5,552 research outputs found

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

    Get PDF
    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

    Asymptotic expansions and fast computation of oscillatory Hilbert transforms

    Full text link
    In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form H+(f(t)eiωt)(x)=−int0∞eiωtf(t)t−xdt,ω>0,x≥0,H^{+}(f(t)e^{i\omega t})(x)=-int_{0}^{\infty}e^{i\omega t}\frac{f(t)}{t-x}dt,\qquad \omega>0,\qquad x\geq 0, where the bar indicates the Cauchy principal value and ff is a real-valued function with analytic continuation in the first quadrant, except possibly a branch point of algebraic type at the origin. When x=0x=0, the integral is interpreted as a Hadamard finite-part integral, provided it is divergent. Asymptotic expansions in inverse powers of ω\omega are derived for each fixed x≥0x\geq 0, which clarify the large ω\omega behavior of this transform. We then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms. Depending on the position of xx, we classify our discussion into three regimes, namely, x=O(1)x=\mathcal{O}(1) or x≫1x\gg1, 0<x≪10<x\ll 1 and x=0x=0. Numerical experiments show that the convergence of the proposed methods greatly improve when the frequency ω\omega increases. Some extensions to oscillatory Hilbert transforms with Bessel oscillators are briefly discussed as well.Comment: 32 pages, 6 figures, 4 table

    Numerical quadrature methods for integrals of singular periodic functions and their application to singular and weakly singular integral equations

    Get PDF
    High accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Such periodic equations are used in the solution of planar elliptic boundary value problems, elasticity, potential theory, conformal mapping, boundary element methods, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples
    • …
    corecore