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 Poissons 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
內容試閱：
This book is focused on the mathematical foundation of the boundary integro-differential equation method. It is well known that the boundary integral equation method (or boundary element method) has become one effective numerical computational method for solving the boundary value problems of partial differential equations， that have been formulated as boundary integral equations including the hypersingular boundary integral equations.~The hypersingular boundary integral operators are derived from the derivative of double layer potential corresponding to the given problem. From the view of science and engineering computing， the appearance of the hypersingular boundary integral operators has caused new difficulties in the boundary integral equation method. One way to resolve the difficulties is to find the computing method for the hypersingular boundary integral operators. In this book， we prefer to resolve the difficulties from a different perspective. The hypersingular boundary integral operators are regularized as boundary integro-differential operators with only weak singular boundary integral operators. It means that each of the hypersingular boundary integral operators is equivalent to a corresponding boundary integro-differential operator with only a weak singular boundary integral operator. The hypersingular boundary integral operators no longer appear; they are replaced by the corresponding integro-differential operators. Therefore the difficulties in computing hypersingular integral operators have completely disappeared. It is the motivation for the title of this book.
The book also pays attention to the boundary integral equations of the first kind and the boundary integro-differential equations of the first kind. They occur in the acoustic scattering theory， as pointed out by Colton and Kress in their book (2013) ``Traditionally， the use of integral equations of the first kind for studying boundary-value problems in acoustic scattering theory has been neglected due to the lack of a Riesz-Fredholm theory for equations of the first kind and the fact that integral equations of the first kind are improperly posed.~\ But in Chapter 5 of this book boundary integral equations of the first kind and boundary integro-differential equations of the first kind are used to study the boundary value problems of the Helmholtz equation， because the Fredholm alternative theorems are established for the corresponding boundary integral equations of the first kind and boundary integro-differential equations of the first kind.
We also discuss in the last two chapters， the application of the boundary integro-differential equation method to nonlinear problems and the coercive and symmetrical coupling method of finite element method and boundary element method based on the boundary integro-differential operators.
This book contains nine Chapters， as listed below.
Chapter 1: Distributions
Chapter 2: Fundamental Solutions of Linear Differential Operators
Chapter 3: Boundary Value Problems of the Laplace Equation
Chapter 4: Boundary Value Problems of Modified Helmholtz Equation
Chapter 5: Boundary Value Problems of Helmholtz Equation
Chapter 6: Boundary Value Problems of the Navier Equations
Chapter 7: Boundary Value Problems of the Stokes Equations
Chapter 8: Some Nonlinear Problems
Chapter 9: Coercive and Symmetrical Coupling Methods of Finite Element Method and Boundary Element Method
Due to the limited knowledge of the authors， errors are inevitable. We would be most grateful to learn of any errors in the book and any suggestions for the revision of a future printing.
We are very grateful for the support and encouragement from our colleagues and friends during the preparing this book. Prof. Hermann Brunner carefully read the whole book and gave helpful suggestions for revision. This book has benefited from the works of many other researchers， including our co-authors: Weijun Tang， Zhi Guan， Bin He， Chongqing Yu， Jeng-Tzong Chen， Ying-Te Lee， Wenjun Ying， etc.
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