# Wavelet Based Approximation Schemes for Singular Integral Equations

Singular Integral EquationApproximate solution of integral equationsThe general scheme of approximationNyström methodCollocation methodGalerkin’s methodQuadratic spline collocation methodMethod based on product integrationKernel with weak (logarithmic and algebraic) singularityIntegral equations with Cauchy singular kernelMethod based on Legendre polynomialsMethod based on Chebyshev polynomialsMethod based on Jacobi polynomialsIntegral equations with hypersingular kernelMultiresolution Analysis of Function SpacesMultiresolution Analysis of L2(R)Multiresolution generatorWaveletsBasis with compact supportProperties of elements in Daubechies familyLimitation of scale functions and wavelets in Daubechies familyMultiresolution Analysis of L2([a, b] ⊂ R)Truncated scale functions and waveletsMultiwaveletsOrthonormal (boundary) scale functions and waveletsOthersSinc functionCoifletAutocorrelation functionApproximations in Multiscale BasisMultiscale Approximation of FunctionsApproximation of f in the basis of Daubechies familyTruncated basisApproximation of f ∈ L2([0, 1]) in multiwavelet basisSparse Approximation of Functions in Higher DimensionsBasis for Ω ⊆ R2Representation of f (x, y)Homogeneous function K (λx, λy) = λμ K (x – y), μ ∈ RNon-smooth function f (x, y) = |x + y|ν, ν ∈ R – {N ⋃ 0}f (x, y) = ln|x ± y| involving logarithmic singularityf ∈ Ω ⊂ R2MomentsScale functions and wavelets in RTruncated scale functions and waveletsBoundary scale functions and waveletsQuadrature RulesDaubechies familyNodes, weights and quadrature rulesFormal orthogonal polynomials, nodes, weights of scale functionsInterior scale functionsFormal orthogonal polynomials, nodes, weights of waveletsAlgorithmError estimatesNumerical illustrationsQuadrature rules for singular integralsQuadrature rule for weakly (algebraic) singular integralsQuadrature rule for Cauchy principal value integralsFinite part integralsComposite quadrature formula for integrals having Cauchy and weak singularityNumerical examplesLogarithmic singular integralsCauchy principal value integralsHypersingular integralsFor multiwavelet familyOthersSinc functionsAutocorrelation functionsRepresentation of function and operator in the basis generated by autocorrelation functionMultiscale Representation of Differential OperatorsRepresentation of the Derivative of a Function in LMW BasisMultiscale Representation of Integral OperatorsIntegral transform of scale function and waveletsRegularization of singular operators in LMW basisPrinciple of regularizationRegularization of convolution operator in LMW basisEstimates of Local Hölder IndicesBasis in Daubechies familyBasis in Multiwavelet familyError Estimates in the Multiscale ApproximationNonlinear/Best n-term ApproximationMultiscale Solution of Integral Equations with Weakly Singular KernelsExistence and UniquenessProjection in multiscale basisLogarithmic Singular KernelLMW basisKernels with Algebraic SingularityExistence and uniquenessApproximation in multiwavelet basisScale functionsScale functions and waveletsWaveletsMultiscale approximation (regularization) of integral operator KA in LMW basisReduction to algebraic equationsMultiscale approximation of solutionError EstimatesApproximation in other basisAn Integral Equation with Fixed SingularityMethod Based on Scale Functions in Daubechies FamilyBasic properties of Daubechies scale function and waveletsMethod of solutionNumerical resultsMultiscale Solution of Cauchy Singular Integral EquationsPrerequisitesBasis Comprising Truncated Scale Functions in Daubechies FamilyEvaluation of matrix elementsEvaluation of fTjEstimate of errorIllustrative examplesMultiwavelet FamilyEquation with constant coefficientsEvaluation of integralsMultiscale representation (regularization) of the operator KC in LMW basisMultiscale approximation of solutionEstimation of errorIllustrative examplesCauchy singular integral equation with variable coefficientsEvaluation of integrals involving function, Cauchy singular kernel and elements in LMW basisEvaluation of the integrals involving product of a(x), scale functions and waveletsMultiscale representation (regularization) of the operator ωKC in LMW basisMultiscale approximation of solutionEstimate of Hölder exponent of u(x) at the boundariesEstimation of errorApplications to problems in elasticityEquation of first kindEvaluation of integrals involving kernel with fixed singularity and elements in the LMW basisEvaluation of integrals involving kernel with fixed singularity and weight factorMultiscale representation (regularization) of the operator ωKF in LMW basisMultiscale approximation of solutionIllustrative examplesAutocorrelation function familyIn RTransformation to the finite range of integrationMultiscale approximation of solutionEstimation of errorIllustrative examplesOther familiesHilbert transformIntegral equation of second kindMultiscale Solution of Hypersingular Integral Equations of Second KindFinite Part Integrals Involving Hypersingular FunctionsExisting MethodsReduction to Cauchy Singular Integro-differential EquationMethod Based on LMW BasisMultiscale approximation of the solutionEstimation of errorIllustrative examplesOther FamiliesAppendicesAuthor IndexSubject Index