The Numerical Solution of Integral Equations of the Second Kind
Cambridg monographs on applied and computational mathematics
Auteur : KENDALL E. ATKINSON
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Preface
1 A brief discussion of integral equations
1.1 Types of integral equations
1.1.1 Volterra integral equations of the second kind
1.1.2 Volterra integral equations of the first kind
1.1.3 Abel integral equations of the first kind
1.1.4 Fredholm integral equations of the second kind
1.1.5 Fredholm integral equations of the first kind
1.1.6 Boundary integral equations
1.1.7 Wiener-Hopf integral equations
1.1.8 Cauchy singular integral equations
1.2 Compact integral operators
1.2.1 Compact integral operators on C(D)
1.2.2 Properties of compact operators
1.2.3 Integral operators on L 2 (a, b)
1.3 The Fredholm alternative theorem
1.4 Additional results on Fredholm integral equations
1.5 Noncompact integral operators
1.5.1 An Abel integral equation
1.5.2 Cauchy singular integral operators
1.5.3 Wiener-Hopf integral operators Discussion of the literature
2 Degenerate kernel methods
2.1 General theory
2.1.1 Solution of degenerate kernel integral equation
2.2 Taylor series approximations
2.2.1 Conditioning of the linear system
2.3 Interpolatory degenerate kernel approximations
2.3.1 Interpolation with respect to the variable t
2.3.2 Interpolation with respect to the variable s
2.3.3 Piecewise linear interpolation
2.3.4 Approximate calculation of the linear system
2.4 Orthonormal expansions Discussion of the literature
3 Projection methods
3.1 General theory
3.1.1 Collocation methods
3.1.2 Galerkin's method
3.1.3 The general framework
3.2 Examples of the collocation method
3.2.1 Piecewise linear interpolation
3.2.2 Collocation with trigonometric polynomials
3.3 Examples of Galerkin's method
3.3.1 Piecewise linear approximations
3.3.2 Galerkin's method with trigonometric polynomials
3.3.3 Uniform convergence
3.4 Iterated projection methods
3.4.1 The iterated Galerkin solution
3.4.2 Uniform convergence of iterated Galerkin approximations
3.4.3 The iterated collocation solution
3.4.4 Piecewise polynomial collocation at Gauss-Legendre nodes
3.4.5 The linear system for the itereted collocation solution
3.5 Regularization of the solution
3.6 Condition numbers
3.6.1 Condition numbers for the collocation method
3.6.2 Condition numbers based on the iterated collocation solution
3.6.3 Condition numbers for the Galerkin method
4 The Nystrom method
4.1 The Nystrom method for continuous kernel functions
4.1.1 Properties and error analysis of the Nystrom method
4.1.2 Collectively compact operator approximations
4.2 Product integration methods
4.2.1 Computation of the quadrature weights
4.2.2 Error analysis
4.2.3 Generalizations to other kernel functions
4.2.4 Improved error results for special kernels
4.2.5 Product integration with graded meshes
Application to integral equations
The relationship of product integration and collocation methods
4.3 Discrete collocation methods
4.3.1 Convergence analysis for {rk} {t, }
4.4 Discrete Galerkin methods
4.4.1 The discrete orthogonal projection operator
4.4.2 An abstract formulation
5 Solving multivariable integral equations
5.1 Multivariable interpolation and numerical integration
5.1.1 Interpolation over triangles
5.1.2 Numerical integration over triangles
5.2 Solving integral equations on polygonal regions
5.2.1 Collocation methods
5.2.2 Galerkin methods
5.2.3 The Nystrom method
5.3 Interpolation and numerical integration on surfaces
5.3.1 Interpolation over a surface
5.3.2 Numerical integration over a surface
5.3.3 Approximating the surface
5.3.4 Nonconforming triangulations
5.4 Boundary element methods for solving integral equations
5.4.1 The Nystrom method
5.4.2 Collocation methods
5.4.3 Galerkin methods
5.5 Global approximation methods on smooth surfaces
5.5.1 Spherical polynomials and spherical harmonics
5.5.2 Numerical integration on the sphere
5.5.3 Solution of integral equations on the unit sphere
6 Iteration methods
6.1 Solving degenerate kernel integral equations by iteration
6.1.1 Implementation
6.2 Two-grid iteration for the Nystrom method
6.2.1 Iteration method 1 for Nystrom's method
6.2.2 Iteration method 2 for Nystrom's method
6.3 Two-grid iteration for collocation methods
6.3.1 Prolongation and restriction operators
6.3.2 The two-grid iteration method
6.4 Multigrid iteration for collocation methods
6.4.1 Operations count
6.5 The conjugate gradient method
6.5.1 The conjugate gradient method for the undiscretized
6.5.2 The conjugate gradient iteration for Nystrom's method
7 Boundary integral equations on a smooth planar boundary
7.1 Boundary integral equations
7.1.1 Green's identities and representation formula
7.1.2 The Kelvin transformation and exterior problems
7.1.3 Boundary integral equations of direct type
The interior Dirichlet problem
The interior Neumann problem
The exterior Neumann problem
The exterior Dirichlet problem
7.1.4 Boundary integral equations of indirect type
7.2 Boundary integral equations of the second kind
7.2.1 Evaluation of the double layer potential
7.2.2 The exterior Neumann problem
7.2.3 Other boundary value problems
7.3 Boundary integral equations of the first kind
7.3.1 Sobolev spaces
The trapezoidal rule and trigonometric interpolation
7.3.2 Some pseudodifferential equations
The Cauchy singular integral operator
A hypersingular integral operator
Pseudodifferential operators
7.3.3 Two numerical methods
A discrete Galerkin method
7.4 Finite element methods
7.4.1 Sobolev spaces - A further discussion
Extensions of boundary integral operators
7.4.2 An abstract framework
A general existence theorem
An abstract finite element theory
The finite element solution as a projection
7.4.3 Boundary element methods for boundary integral equations
8 Boundary integral equations on a piecewise smooth planar boundary
8.1 Theoretical behavior
8.1.1 Boundary integral equations for the interior Dirichlet problem
8.1.2 An indirect method for the Dirichlet problem
8.1.3 A BIE on an open wedge
8.1.4 A decomposition of the boundary integral equation
8.2 The Galerkin method
8.2.1 Superconvergence results
8.3 The collocation method
8.3.1 Preliminary definitions and assumptions
8.3.2 The collocation method
A modified collocation method
8.4 The Nystrom method
8.4.1 Error analysis
Discussion of the literature
9 Boundary integral equations in three dimensions
9.1 Boundary integral representations
9.1.1 Green's representation formula
The existence of the single and double layer potentials
Exterior problems and the Kelvin transform
Green's representation formula for exterior regions
9.1.2 Direct boundary integral equations
9.1.3 Indirect boundary integral equations
9.1.4 Properties of the integral operators
9.1.5 Properties of K and S when S is only piecewise smooth
9.2 Boundary element collocation methods on smooth surfaces
9.2.1 The linear system
Numerical integration of singular integrals
Numerical integration of nonsingular integrals
9.2.2 Solving the linear system
9.2.3 Experiments for a first kind equation
9.3.1 The collocation method
Applications to various interpolatory projections
Numerical integration and surface approximation
9.3.2 Iterative solution of the linear system
9.3.3 Collocation methods for polyhedral regions
9.4 Boundary element Galerkin methods
9.4.1 A finite element method for an equation of the first kind
Generalizations to other boundary integral equations
9.5 Numerical methods using spherical polynomial approximations
9.5.1 The linear system for (27r + Pn1C) p = Pn f
9.5.2 Solution of an integral equation of the first kind
Implementation of the Galerkin
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