扫一扫
关注中图网
官方微博
本类五星书更多>
-
>
宇宙、量子和人类心灵
-
>
(精)BBC地球故事系列-星际旅行
-
>
从一到无穷大
-
>
图说相对论(32开平装)
-
>
一本有趣又有料的化学书
-
>
刘薰宇的数学三书:原来数学可以这样学全3册
-
>
光学零件制造工艺学
数学物理方程及相关分析工具简明教程(英文版) 版权信息
- ISBN：9787030706973
- 条形码：9787030706973 ; 978-7-03-070697-3
- 装帧：一般胶版纸
- 册数：暂无
- 重量：暂无
- 所属分类：>>
数学物理方程及相关分析工具简明教程(英文版) 本书特色
适读人群 ：力学、物理学、工程科学及相关专业有双语或全英文教学需求的本科生、研究生以及教师，需使用偏微分方程分析物理问题之科研人员该书在阐述一个知识点的时候，尽量从方法内在规律的角度提及该知识点在未来可能的应用情况。
数学物理方程及相关分析工具简明教程(英文版) 内容简介
本书旨在帮助理工科专业本科生初步形成对物理问题进行有效量化分析的能力。其主要内容包括如何基于偏微分方程对物理问题开展数学建模，以及如何利用进阶分析工具对二阶线性偏微分方程进行求解与分析。具体涉及的物理过程包括弦/薄膜振动过程、热传导过程、物质扩散过程以及静电平衡问题。所涉及的常用分析方法包括波传播法、积分变换法、分离变量法、格林函数法、变分原理等。所涉及的高等数学知识未涵盖到的进阶分析工具包括贝塞尔函数、傅里叶变换和卷积等。此外，本书还将探讨二阶线性偏微分方程的一些定性理论，例如各类二阶线性偏微分方程解的专享性问题等。在对上述物理问题与方法进行阐述之后，本书也将讨论二阶线性偏微分方程的分类原则，以期帮助学生从数学结构的视角更好地理解物理问题，也为在未来学习中会遇到更具专业特点偏微分方程之学生可以有所预备。
数学物理方程及相关分析工具简明教程(英文版) 目录
Contents
Part I Second-Order Linear Partial Differential Equations
1 The Wave Equation 3
1.1 Equation for String Vibration 4
1.1.1 Derivation of the Equation for String Vibration 4
1.1.2 Initial and Boundary Conditions 9
1.1.3 Terminology 11
1.2 D’Alembert’s Formula 12
1.2.1 D’Alembert’s Formula 13
1.2.2 Characteristics 15
1.2.3 The Case of a Semi-infinitely Long String 17
1.2.4 DuhamePs Principle 22
1.3 Method of Separation of Variables 24
1.3.1 Making Boundary Conditions Homogeneous 24
1.3.2 Method of Separation of Variables—Its Procedure 25
1.3.3 Physical Implications 31
1.3.4 Inhomogeneous Governing Equations 32
1.4 Wave Equation in Higher Dimensions 35
1.4.1 Small and Transverse Vibration of a Membrane 35
1.4.2 Definite Problems 40
1.4.3 Solutions for Cauchy Problems 41
1.5 Solution Properties 46
1.5.1 The Energy of a Vibrating Membrane 47
1.5.2 Solution Uniqueness of Problems for the Wave Equation 48
2 The Heat Equation 57
2.1 Modelling Heat Conduction 57
2.1.1 Derivation of the Heat Equation 57
2.1.2 Initial and Boundary Conditions 61
2.1.3 Physical Analogies 62
2.2 Method of Integral Transform 63
2.2.1 Convolution and Fourier Transform 64
2.2.2 Solution for Cauchy Problems 67
2.2.3 Solution Properties 70
2.2.4 Inhomogeneous Governing Equations 72
2.3 A Revisit to the Method of Separation of Variables 75
2.3.1 An Example with the Heat Equation 75
2.3.2 Sturm-Liouville System 81
2.3.3 Inhomogeneous Governing Equations 88
2.4 Solution Properties 89
2.4.1 Maximum Principle 89
2.4.2 Solution Uniqueness 91
2.4.3 Stability 92
3 Poisson’s Equation 99
3.1 Poisson’s Equation and Harmonic Equation 99
3.1.1 Definitions 99
3.1.2 Motivation from Physics 101
3.1.3 Boundary Conditions 106
3.2 Variational Principle 108
3.3 Harmonic Functions in Polar System 112
3.3.1 Laplace’s Equation in Polar System 113
3.3.2 Radial Solutions to Laplace’s Equation 115
3.4 The Method of Green^ Function 117
3.4.1 Green’s Formulae Related to Laplacian Operator 117
3.4.2 Fundamental Solution 118
3.4.3 Derivation of Green’s Function 120
3.4.4 Properties of Green’s Function 122
3.4.5 Problems for Poisson’s Equation 123
3.4.6 Final Remarks 125
3.5 Image Method for Electric Potentials 125
3.5.1 The Case in Three-Dimensional Half Space 126
3.5.2 The Case in a Spherical Domain 127
3.6 Solution Uniqueness 130
3.6.1 Mean-Value Formula 130
3.6.2 Maximum Principle 132
3.6.3 Strong Maximum Principle 133
3.6.4 Energy Method 134
4 Summary over Second-Order Linear Partial Differential Equation 141
4.1 Classification of Second-Order Linear Partial Differential Equations 141
4.1.1 Cases with Two Variables 141
4.1.2 Summary and Examples 148
4.1.3 Multivariable Situations 150
4.2 Topical Discussion 155
Part II Special Functions
5 Bessel Functions 165
5.1 Bessel Equation and Bessel Functions 165
5.1.1 Physical Motivation 165
5.1.2 Bessel Function of the First Kind 169
5.1.3 Bessel Function of the Second Kind 173
5.2 Properties of Bessel Functions 177
5.2.1 Recurrence Formulae 177
5.2.2 Zeros 179
5.2.3 Approximating Formula 181
5.2.4 Orthogonality 181
5.2.5 Analogies with Sinusoidal Functions 184
5.3 Solving PDE Problems with Bessel Functions 186
5.4 Generalisation 195
5.4.1 Hankel Functions 195
5.4.2 Modified Bessel Functions 195
6 Legendre Polynomial 199
6.1 Legendre Equation 199
6.2 Legendre Polynomial 204
6.2.1 Series Solution to the Legendre Equation 204
6.2.2 Legendre Polynomial 207
6.3 Properties of Legendre Polynomial 210
6.3.1 Rodrigues Formula 210
6.3.2 Key Properties at a Glance 210
6.3.3 Orthogonal Systems 212
6.3.4 Discussion on General Orthogonal Polynomials 218
6.4 Applications with Legendre Polynomials 220
6.4.1 Solving PDEs Defined in a Sphere 220
6.4.2 Legendre-Gauss Quadrature 222
6.5 The Associated Legendre Functions 225
7 Introduction of Hypergeometric Function 231
7.1 Commonalities of Bessel and Legendre Functions 231
7.2 Gauss Hypergeometric Function 234
7.2.1 Definition 234
7.2.2 Hypergeometric Differential Equation 235
7.2.3 Legendre Function and Legendre PolynomialA Revisit 239
7.3 Confluent Hypergeometric Function 241
7.3.1 Kummer Differential Equation and Its Solutions 241
7.3.2 Special Cases 242
7.4 Final Remarks on Hypergeometric Functions 244
Bibliography 247
Index 249
Part I Second-Order Linear Partial Differential Equations
1 The Wave Equation 3
1.1 Equation for String Vibration 4
1.1.1 Derivation of the Equation for String Vibration 4
1.1.2 Initial and Boundary Conditions 9
1.1.3 Terminology 11
1.2 D’Alembert’s Formula 12
1.2.1 D’Alembert’s Formula 13
1.2.2 Characteristics 15
1.2.3 The Case of a Semi-infinitely Long String 17
1.2.4 DuhamePs Principle 22
1.3 Method of Separation of Variables 24
1.3.1 Making Boundary Conditions Homogeneous 24
1.3.2 Method of Separation of Variables—Its Procedure 25
1.3.3 Physical Implications 31
1.3.4 Inhomogeneous Governing Equations 32
1.4 Wave Equation in Higher Dimensions 35
1.4.1 Small and Transverse Vibration of a Membrane 35
1.4.2 Definite Problems 40
1.4.3 Solutions for Cauchy Problems 41
1.5 Solution Properties 46
1.5.1 The Energy of a Vibrating Membrane 47
1.5.2 Solution Uniqueness of Problems for the Wave Equation 48
2 The Heat Equation 57
2.1 Modelling Heat Conduction 57
2.1.1 Derivation of the Heat Equation 57
2.1.2 Initial and Boundary Conditions 61
2.1.3 Physical Analogies 62
2.2 Method of Integral Transform 63
2.2.1 Convolution and Fourier Transform 64
2.2.2 Solution for Cauchy Problems 67
2.2.3 Solution Properties 70
2.2.4 Inhomogeneous Governing Equations 72
2.3 A Revisit to the Method of Separation of Variables 75
2.3.1 An Example with the Heat Equation 75
2.3.2 Sturm-Liouville System 81
2.3.3 Inhomogeneous Governing Equations 88
2.4 Solution Properties 89
2.4.1 Maximum Principle 89
2.4.2 Solution Uniqueness 91
2.4.3 Stability 92
3 Poisson’s Equation 99
3.1 Poisson’s Equation and Harmonic Equation 99
3.1.1 Definitions 99
3.1.2 Motivation from Physics 101
3.1.3 Boundary Conditions 106
3.2 Variational Principle 108
3.3 Harmonic Functions in Polar System 112
3.3.1 Laplace’s Equation in Polar System 113
3.3.2 Radial Solutions to Laplace’s Equation 115
3.4 The Method of Green^ Function 117
3.4.1 Green’s Formulae Related to Laplacian Operator 117
3.4.2 Fundamental Solution 118
3.4.3 Derivation of Green’s Function 120
3.4.4 Properties of Green’s Function 122
3.4.5 Problems for Poisson’s Equation 123
3.4.6 Final Remarks 125
3.5 Image Method for Electric Potentials 125
3.5.1 The Case in Three-Dimensional Half Space 126
3.5.2 The Case in a Spherical Domain 127
3.6 Solution Uniqueness 130
3.6.1 Mean-Value Formula 130
3.6.2 Maximum Principle 132
3.6.3 Strong Maximum Principle 133
3.6.4 Energy Method 134
4 Summary over Second-Order Linear Partial Differential Equation 141
4.1 Classification of Second-Order Linear Partial Differential Equations 141
4.1.1 Cases with Two Variables 141
4.1.2 Summary and Examples 148
4.1.3 Multivariable Situations 150
4.2 Topical Discussion 155
Part II Special Functions
5 Bessel Functions 165
5.1 Bessel Equation and Bessel Functions 165
5.1.1 Physical Motivation 165
5.1.2 Bessel Function of the First Kind 169
5.1.3 Bessel Function of the Second Kind 173
5.2 Properties of Bessel Functions 177
5.2.1 Recurrence Formulae 177
5.2.2 Zeros 179
5.2.3 Approximating Formula 181
5.2.4 Orthogonality 181
5.2.5 Analogies with Sinusoidal Functions 184
5.3 Solving PDE Problems with Bessel Functions 186
5.4 Generalisation 195
5.4.1 Hankel Functions 195
5.4.2 Modified Bessel Functions 195
6 Legendre Polynomial 199
6.1 Legendre Equation 199
6.2 Legendre Polynomial 204
6.2.1 Series Solution to the Legendre Equation 204
6.2.2 Legendre Polynomial 207
6.3 Properties of Legendre Polynomial 210
6.3.1 Rodrigues Formula 210
6.3.2 Key Properties at a Glance 210
6.3.3 Orthogonal Systems 212
6.3.4 Discussion on General Orthogonal Polynomials 218
6.4 Applications with Legendre Polynomials 220
6.4.1 Solving PDEs Defined in a Sphere 220
6.4.2 Legendre-Gauss Quadrature 222
6.5 The Associated Legendre Functions 225
7 Introduction of Hypergeometric Function 231
7.1 Commonalities of Bessel and Legendre Functions 231
7.2 Gauss Hypergeometric Function 234
7.2.1 Definition 234
7.2.2 Hypergeometric Differential Equation 235
7.2.3 Legendre Function and Legendre PolynomialA Revisit 239
7.3 Confluent Hypergeometric Function 241
7.3.1 Kummer Differential Equation and Its Solutions 241
7.3.2 Special Cases 242
7.4 Final Remarks on Hypergeometric Functions 244
Bibliography 247
Index 249
展开全部
数学物理方程及相关分析工具简明教程(英文版) 节选
Part I Second-Order Linear Partial Differential Equations Chapter 1 The Wave Equation In this chapter， partial differential equations that govern many featured wave propagation phenomena are discussed. We start with modelling the process of string vibration through a partial differential equation， known as the wave equation， or the equation for string vibration. Then the concepts of initial and boundary conditions are introduced. This is followed by the presentation of a number of relevant methods widely used for solving problems governed by the wave equation. Our scope is then extended to cover two-/three-dimensional cases of wave propagation. The chapter concludes with discussion over the properties of solutions for problems governed by the wave equation. The contents in this chapter are summarised in the knowledge map given in Fig. 1.1. Note that the contents in Fig. 1.1 are classified into several groups identified by the shapes of their frames. “Modelling” corresponds to a stage where a physical phenomenon is being represented by means of certain mathematical formulations. “Analysis” corresponds to a stage where certain mathematical tools are introduced so as to analyse the derived formulations (from physics). “Interpretation” corresponds to a stage where the mathematical results obtained through the analysis stage are employed to offer (often more insightful) interpretations to the physical phenomena that are modelled. For developing solid understanding of the course materials， certain mathematical knowledge that is supposed to be covered in the prerequisite courses， e.g. calculus， linear algebra， etc” is required， and these contents are summarised in the “prerequisite” stage. The linkages of the introduced contents with other topics likely to appear for future studies are also mentioned， and these “related topics” are identified in hexagonal frames. The setting here also applies for the knowledge map charted for other chapters in this book. Fig. 1.1 Knowledge map~wave equation 1.1 Equation for String Vibration 1.1.1 Derivation of the Equation for String Vibration In this section， we consider deriving a mathematical model to describe string vibration. To this end， we refer to a string， as shown in Fig. 1.2， of initial length L and of (linear) mass density p. As for modelling， one often needs to specify certain physically reasonable presumptions from the key features of the process of interest. For the case of string vibration， three assumptions are given as follows. 1. The diameter of a string should be far shorter than its length. This agrees with the common understanding of a “string”. But we need to express such a fact in mathematical terms. If we use d to denote the string diameter， then d《L， Hence， the string can be fully represented by one spatial variable， say， jc. As shown in Fig. 1.2， the x-axis is set in parallel with the configuration when the string is straight， and any point on the string should correspond to a value of x G [0， L]. 2. The string undergoes small and transverse vibration. Here the term “trans-verse” means that each point on the string only moves along the direction that is perpendicular to the x-axis. Thus， a function u(x， t) can be introduced to denote the (transverse) displacement of the string section around x9 away from the x-axis at time t. The term “small vibration” means that the magnitude of the displacement u is far smaller than that of the string length L. Note that “small” is meant in a relative sense. For instance， if the string of interest is from a guitar， then a displacement of 1 dm is quite large. But if the string of interest is a cable upholding a suspension bridge， then a displacement of 1 dm can still be considered as “small”. Mathematically， the non-dimensional quantity of is employed to measure the degree of string vibration. Under small vibration， we write ，indicating that is small in magnitude， or we can simply require (1.1) 3. A string resists elongation by a line tension following Hooke，s law. A string is expected to resist elongation，leading to a state of tension. The resisting force， or the line tension denoted by T(x， t)， is supposed to follow Hooke’s law， i.e. the magnitude of the line tension at (jc， t) is proportional to the local increment in the string arc-length denoted by s{x， t). Moreover， a transversely vibrating string may also experience an external driving force，whose distribution can be denoted by F(x， t). Here F(x， t) is measured in unit “Newton per metre”. In Fig. 1.1，the key physical quantities that are needed for modelling string vibration are summarised. For a transversely vibrating string， its behaviour can be dictated through the displacement function u(x， t). Thus， our modelling target is to set up mathematical formulations for m(jc， t). Here the modelling procedure encompasses the use of the above-listed three assumptions and the law of conservation in momentu
书友推荐
- >
推拿
推拿
¥12.2¥32.0 - >
有舍有得是人生
有舍有得是人生
¥31.5¥45.0 - >
伊索寓言-世界文学名著典藏-全译本
伊索寓言-世界文学名著典藏-全译本
¥6.1¥19.0 - >
罗曼·罗兰读书随笔-精装
罗曼·罗兰读书随笔-精装
¥40.6¥58.0 - >
诗经-先民的歌唱
诗经-先民的歌唱
¥15.1¥39.8 - >
龙榆生：词曲概论/大家小书
龙榆生：词曲概论/大家小书
¥9.1¥24.0 - >
上帝之肋:男人的真实旅程
上帝之肋:男人的真实旅程
¥19.3¥35.0 - >
烟与镜
烟与镜
¥20.6¥48.0
本类畅销
-
相对论
¥10.2¥32 -
数学物理方法
¥13.2¥27 -
2022图书×抽奖盲袋
¥9.9¥25 -
2023读书月阅读盲盒——天黑，闭眼，刀谁？
¥42.3¥158 -
-
2022读者节纪念徽章-三星会员专属
¥45¥45.6
