A direct method to quadrature rules for a certain class of singular integrals with logarithmic, Cauchy, or Hadamard-type singularities

作者：Tsalamengas John L^*

来源：International Journal of Numerical Modelling: Electronic Networks, Devices and Fields , 2012, 25(5-6): 512-524.

DOI：10.1002/jnm.1863

• quadrature rules
• singular integrals
• integral equations
• Nystrom method

Singular integrals of the form ?-11w(t)f(t)K(t,t)dt with logarithmic, Cauchy, or Hadamard-type singularities in addition to endpoint algebraic singularities w(t)?=?(1?-?t2)+/-?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.

• 出版日期2012-12