Computational methods for integral equations
Computational methods for integral equations
Mathematica: a system for doing mathematics by computer (2nd ed.)
Mathematica: a system for doing mathematics by computer (2nd ed.)
Numerical evaluation of hypersingular integrals
ICCAM'92 Proceedings of the fifth international conference on Computational and applied mathematics
Generalized Gaussian quadrature rules for systems of arbitrary functions
SIAM Journal on Numerical Analysis
Gauss-Chebyshev quadrature formulae for strongly singular integrals
Quarterly of Applied Mathematics
Numerical Quadratures for Singular and Hypersingular Integrals in Boundary Element Methods
SIAM Journal on Scientific Computing
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Singular integrals of the form with logarithmic, Cauchy, or Hadamard-type singularities in addition to endpoint algebraic singularities w(τ) = (1 − τ2)± 1/2 are frequently encountered in integral equation formulations of potential problems. Some of the existing quadratures for the evaluation of such integrals only apply to preassigned values of the external variable t. Other fairly general rules suffer from loss of accuracy when t is close to any of the nodes of the quadrature. Finally, derivation of the rules is in general multistage, and thus, considerable analytical preprocessing is required. In this paper, a straightforward direct method is presented, which demonstrates the derivation of numerical quadrature rules for Cauchy-type and Hadamard-type integrals from corresponding quadratures pertinent to logarithmically singular integrals. The proposed rules share the following characteristics: (1) their derivation, based on first principles, is remarkably simple; (2) the external variable t may be arbitrarily selected; and (3) in their framework, the loss of accuracy referred to earlier is fully remedied. Numerical examples and case studies illustrate the simplicity, flexibility, and high accuracy of the algorithms. Application to the solution of an integral equation associated with wave diffraction by a perfectly conducting strip is exemplified. Copyright © 2012 John Wiley & Sons, Ltd.