Chapter 1 Distributions 1.1 Space of Test Functions 1.2 Definition of Distributions and Their Operations 1.3 Direct Products and Convolution of Distributions 1.4 Tempered Distributions and Fourier Transform References Chapter 2 Fundamental Solutions of Linear Differential Operators 2.1 Definition of Fundamental Solution 2.2 Elliptic Operators 2.2.1 Laplace Operator 2.2.2 Helmholtz Operator 2.2.3 Biharmonic Operator 2.3 Transient Operator 2.3.1 Heat Conduction Operator 2.3.2 Schr?dinger Operator 2.3.3 Wave Operator 2.4 Matrix Operator 2.4.1 Steady-State Navier Operator 2.4.2 Harmonic Navier Operator 2.4.3 Steady-State Stokes Operator 2.4.4 Steady-State Oseen Operator References Chapter 3 Boundary Value Problems of the Laplace Equation 3.1 Function Spaces 3.1.1 Continuous and Continuously Differential Function Spaces 3.1.2 H?lder Spaces 3.1.3 The Spaces 3.1.4 Sobolev Spaces 3.2 The Dirichlet and Neumann Problems of the Laplace Equation 3.2.1 Classical Solutions 3.2.2 Generalized Solutions and Variational Problems 3.3 Single Layer and Double Layer Potentials 3.3.1 Weakly Singular Integral Operators on 3.3.2 Double Layer Potentials 3.3.3 Single Layer Potentials 3.3.4 The Derivatives of Single Layer Potentials 3.3.5 The Derivatives of Double Layer Potentials 3.3.6 The Single and Double Layer Potentials in Sobolev Spaces 3.4 Boundary Reduction 3.4.1 Boundary Integral (Integro-Differential) Equations of the First Kind 3.4.2 Solvability of First Kind Integral Equation with n=2 and the Degenerate Scale 3.4.3 Boundary Integral Equations of the Second Kind References Chapter 4 Boundary Value Problems of Modified Helmholtz Equation 4.1 The Dirichlet and Neumann Boundary Problems of Modified Helmholtz Equation 4.2 Single and Double Layer Potentials of Modified Helmholtz Operator for the Continuous Densities 4.3 Single Layer Potential and Double Layer Potential in Soblov Spaces 4.4 Boundary Reduction for the Boundary Value Problems of Modified Helmholtz Equation 4.4.1 Boundary Integral Equation and Integro-Differential Equation of the First Kind 4.4.2 Boundary Integral Equations of the Second Kind
References Chapter 5 Boundary Value Problems of Helmholtz Equation 5.1 Interior and Exterior Boundary Value Problems of Helmholtz Equation 5.2 Single and Double Layers Potentials of Helmholtz Equation 5.2.1 Single Layer Potential 5.2.2 The Double Layer Potential 5.3 Boundary Reduction for the Principal Boundary Value Problems of Helmholtz Equation 5.3.1 Boundary Integral Equation of the First Kind 5.3.2 Boundary Integro-Differential Equations of the First Kind 5.3.3 Boundary Integral Equations of the Second Kind 5.3.4 Modified Integral and Integro-Differential Equations 5.4 The Boundary Integro-Differential Equation Method for Interior Dirichlet and Neumann Eigenvalue Problems of Laplace Operator 5.4.1 Interior Dirichlet Eigenvalue Problems of Laplace Operator 5.4.2 Interior Neuamann Eigenvalue Problem of Laplace Operator References Chapter 6 Boundary Value Problems of the Navier Equations 6.1 Some Basic Boundary Value Problems 6.2 Single and Double Layer Potentials of the Navier System 6.2.1 Single Layer Potential 6.2.2 Double Layer Potential 6.2.3 The Derivatives of the Single Layer Potential 6.2.4 The Derivatives of the Double Layer Potential 6.2.5 The Layer Potentials and in Sobolev Spaces 6.3 Boundary Reduction for the Boundary Value Problems of the Navier System 6.3.1 First Kind Integral (Differential-integro-differential) Equations of the Boundary Value Problems of the Navier System 6.3.2 Solvability of the First Kind Integral Equations with n = 2 and the Degenerate Scales 6.3.3 The Second Kind Integral Equations of the Boundary Value Problems of the Navier System References Chapter 7 Boundary Value Problems of the Stokes Equations 7.1 Principal Boundary Value Problems of Stokes equations 7.2 Single Layer Potential and Double Layer Potential of Stokes Operator 7.3 Boudary Reduction of the Boundary Value Problems of Stokes Equations References Chapter 8 Some Nonlinear Problems 8.1 Heat Radiation Problems 8.1.1 Boundary Condition of Nonlinear Boundary Problem (8.1.1) 8.1.2 Equivalent Formula of Problem (8.1.1) 8.1.3 Equivalent Saddle-point Problem 8.1.4 The Numerical Solutions of Nonlinear Boundary Variational Problem (8.1.17) 8.2 Variational Inequality (I)-Laplace Equation with Unilateral Boundary Conditions 8.2.1 Equivalent Boundary Variational Inequality of Problem (8.2.2) 8.2.2 Abstract Error Estimate of the Numerical Solution of Boundary Variational Inequality (8.2.9) 8.3 Variational Inequality (II)-Signorini Problems in Linear Elasticity 8.3.1 Signorini Problems in Linear Elasticity 8.3.2 An Equivalent Boundary Variational Inequality of Problem (8.3.3)
Chapter 9 Coercive and Symmetrical Coupling Methods of Finite Element Method and Boundary Element Method 9.1 Exterior Dirichelet Problem of Poisson's Equation (I) 9.1.1 The Symmetric and Coercive Coupling Formula of Problem (9.1.1) 9.1.2 The Numerical Solutions of Problem (9.1.1) Based on the Symmetric and Coercive Coupling Formula 9.2 Exterior Dirichlet Problem of Poisson Equation (II) 9.3 An Exterior Displacement Problem of Nonhomogeneous Navier System 9.3.1 The Coercive and Symmetrical Variational Formulation of Problem (9.3.1 ) on Bounded Domain 9.3.2 The Discrete Approximation of Problem (9.3.19) and (9.3.20) References
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