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
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