数学物理方程及相关分析工具简明教程 朱一超 编 下载 pdf 百度网盘 epub 免费 2025 电子版 mobi 在线

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。

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

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书籍介绍

偏微分方程是描述在变化中有守恒之物理世界诸多机制的重要手段。《数学物理方程及相关分析工具简明教程（英文版）》将围绕波动、热传导以及泊松方程三类*典型的二阶偏微分方程展开讨论，同时介绍特殊函数这一可用于求解偏微分方程的分析工具。《数学物理方程及相关分析工具简明教程（英文版）》旨在帮助读者初步形成综合运用偏微分方程分析解决物理问题的能力。