Discrete Projection Methods for Integral Equations

Transcription

1 SUB Gttttingen A 5141 Discrete Projection Methods for Integral Equations M.A. Golberg & C.S. Chen TM Computational Mechanics Publications Southampton UK and Boston USA

2 Contents Sources of Integral Equations Introduction Initial value problems Boundary value problems for ordinary differential equations Integral equations for partial differential equations The direct method The Abel equation References 12 Classification of Integral Equations Introduction Linear equations Kind and type Kernel classification Nonlinear equations Special equations Abel equations Cauchy singular equations Hypersingular equations Boundary integral equations References 22 Some Analytic Methods for Solving Integral Equations Introduction The Abel equation The generalized Abel equation Volterra equations of the second kind Convolution kernels Degenerate kernels Continuous kernels FVedholm equations of the second kind Degenerate kernels Semidegenerate kernels The boundary value method 37

3 3.4.4 Continuous kernels FYedholm equations of the first kind The airfoil equation References 47 Functional Analysis Introduction Vector spaces Subspaces Linear combinations Spanning sets, bases and dimension Linear transformations Properties of linear transformations Powers of T Inverses Normed vector spaces Properties of inner product spaces Metric spaces and convergence Metric convergence Cauchy sequences and completeness Equivalent norms The contraction mapping theorem Some applications of the CMT Bounded linear operators The dual space Banach's lemma Split equations of the first kind The Predholm alternative Hilbert-Schmidt operators Approximation theory Projection methods Projection methods for Hu = Ku + f Iterated projection methods Superconvergence of linear functionals Perturbed projection methods Brakhage's lemma Further variants of projection methods Conditioning Stability of Galerkin's method Stability of projection methods References 115

4 Approximation and Numerical Integration Introduction One dimensional approximation Taylor polynomials Polynomial interpolation Hermite interpolation Piecewise polynomial interpolation Trigonometric interpolation Other approximation methods Multivariate approximation Multivariate interpolation Radial basis function interpolation Approximation on the unit sphere in R Numerical integration Taylor polynomial rules Interpolatory rules Error analysis for interpolatory rules Compound rules Gaussian integration Orthogonal interpolation Numerical approximation of Fourier coefficients Numerical approximation of integral transforms Cauchy singular integrals Numerical approximation of multiple integrals Numerical integration on triangles Numerical integration on the sphere References 170 Discrete Projection Methods for Fredholm Equations Introduction Projection methods for one-dimensional equations Galerkin's method - orthogonal polynomial bases The discrete Galerkin method Trigonometric approximations Piecewise polynomial approximation Collocation Orthogonal collocation Discrete orthogonal collocation Trigonometric collocation Piecewise polynomial collocation Superconvergence of collocation for Volterra equations Multiquadric collocation Stability of collocation Quadrature methods 204

5 6.3.1 The Nystrom method: continuous kernels Product quadrature Stability of the Nystrom method Multivariable equations Galerkin's method Direct analysis of the discrete Galerkin method Piecewise polynomial collocation Direct analysis of the discrete collocation method Rbf collocation Wiener-Hopf equations Piecewise polynomial collocation Extensions to CJ 1 " Nonlinear Equations Iterated projection methods Discrete projection methods Hammerstein equations - the method of Kumar and Sloan References 246 Discrete Projection Methods for Cauchy Singular Equations Introduction Numerical methods for the GAE Operator formulation of the GAE Degenerate kernel methods Galerkin's method: v = The Sloan iterate Logarithmically singular kernels The flap problem Discrete Galerkin methods: v = Galerkin's method: v = ' Collocation v = 0: continuous data Discrete collocation methods: u = Collocation: v = ± Quadrature methods CSIEs of the second kind with constant coefficients The standard polynomial algorithms Galerkin's method Collocation Quadrature methods Further convergence results Galerkin's method: v = The discrete Galerkin method: v = Galerkin's method: v = Discrete Galekin's method: v = Collocation: v = 0 291

6 7.4.6 Discrete collocation: v = Discrete collocation: v \ Chebyshev collocation Stability Convergence of collocation with logarimithcally singular kernels Convergence of the Lagrange interpolant of the Gaussian quadrature method The discrete Sloan iterate Hypersingular equations References 309 Boundary Integral Equations in R Introduction Partial differential equations equivalent to Laplace's equation Poisson's equation Rbf approximations for particular solutions Semilinear equations The Kirchhoff transformation Boundary integral equation solution of Laplace's equation-smooth domains The Dirichlet problem - double layer potential The Dirichlet problem - single layer potential Galerkin's method A quadrature method The delta-trigonometric method Other boundary value problems for Laplace's equation The method of fundamental solutions MFS for Poisson's equation A thermal explosion problem Further aspects of the MFS Boundaries with corners Cea's lemma References 364 Boundary Integral Equations in R Introduction Integral equations on the sphere A Galerkin method A discrete Galerkin method A Nystrom method A collocation method The BEM for boundary integral equations Piecewise quadratic approximation Discrete collocation methods 384

7 9.3.3 The approximate surface Boundary value problems for Laplace's equation Time dependent problems and the Helmholtz equation The Dirichlet problem - double layer potential Spherical harmonic approximation The single layer potential Boundary element approximation Boundary integral equation solution of the Helmholtz equation The method of fundamental solutions Laplace's equation MFS for Poisson's equation References 406