2.5 Body Weight and Height, Strength and Agility 3 Model Fitting 　3.1 Fitting Models to Data Graphically 　3.2 Analytic Methods of Model Fitting 　3.3 Applying the Least-Squares Criterion 　3.4 Choosing a Best Model 4 Experimental Modeling 　4.1 Harvesting in the Chesapeake Bay and Other One-Term Models  　4.2 High-Order Polynomial Models　4.3 Smoothing: Low-Order Polynomial Models　4.4 Cubic Spline Models5　Simulation Modeling　Introduction　5.1 Simulating Deterministic Behavior: Area Under a Curve　5.2 Generating Random Numbers  　5.3 Simulating Probabilistic Behavior  　5.4 Inventory Model: Gasoline and Consumer Demand  　5.5 Queuing Models  6　Discrete Probabilistic Modeling7 Optimization of Discrete Models8 Modeling Using Graph Theory9 Dimensional Analysis and Similitude10 Graphs of Functions as Models11 Modeling with a Differential Equation12 Modeling with Systems of Differential Equations13 Optimization of Continuous Models

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