边界积分-微分方程方法的数学基础(英文版)(精)

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。 本书可供计算数学与机 械工程相关领域的研究人员 和研究生参考使用。

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