作者：韩厚德、殷东生
页数：320
出版社：清华大学出版社
出版日期：2024
ISBN：9787302664734
电子书格式：pdf/epub/txt
内容简介
本书主要讨论边界积分-微分方程的数学基础理论，主要集中于把传统的边界积分方程中的超奇异积分转化为带弱奇性的边界积分-微分方程。本书简要地介绍了分布理论，而边界积分方程方法是基于线性偏微分方程基本解的，所以对微分方程的基本解也做了较为详细的介绍。在余下的章节里，本书依次讨论了Laplace方程、Helmholtz方程、Navier方程组、Stokes方程等边界积分-微分方程方法和理论。还讨论了某系非线性方程如：如热辐射、变分不等式和Steklov特征值问题的边界积分-微分方程理论。最后讨论了有限元和边界元的对称耦合问题。
作者简介
韩厚德，清华大学教授，长期从事计算数学研究工作。在有限元方法、无限元方法、边界元方法以及无界区域上偏微分方程的数值解等领域取得了一系列的重要研究成果。曾获得国家科学大会奖（1978），国家二等奖（1988）和一等奖（1995），北京市科技进步二等奖（2002），Hermker奖（2008），国家自然科学二等奖（2008）等多项奖励。
殷东生，清华大学副教授，主要研究方向为高频波、无界域上的偏微分方程和分数阶微分方程。
本书特色
《边界积分-微分方程方法的数学基础（英文版）》主要讨论边界积分-微分方程的数学基础理论，可供计算数学与机械工程相关领域的研究人员和研究生参考使用。
目录
Chapter 1 Distributions 1
1.1 Space of Test Functions 2
1.2 Definition of Distributions and Their Operations 3
1.3 Direct Products and Convolution of Distributions 8
1.4 Tempered Distributions and Fourier Transform 11
References 15
Chapter 2 Fundamental Solutions of Linear Differential Operators 16
2.1 Definition of Fundamental Solution 16
2.2 Elliptic Operators 19
2.2.1 Laplace Operator 19
2.2.2 Helmholtz Operator 20
2.2.3 Biharmonic Operator 24
2.3 Transient Operator 25
2.3.1 Heat Conduction Operator 25
2.3.2 Schr?dinger Operator 26
2.3.3 Wave Operator 27
2.4 Matrix Operator 28
2.4.1 Steady-State Navier Operator 29
2.4.2 Harmonic Navier Operator 33
2.4.3 Steady-State Stokes Operator 37
2.4.4 Steady-State Oseen Operator 40
References 43
Chapter 3 Boundary Value Problems of the Laplace Equation 44
3.1 Function Spaces 44
3.1.1 Continuous and Continuously Differential Function Spaces 44
3.1.2 H?lder Spaces 45
3.1.3 The Spaces 46
3.1.4 Sobolev Spaces 47
3.2 The Dirichlet and Neumann Problems of the Laplace Equation 49
3.2.1 Classical Solutions 50
3.2.2 Generalized Solutions and Variational Problems 52
3.3 Single Layer and Double Layer Potentials 54
3.3.1 Weakly Singular Integral Operators on 55
3.3.2 Double Layer Potentials 56
3.3.3 Single Layer Potentials 62
3.3.4 The Derivatives of Single Layer Potentials 64
3.3.5 The Derivatives of Double Layer Potentials 67
3.3.6 The Single and Double Layer Potentials in Sobolev Spaces 70
3.4 Boundary Reduction 73
3.4.1 Boundary Integral (Integro-Differential) Equations of the First Kind 73
3.4.2 Solvability of First Kind Integral Equation with n=2 and the Degenerate
Scale 79
3.4.3 Boundary Integral Equations of the Second Kind 84
References 93
Chapter 4 Boundary Value Problems of Modified Helmholtz Equation 95
4.1 The Dirichlet and Neumann Boundary Problems of Modified Helmholtz Equation 95
4.2 Single and Double Layer Potentials of Modified Helmholtz
Operator for the Continuous Densities 98
4.3 Single Layer Potential and Double Layer Potential
in Soblov Spaces 106
4.4 Boundary Reduction for the Boundary Value Problems of Modified
Helmholtz Equation 115
4.4.1 Boundary Integral Equation and Integro-Differential Equation of
the First Kind 115
4.4.2 Boundary Integral Equations of the Second Kind 118
References 125
Chapter 5 Boundary Value Problems of Helmholtz Equation 127
5.1 Interior and Exterior Boundary Value Problems of Helmholtz Equation 128
5.2 Single and Double Layers Potentials of Helmholtz Equation 133
5.2.1 Single Layer Potential 136
5.2.2 The Double Layer Potential 142
5.3 Boundary Reduction for the Principal Boundary Value Problems
of Helmholtz Equation 149
5.3.1 Boundary Integral Equation of the First Kind 151
5.3.2 Boundary Integro-Differential Equations of the First Kind 156
5.3.3 Boundary Integral Equations of the Second Kind 162
5.3.4 Modified Integral and Integro-Differential Equations 176
5.4 The Boundary Integro-Differential Equation Method for Interior
Dirichlet and Neumann Eigenvalue Problems of Laplace Operator 179
5.4.1 Interior Dirichlet Eigenvalue Problems of Laplace Operator 179
5.4.2 Interior Neuamann Eigenvalue Problem of Laplace Operator 182
References 185
Chapter 6 Boundary Value Problems of the Navier Equations 186
6.1 Some Basic Boundary Value Problems 186
6.2 Single and Double Layer Potentials of the Navier System 191
6.2.1 Single Layer Potential 191
6.2.2 Double Layer Potential 192
6.2.3 The Derivatives of the Single Layer Potential 195
6.2.4 The Derivatives of the Double Layer Potential 197
6.2.5 The Layer Potentials and in Sobolev Spaces 202
6.3 Boundary Reduction for the Boundary Value Problems of the Navier System 204
6.3.1 First Kind Integral (Differential-integro-differential) Equations of
the Boundary Value Problems of the Navier System 205
6.3.2 Solvability of the First Kind Integral Equations with n = 2 and
the Degenerate Scales 212
6.3.3 The Second Kind Integral Equations of the Boundary Value
Problems of the Navier System 218
References 225
Chapter 7 Boundary Value Problems of the Stokes Equations 227
7.1 Principal Boundary Value Problems of Stokes equations 227
7.2 Single Layer Potential and Double Layer Potential of Stokes Operator 234
7.3 Boudary Reduction of the Boundary Value Problems of Stokes Equations 243
References 247
Chapter 8 Some Nonlinear Problems 248
8.1 Heat Radiation Problems 248
8.1.1 Boundary Condition of Nonlinear Boundary Problem (8.1.1) 249
8.1.2 Equivalent Formula of Problem (8.1.1) 250
8.1.3 Equivalent Saddle-point Problem 255
8.1.4 The Numerical Solutions of Nonlinear Boundary
Variational Problem (8.1.17) 257
8.2 Variational Inequality (I)-Laplace Equation with Unilateral
Boundary Conditions 259
8.2.1 Equivalent Boundary Variational Inequality of Problem (8.2.2) 260
8.2.2 Abstract Error Estimate of the Numerical Solution of
Boundary Variational Inequality (8.2.9) 262
8.3 Variational Inequality (II)-Signorini Problems in Linear Elasticity 264
8.3.1 Signorini Problems in Linear Elasticity 264
8.3.2 An Equivalent Boundary Variational Inequality of Problem (8.3.3) 265
8.4 Steklov Eigenvalue Problems 268
8.4.1 The Boundary Reduction of Steklov Eigenvalue Problem 270
8.4.2 The Numerical Solutions of Steklov Eigenvalue Problem Based
on the Variational Form (8.4.13) 272
8.4.3 The Error Estimate of Numerical Solution of Steklov
Eigenvalue Problem 273
References 282
Chapter 9 Coercive and Symmetrical Coupling Methods of Finite
Element Method and Boundary Element Method 285
9.1 Exterior Dirichelet Problem of Poisson’s Equation (I) 286
9.1.1 The Symmetric and Coercive Coupling Formula of Problem (9.1.1) 286
9.1.2 The Numerical Solutions of Problem (9.1.1) Based on the
Symmetric and Coercive Coupling Formula 291
9.2 Exterior Dirichlet Problem of Poisson Equation (II) 292
9.3 An Exterior Displacement Problem of Nonhomogeneous Navier System 298
9.3.1 The Coercive and Symmetrical Variational Formulation
of Problem (9.3.1) on Bounded Domain 298
9.3.2 The Discrete Approximation of Problem (9.3.19) and (9.3.20) 303
References 304
1.1 Space of Test Functions 2
1.2 Definition of Distributions and Their Operations 3
1.3 Direct Products and Convolution of Distributions 8
1.4 Tempered Distributions and Fourier Transform 11
References 15
Chapter 2 Fundamental Solutions of Linear Differential Operators 16
2.1 Definition of Fundamental Solution 16
2.2 Elliptic Operators 19
2.2.1 Laplace Operator 19
2.2.2 Helmholtz Operator 20
2.2.3 Biharmonic Operator 24
2.3 Transient Operator 25
2.3.1 Heat Conduction Operator 25
2.3.2 Schr?dinger Operator 26
2.3.3 Wave Operator 27
2.4 Matrix Operator 28
2.4.1 Steady-State Navier Operator 29
2.4.2 Harmonic Navier Operator 33
2.4.3 Steady-State Stokes Operator 37
2.4.4 Steady-State Oseen Operator 40
References 43
Chapter 3 Boundary Value Problems of the Laplace Equation 44
3.1 Function Spaces 44
3.1.1 Continuous and Continuously Differential Function Spaces 44
3.1.2 H?lder Spaces 45
3.1.3 The Spaces 46
3.1.4 Sobolev Spaces 47
3.2 The Dirichlet and Neumann Problems of the Laplace Equation 49
3.2.1 Classical Solutions 50
3.2.2 Generalized Solutions and Variational Problems 52
3.3 Single Layer and Double Layer Potentials 54
3.3.1 Weakly Singular Integral Operators on 55
3.3.2 Double Layer Potentials 56
3.3.3 Single Layer Potentials 62
3.3.4 The Derivatives of Single Layer Potentials 64
3.3.5 The Derivatives of Double Layer Potentials 67
3.3.6 The Single and Double Layer Potentials in Sobolev Spaces 70
3.4 Boundary Reduction 73
3.4.1 Boundary Integral (Integro-Differential) Equations of the First Kind 73
3.4.2 Solvability of First Kind Integral Equation with n=2 and the Degenerate
Scale 79
3.4.3 Boundary Integral Equations of the Second Kind 84
References 93
Chapter 4 Boundary Value Problems of Modified Helmholtz Equation 95
4.1 The Dirichlet and Neumann Boundary Problems of Modified Helmholtz Equation 95
4.2 Single and Double Layer Potentials of Modified Helmholtz
Operator for the Continuous Densities 98
4.3 Single Layer Potential and Double Layer Potential
in Soblov Spaces 106
4.4 Boundary Reduction for the Boundary Value Problems of Modified
Helmholtz Equation 115
4.4.1 Boundary Integral Equation and Integro-Differential Equation of
the First Kind 115
4.4.2 Boundary Integral Equations of the Second Kind 118
References 125
Chapter 5 Boundary Value Problems of Helmholtz Equation 127
5.1 Interior and Exterior Boundary Value Problems of Helmholtz Equation 128
5.2 Single and Double Layers Potentials of Helmholtz Equation 133
5.2.1 Single Layer Potential 136
5.2.2 The Double Layer Potential 142
5.3 Boundary Reduction for the Principal Boundary Value Problems
of Helmholtz Equation 149
5.3.1 Boundary Integral Equation of the First Kind 151
5.3.2 Boundary Integro-Differential Equations of the First Kind 156
5.3.3 Boundary Integral Equations of the Second Kind 162
5.3.4 Modified Integral and Integro-Differential Equations 176
5.4 The Boundary Integro-Differential Equation Method for Interior
Dirichlet and Neumann Eigenvalue Problems of Laplace Operator 179
5.4.1 Interior Dirichlet Eigenvalue Problems of Laplace Operator 179
5.4.2 Interior Neuamann Eigenvalue Problem of Laplace Operator 182
References 185
Chapter 6 Boundary Value Problems of the Navier Equations 186
6.1 Some Basic Boundary Value Problems 186
6.2 Single and Double Layer Potentials of the Navier System 191
6.2.1 Single Layer Potential 191
6.2.2 Double Layer Potential 192
6.2.3 The Derivatives of the Single Layer Potential 195
6.2.4 The Derivatives of the Double Layer Potential 197
6.2.5 The Layer Potentials and in Sobolev Spaces 202
6.3 Boundary Reduction for the Boundary Value Problems of the Navier System 204
6.3.1 First Kind Integral (Differential-integro-differential) Equations of
the Boundary Value Problems of the Navier System 205
6.3.2 Solvability of the First Kind Integral Equations with n = 2 and
the Degenerate Scales 212
6.3.3 The Second Kind Integral Equations of the Boundary Value
Problems of the Navier System 218
References 225
Chapter 7 Boundary Value Problems of the Stokes Equations 227
7.1 Principal Boundary Value Problems of Stokes equations 227
7.2 Single Layer Potential and Double Layer Potential of Stokes Operator 234
7.3 Boudary Reduction of the Boundary Value Problems of Stokes Equations 243
References 247
Chapter 8 Some Nonlinear Problems 248
8.1 Heat Radiation Problems 248
8.1.1 Boundary Condition of Nonlinear Boundary Problem (8.1.1) 249
8.1.2 Equivalent Formula of Problem (8.1.1) 250
8.1.3 Equivalent Saddle-point Problem 255
8.1.4 The Numerical Solutions of Nonlinear Boundary
Variational Problem (8.1.17) 257
8.2 Variational Inequality (I)-Laplace Equation with Unilateral
Boundary Conditions 259
8.2.1 Equivalent Boundary Variational Inequality of Problem (8.2.2) 260
8.2.2 Abstract Error Estimate of the Numerical Solution of
Boundary Variational Inequality (8.2.9) 262
8.3 Variational Inequality (II)-Signorini Problems in Linear Elasticity 264
8.3.1 Signorini Problems in Linear Elasticity 264
8.3.2 An Equivalent Boundary Variational Inequality of Problem (8.3.3) 265
8.4 Steklov Eigenvalue Problems 268
8.4.1 The Boundary Reduction of Steklov Eigenvalue Problem 270
8.4.2 The Numerical Solutions of Steklov Eigenvalue Problem Based
on the Variational Form (8.4.13) 272
8.4.3 The Error Estimate of Numerical Solution of Steklov
Eigenvalue Problem 273
References 282
Chapter 9 Coercive and Symmetrical Coupling Methods of Finite
Element Method and Boundary Element Method 285
9.1 Exterior Dirichelet Problem of Poisson’s Equation (I) 286
9.1.1 The Symmetric and Coercive Coupling Formula of Problem (9.1.1) 286
9.1.2 The Numerical Solutions of Problem (9.1.1) Based on the
Symmetric and Coercive Coupling Formula 291
9.2 Exterior Dirichlet Problem of Poisson Equation (II) 292
9.3 An Exterior Displacement Problem of Nonhomogeneous Navier System 298
9.3.1 The Coercive and Symmetrical Variational Formulation
of Problem (9.3.1) on Bounded Domain 298
9.3.2 The Discrete Approximation of Problem (9.3.19) and (9.3.20) 303
References 304
