Boundary integral equations / George C. Hsiao, Wolfgang L. Wendland.

Author
Hsiao, G. C. (George C.) [Browse]
Format
Book
Language
English
Εdition
Second edition.
Published/​Created
  • Cham, Switzerland : Springer, [2021]
  • ©2021
Description
1 online resource (xx, 783 pages) : illustrations.

Details

Subject(s)
Author
Series
  • Applied mathematical sciences (Springer-Verlag New York Inc.) ; volume 164. [More in this series]
  • Applied mathematical sciences ; Volume 164
Bibliographic references
Includes bibliographical references and index.
Source of description
Description based on print version record.
Contents
  • Intro
  • Preface to the Second Edition
  • Preface to the First Edition
  • Acknowledgements
  • Table of Contents
  • 1. Introduction
  • 1.1 The Green Representation Formula
  • 1.2 Boundary Potentials and Calderón's Projector
  • 1.3 Boundary Integral Equations
  • 1.3.1 The Dirichlet Problem
  • 1.3.2 The Neumann Problem
  • 1.4 Exterior Problems
  • 1.4.1 The Exterior Dirichlet Problem
  • 1.4.2 The Exterior Neumann Problem
  • 1.5 Remarks
  • 2. Boundary Integral Equations
  • 2.1 The Helmholtz Equation
  • 2.1.1 Low Frequency Behaviour
  • 2.2 The Lamé System
  • 2.2.1 The Interior Displacement Problem
  • 2.2.2 The Interior Traction Problem
  • 2.2.3 Some Exterior Fundamental Problems
  • 2.2.4 The Incompressible Material
  • 2.3 The Stokes Equations
  • 2.3.1 Hydrodynamic Potentials
  • 2.3.2 The Stokes Boundary Value Problems
  • 2.3.3 The Incompressible Material - Revisited
  • 2.4 The Biharmonic Equation
  • 2.4.1 Calderón's Projector
  • 2.4.2 Boundary Value Problems and Boundary Integral Equations
  • 2.5 Remarks
  • 3. Representation Formulae, Local Coordinates and Direct Boundary Integral Equations
  • 3.1 Classical Function Spaces and Distributions
  • 3.2 Hadamard's Finite Part Integrals
  • 3.3 Local Coordinates
  • 3.4 Short Excursion to Elementary Differential Geometry
  • 3.4.1 Second Order Differential Operators in Divergence Form
  • 3.5 Distributional Derivatives and Abstract Green's Second Formula
  • 3.6 The Green Representation Formula
  • 3.7 Green's Representation Formulae in Local Coordinates
  • 3.8 Multilayer Potentials
  • 3.9 Direct Boundary Integral Equations
  • 3.9.1 Boundary Value Problems
  • 3.9.2 Transmission Problems
  • 3.10 Remarks
  • 4. Sobolev Spaces
  • 4.1 The Spaces Hs(Ω)
  • 4.2 The Trace Spaces Hs(Γ)
  • 4.2.1 Trace Spaces for Periodic Functions on a Smooth Curve in IR
  • 4.2.2 Trace Spaces on Curved Polygons in IR.
  • 4.3 The Trace Spaces on an Open Surface
  • 4.4 The Weighted Sobolev Spaces Hm(Ωc
  • λ) and Hm(IRn
  • λ)
  • 4.5 Function Spaces H( div ,Ω) and H( curl,Ω)
  • 5. Variational Formulations
  • 5.1 Partial Differential Equations of Second Order
  • 5.1.1 Interior Problems
  • 5.1.2 Exterior Problems
  • 5.1.3 Transmission Problems
  • 5.2 Abstract Existence Theorems for Variational Problems
  • 5.2.1 The Lax-Milgram Theorem
  • 5.3 The Fredholm-Nikolski Theorems
  • 5.3.1 Fredholm's Alternative
  • 5.3.2 The Riesz-Schauder and the Nikolski Theorems
  • 5.3.3 Fredholm's Alternative for Sesquilinear Forms
  • 5.3.4 Fredholm Operators
  • 5.4 Gårding's Inequality for Boundary Value Problems
  • 5.4.1 Gårding's Inequality for Second Order Strongly Elliptic Equations in Ω
  • 5.4.2 The Stokes System
  • 5.4.3 Gårding's Inequality for Exterior Second Order Problems
  • 5.4.4 Gårding's Inequality for Second Order Transmission Problems
  • 5.5 Existence of Solutions to Strongly Elliptic Boundary Value Problems
  • 5.5.1 Interior Boundary Value Problems
  • 5.5.2 Exterior Boundary Value Problems
  • 5.5.3 Transmission Problems
  • 5.6 Solutions of Certain Boundary Integral Equations and Associated Boundary Value Problems
  • 5.6.1 The Generalized Representation Formula for Second Order Systems
  • 5.6.2 Continuity of Some Boundary Integral Operators
  • 5.6.3 Continuity Based on Finite Regions
  • 5.6.4 Continuity of Hydrodynamic Potentials
  • 5.6.5 The Equivalence Between Boundary Value Problems and Integral Equations
  • 5.6.6 Variational Formulation of Direct Boundary Integral Equations
  • 5.6.7 Positivity and Contraction of Boundary Integral Operators
  • 5.6.8 The Solvability of Direct Boundary Integral Equations
  • 5.6.9 Positivity of the Boundary Integral Operators of the Stokes System
  • 5.7 Partial Differential Equations of Higher Order
  • 5.8 Remarks
  • 5.8.1 Assumptions on Γ.
  • 5.8.2 Higher Regularity of Solutions
  • 5.8.3 Mixed Boundary Conditions and Crack Problem
  • 6. Electromagnetic Fields
  • 6.1 Introduction
  • 6.2 Maxwell Equations
  • 6.3 Constitutive Equations
  • 6.4 Time Harmonic Fields
  • 6.4.1 Plane waves
  • 6.5 Electromagnetic potentials
  • 6.6 Transmission and Boundary Conditions
  • 6.7 Boundary Value Problems
  • 6.7.1 Scattering problems
  • 6.7.2 Eddy current problems
  • 6.8 Uniqueness
  • 6.8.1 The cavity problem
  • 6.8.2 Exterior problems
  • 6.8.3 The transmission problem
  • 6.9 Representation Formulae
  • 6.10 Boundary Integral Equations for Electromagnetic fields
  • 6.10.1 The Calderon projector and the capacity operators
  • 6.10.2 Weak solutions for a fundamental problem
  • 6.10.2.1 Interior Dirichlet problem in Ω.
  • 6.10.2.2 A reduction to boundary integral equations.
  • 6.11 Application of the Electromagnetic Potentials to Eddy Current Problems
  • 6.11.1 The '(A, ϕ) − (A) − (ψ)' formulation in the bounded domain
  • 6.11.2 The '(A, ϕ) − (ψ)' formulation in an unbounded domain
  • 6.11.3 Electric field in the dielectric domain ΩD.
  • 6.11.4 Vector potentials - revisited
  • 6.12 Applications of boundary integral equations to scattering problems
  • 6.12.1 Scattering by a perfect electric conductor, EFIE and MFIE
  • 6.12.2 Scattering by a dielectric body
  • 6.12.3 Scattering by objects with impedance boundary conditions
  • 7. Introduction to Pseudodifferential Operators
  • 7.1 Basic Theory of Pseudodifferential Operators
  • 7.2 Elliptic Pseudodifferential Operators on Ω ⊂ IRn
  • 7.2.1 Systems of Pseudodifferential Operators
  • 7.2.2 Parametrix and Fundamental Solution
  • 7.2.3 Levi Functions for Scalar Elliptic Equations
  • 7.2.4 Levi Functions for Elliptic Systems
  • 7.2.5 Strong Ellipticity and Gårding's Inequality
  • 7.3 Review on Fundamental Solutions
  • 7.3.1 Local Fundamental Solutions.
  • 7.3.2 Fundamental Solutions in IRn for Operators with Constant Coefficients
  • 7.3.3 Existing Fundamental Solutions in Applications
  • 8. Pseudodifferential Operators as Integral Operators
  • 8.1 Pseudohomogeneous Kernels
  • 8.1.1 Integral Operators as Pseudodifferential Operators of Negative Order
  • 8.1.2 Non-Negative Order Pseudodifferential Operators as Hadamard Finite Part Integral Operators
  • 8.1.3 Parity Conditions
  • 8.1.4 A Summary of the Relations between Kernels and Symbols
  • 8.2 Coordinate Changes and Pseudohomogeneous Kernels
  • 8.2.1 The Transformation of General Hadamard Finite Part Integral Operators under Change of Coordinates
  • 8.2.2 The Class of Invariant Hadamard Finite Part Integral Operators under Change of Coordinates
  • 9. Pseudodifferential and Boundary Integral Operators
  • 9.1 Pseudodifferential Operators on Boundary Manifolds
  • 9.1.1 Ellipticity on Boundary Manifolds
  • 9.1.2 Schwartz Kernels on Boundary Manifolds
  • 9.2 Boundary Operators Generated by Domain Pseudodifferential Operators
  • 9.3 Surface Potentials on the Plane IRn−1
  • 9.4 Pseudodifferential Operators with Symbols of Rational Type
  • 9.5 Surface Potentials on the Boundary Manifold Γ
  • 9.6 Volume Potentials
  • 9.7 Strong Ellipticity and Fredholm Properties
  • 9.8 Strong Ellipticity of Boundary Value Problems and Associated Boundary Integral Equations
  • 9.8.1 The Boundary Value and Transmission Problems
  • 9.8.2 The Associated Boundary Integral Equations of the First Kind
  • 9.8.3 The Transmission Problem and Gårding's inequality
  • 9.9 Remarks
  • 10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations
  • 10.1 Newton Potential Operators for Elliptic Partial Differential Equations and Systems
  • 10.1.1 Generalized Newton Potentials for the Helmholtz Equation
  • 10.1.2 The Newton Potential for the Lamé System.
  • 10.1.3 The Newton Potential for the Stokes System
  • 10.2 Surface Potentials for Second Order Equations
  • 10.2.1 Strongly Elliptic Differential Equations
  • 10.2.2 Surface Potentials for the Helmholtz Equation
  • 10.2.3 Surface Potentials for the Lamé System
  • 10.2.4 Surface Potentials for the Stokes System
  • 10.3 Invariance of Boundary Pseudodifferential Operators
  • 10.3.1 The Hypersingular Boundary Integral Operators for the Helmholtz Equation
  • 10.3.2 The Hypersingular Operator for the Lamé System
  • 10.3.3 The Hypersingular Operator for the Stokes System
  • 10.4 Derivatives of Boundary Potentials
  • 10.4.1 Derivatives of the Solution to the Helmholtz Equation
  • 10.4.2 Computation of Stress and Strain on the Boundary for the Lamé System
  • 10.5 Remarks
  • 11. Boundary Integral Equations on Curves in IR2
  • 11.1 Representation of the basic operators for the 2D-Laplacian in terms of Fourier series
  • 11.2 The Fourier Series Representation of Periodic Operators A ∈ L m cl(Γ)
  • 11.3 Ellipticity Conditions for Periodic Operators on Γ
  • 11.3.1 Scalar Equations
  • 11.3.2 Systems of Equations
  • 11.3.3 Multiply Connected Domains
  • 11.4 Fourier Series Representation of some Particular Operators
  • 11.4.1 The Helmholtz Equation
  • 11.4.2 The Lamé System
  • 11.4.3 The Stokes System
  • 11.4.4 The Biharmonic Equation
  • 11.5 Remarks
  • 12. Remarks on Pseudodifferential Operators Related to the Time Harmonic Maxwell Equations
  • 12.1 Introduction
  • 12.2 Symbols of P and the corresponding Newton potentials
  • 12.3 Representation formulae
  • 12.4 Symbols of the Electromagnetic Boundary Potentials
  • 12.5 Symbols of boundary integral operators
  • 12.6 Symbols of the Capacity Operators
  • 12.7 Boundary Integral Operators for the Fundamental Boundary Value Problems
  • 12.8 Coerciveness and Strong Ellipticity.
  • 12.9 Gårding's inequality for the sesquilinear form A in (6.12.23).
ISBN
3-030-71127-7
OCLC
1245857973
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