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数学

常微分方程及其应用——理论与模型

作者：周宇虹，罗建书

出版社：科学出版社出版时间：2022-01-01

开本： B5 页数： 224

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常微分方程及其应用——理论与模型版权信息

ISBN：9787030301253
条形码：9787030301253 ; 978-7-03-030125-3
装帧：一般胶版纸
册数：暂无
重量：暂无
所属分类：
自然科学
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数学

常微分方程及其应用——理论与模型内容简介

本书是常微分方程课程的英文教材，是作者结合多年的双语教学经验编写而成.本书共5章，包括一阶线性微分方程，高阶线性微分方程，线性微分方程组，Laplace变换及其在微分方程求解中的应用，微分方程的稳定性理论.书中配有大量的实用实例和用Matlab软件绘制的微分方程解的相图，并介绍了绘制相图的程序.本书可作为高等院校理工科非数学专业的本科双语教材,也可供有关专业的研究生、教师和广大科技人员参考.

常微分方程及其应用——理论与模型目录

Contents
Chapter 1 First-order Dierential Equations 1
1.1 Introduction 1
Exercise 1.1 7
1.2 First-order Linear Dierential Equations 8
1.2.1 First-order Homogeneous Linear Dierential Equations 8
1.2.2 First-order Nonhomogeneous Linear Dierential Equations 11
1.2.3 Bernoulli Equations 16
Exercise 1.2 18
1.3 Separable Equations 19
1.3.1 Separable Equations 19
1.3.2 Homogeneous Equations 23
Exercise 1.3 26
1.4 Applications 27
Module 1 The Spread of Technological Innovations 27
Module 2 The Van Meegeren Art Forgeries 30
1.5 Exact Equations 35
1.5.1 Criterion for Exactness 35
1.5.2 Integrating Factor 39
Exercise 1.5 42
1.6 Existence and Uniqueness of Solutions 43
Exercise 1.6 50
Chapter 2 Second-order Dierential Equations 51
2.1 General Solutions of Homogeneous Second-order Linear Equations 51
Exercise 2.1 59
2.2 Homogeneous Second-order Linear Equations with Constant Coe±cients 60
2.2.1 The Characteristic Equation Has Distinct Real Roots 61
2.2.2 The Characteristic Equation Has Repeated Roots 62
2.2.3 The Characteristic Equation Has Complex Conjugate Roots 63
Exercise 2.2 65
2.3 Nonhomogeneous Second-order Linear Equations 66
2.3.1 Structure of General Solutions 66
2.3.2 Method of Variation of Parameters 68
2.3.3 Methods for Some Special Form of the Nonhomogeneous Term g(t) 70
Exercise 2.3 76
2.4 Applications 77
Module 1 An Atomic Waste Disposal Problem 77
Module 2 Mechanical Vibrations 82
Chapter 3 Linear Systems of Dierential Equations 90
3.1 Basic Concepts and Theorems 90
Exercise 3.1 98
3.2 The Eigenvalue-Eigenvector Method of Finding Solutions 99
3.2.1 The Characteristic Polynomial of A Has n Distinct Real Eigenvalues 100
3.2.2 The Characteristic Polynomial of A Has Complex Eigenvalues 101
3.2.3 The Characteristic Polynomial of A Has Equal Eigenvalues 104
Exercise 3.2 108
3.3 Fundamental Matrix Solution; Matrix-valued Exponential Function eAt 109
Exercise 3.3 113
3.4 Nonhomogeneous Equations; Variation of Parameters 115
Exercise 3.4 120
3.5 Applications 121
Module 1 The Principle of Competitive Exclusion in Population Biology 121
Module 2 A Model for the Blood Glucose Regular System 127
Chapter 4 Laplace Transforms and Their Applications in Solving Dierential Equations 136
4.1 Laplace Transforms 136
Exercise 4.1 138
4.2 Properties of Laplace Transforms 138
Exercise 4.2 145
4.3 Inverse Laplace Transforms 146
Exercise 4.3 148
4.4 Solving Dierential Equations by Laplace Transforms 148
4.4.1 The Right-Hand Side of the Dierential Equation is Discontinuous 152
4.4.2 The Right-Hand Side of Dierential Equation is an Impulsive Function 154
Exercise 4.4 156
4.5 Solving Systems of Dierential Equations by Laplace Transforms 157
Exercise 4.5 159
Chapter 5 Introduction to the Stability Theory 161
5.1 Introduction 161
Exercise 5.1 164
5.2 Stability of the Solutions of Linear System 164
Exercise 5.2 171
5.3 Geometrical Characteristics of Solutions of the System of Dierential Equations 173
5.3.1 Phase Space and Direction Field 173
5.3.2 Geometric Characteristics of the Orbits near a Singular Point 176
5.3.3 Stability of Singular Points 180
Exercise 5.3 183
5.4 Applications 183
Module 1 Volterra's Principle 183
Module 2 Mathematical Theories of War 188
Answers to Selected Exercises 196
References 209
附录软件包Iode简介 210

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常微分方程及其应用——理论与模型节选

Chapter 1 First-order Differen-tial Equations 1.1 Introduction Example 1.1.1 Dating of art works On May 29， 1945， H.A.Van Meegeren， a third rate Dutch painter[3]， was arrested on the charge of collaborating with the enemy for [4] the sale to Goering of a painting of famed 17th century Dutch painter. Van Meegeren refused to accept the charge and an- nounced， in his prison cell [5]， that he had never sold painting to Goering. He stated that all the questioned paintings[6] were his own works. To settle the question an international panel of dis- tinguished chemists， physicists and art historians was appointed to investigate the matter. The panel took X-rays of the paint- ings to determine whether other paintings were underneath those paintings， analyzed the pigments[7] (coloring materials) used in the paintings， and examined the paintings for certain signs of old age[8]. The panel of experts found traces of the modern pigment cobalt blue[9] in some paintings. In addition， they also detected phenoformaldehyde[10]， which was not discovered until the turn of the 19th century[11]， in several paintings. On the basis of these evidences Van Meegeren was convicted， of forgery[12]， on October 12， 1947 and sentenced to one year in prison[13]. Two months later， he died of a heart attack. However， many people refused to believe that the famed \Dis- ciples at Emmaus[14]" was a forgery[15]. In 1967， almost twenty years later， scientists at Carnegie Mellon University proved that the \Disciples at Emmaus" was indeed a forgery. The key to the dating of materials lies in the phenomenon of radioactivity[16] discovered at the turn of the 20th century by the physicist Rutherford and his colleagues. They showed that the atoms of certain \radioactive" elements are unstable[17] and that within a given time period a fixed proportion of the atoms spontaneously disintegrates to form atoms of a new element[18]. Rutherford also showed that the radioactivity of a substance is directly proportional to[19] the number of atoms of the substance present. Let N(t) denotes the number of atoms present at time t， then dN/dt， the number of atoms that disintegrate per unit time， is proportional to N， thus we have the following equation (1.1.1) where[20] constant ， is positive and is known as the decay con- stant of the substance[21]. Usually， we use half-life[22]， the time required for half of a given quantity of radioactive atoms to decay， to measure the rate of disintegration of a substance. Assume that N(t0) = N0， then we have the mathematical model for computing half-life (1.1.2) By evaluating the present disintegration rates of the radioac- tive pigments in Van Meegeren's questioned paintings， the experts concluded that the paintings \Disciples at Emmaus"， \Woman Reading Music[23]" and \Woman Playing Mandolin[24]" must be modern forgeries. Example 1.1.2 Detection of diabetes[25] Diabetes mellitus[26] is a disease of metabolism which is charac- terized by too much sugar in the blood and urine[27]. Glucose tolerance test[28] (GTT) is a commonly used method to diagnose the disease. In this test， the patient is asked to take a large dose of glucose after an overnight fast[29]. During the next three to five hours， several measurements of the concentration of glu- cose are made in the patient's blood， and these measurements are used in the diagnosis of diabetes. Unfortunately， there is no universally accepted criterion[30] exist for interpreting the results of a GTT. Different physicians[31] interpreting the results of a GTT may come up with[32] different diagnoses. Here is a case. A Rhode Island physician， after reviewing the results of a GTT， came up with a diagnosis of diabetes. But another physician de- clared the patient to be normal after reviewing the results of the same GTT. To settle the question， the results of the GTT were sent to a specialist in Boston. After examining these results， the specialist concluded that the patient was suffering from a pituitary tumor[33]. In the mid-1960's， Drs. Rosevear and Molnar of the Mayo Clinic and Drs. Ackerman and Gatewood of the University of Min- nesota discovered a fairly reliable criterion for interpreting the re- sults of a GTT. They constructed a model which could accurately describe the blood glucose regulatory system[34] during a glucose tolerance test and in which one or two parameters[35] would yield criteria for distinguishing normal individuals from mild diabetics and prediabetics[36]. The basic model is described analytically[37] by following sys- tem of equations[38] (1.1.3) (1.1.4) where G denotes the concentration of glucose in the blood and H denotes the concentration of the net hormonal[39] concentration. The function J(t) is the external rate at which the blood glucose concentration is being increased. Definition 1.1.1 An equation relating an unknown function， its derivatives[40] and independent

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