Quadrature Formulas of Quasi-Interpolation Type for Singular Integrals with Hilbert Kernel

Numerical quadrature for computing of singular integrals

In the present work we have studied superconvergence of Hadamard finite-part integral. We have studied the second-order and the third-order quadrature formulae of Newton-Cotes type. We follow works [Sun, Wu, 2005b], [Lü, Wu, 2005] and work [Wu, Yu and Zhang, 2009] and introduce new rule which gives the same convergence rate as rules in [Lü and Wu, 2005] and [Wu, Yu and Zhang, 2009] but in more ...

Quadrature methods for highly oscillatory singular integrals

We study asymptotic expansions, Filon-type methods and complex-valued Gaussian quadrature for highly oscillatory integrals with power-law and logarithmic singularities. We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand, the stationary points and the endpoints of the integral. A truncated asymptotic expansion...

Hierarchical quadrature for multidimensional singular integrals

We introduce a method for the evaluation of singular integrals arising in the discretisation of integral equations. The method is based on the idea of repeated subdivision of domains. The integrals defined on these subdomains are classified. Each class can be expressed as a sum of regular integrals and representatives of other classes. A system of equations describes the relations between the c...

Singular Integrals on Hilbert Space

exists as a bounded operator on L(H) as ô tends to zero and p tends to infinity. A theorem of this type extends the Calderon-Zygmund theory of singular integral operators on En to infinite dimensions. For if fc(#)||x||-" is a Calderon-Zygmund kernel and if £ is a bounded Borel set which is disjoint from a neighborhood of the origin then v{E) =fEk(x)\\x\\~ dx satisfies v(tE) = v(E) for £>0; if g...

Quadrature Rules for Fuzzy Integrals