4,671 research outputs found

    A fast and well-conditioned spectral method for singular integral equations

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    We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in O(m2n){\cal O}(m^2n) operations using an adaptive QR factorization, where mm is the bandwidth and nn is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to O(mn){\cal O}(m n) operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The Julia software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface

    On the convergence of local expansions of layer potentials

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    In a recently developed quadrature method (quadrature by expansion or QBX), it was demonstrated that weakly singular or singular layer potentials can be evaluated rapidly and accurately on surface by making use of local expansions about carefully chosen off-surface points. In this paper, we derive estimates for the rate of convergence of these local expansions, providing the analytic foundation for the QBX method. The estimates may also be of mathematical interest, particularly for microlocal or asymptotic analysis in potential theory

    The Nystrom plus Correction Method for Solving Bound State Equations in Momentum Space

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    A new method is presented for solving the momentum-space Schrodinger equation with a linear potential. The Lande-subtracted momentum space integral equation can be transformed into a matrix equation by the Nystrom method. The method produces only approximate eigenvalues in the cases of singular potentials such as the linear potential. The eigenvalues generated by the Nystrom method can be improved by calculating the numerical errors and adding the appropriate corrections. The end results are more accurate eigenvalues than those generated by the basis function method. The method is also shown to work for a relativistic equation such as the Thompson equation.Comment: Revtex, 21 pages, 4 tables, to be published in Physical Review

    A note on the integral equation for the Wilson loop in N = 2 D=4 superconformal Yang-Mills theory

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    We propose an alternative method to study the saddle point equation in the strong coupling limit for the Wilson loop in N=2\mathcal{N}=2 D=4 super Yang-Mills with an SU(N) gauge group and 2N hypermultiplets. This method is based on an approximation of the integral equation kernel which allows to solve the simplified problem exactly. To determine the accuracy of this approximation, we compare our results to those obtained recently by Passerini and Zarembo. Although less precise, this simpler approach provides an explicit expression for the density of eigenvalues that is used to derive the planar free energy.Comment: 12 pages, v2: section 2.5 (Free Energy) amended and reference added, to appear in J. Phys.

    On direct numerical treatment of hypersingular integral equations arising in mechanics and acoustics

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    In this paper we present a treatment of hypersingular integral equations, which have relevant applications in many problems of wave dynamics, elasticity and fluid mechanics with mixed boundary conditions. The main goal of the present work is the development of an efficient direct numerical collocation method. The paper is completed with two examples taken from crack theory and acoustics: the study of a single crack in a linear isotropic elastic medium, and diffraction of a plane acoustic wave by a thin rigid screen.Comment: accepted by Acta Mechanica, 19 pages, 3 figure

    Unsteady two dimensional airloads acting on oscillating thin airfoils in subsonic ventilated wind tunnels

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    The numerical calculation of unsteady two dimensional airloads which act upon thin airfoils in subsonic ventilated wind tunnels was studied. Neglecting certain quadrature errors, Bland's collocation method is rigorously proved to converge to the mathematically exact solution of Bland's integral equation, and a three way equivalence was established between collocation, Galerkin's method and least squares whenever the collocation points are chosen to be the nodes of the quadrature rule used for Galerkin's method. A computer program displayed convergence with respect to the number of pressure basis functions employed, and agreement with known special cases was demonstrated. Results are obtained for the combined effects of wind tunnel wall ventilation and wind tunnel depth to airfoil chord ratio, and for acoustic resonance between the airfoil and wind tunnel walls. A boundary condition is proposed for permeable walls through which mass flow rate is proportional to pressure jump
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