Details

Subject(s)

• Boundary element methods [Browse]
• Integral equations [Browse]

Author

• Wendland, W. L. (Wolfgang L.), 1936- [Browse]

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).

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ISBN

3-030-71127-7

OCLC

1245857973

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