Exact Solution of Some Integral Equations over a Circular Disc

Two-dimensional integral equations of the first kind over a circular disc are considered. The kernels involve the distance between two points on the disc raised to an arbitrary power. A review is given, comparing several published exact solutions for weakly-singular equations: these solutions are complicated, but three of them are shown to be equivalent. Some extensions to hypersingular equations are discussed.

[1]  Xian‐Fang Li,et al.  Solution of a class of two-dimensional integral equations , 2002 .

[2]  M. Abdou,et al.  Integral equation of mixed type and integrals of orthogonal polynomials , 2002 .

[3]  M. A. Darwish,et al.  Integral equations and potential-theortic type integrals of orthogonal polynomials , 1999 .

[4]  Paul A. Martin,et al.  Mapping flat cracks onto penny-shaped cracks, with application to somewhat circular tensile cracks , 1996 .

[5]  J. Boersma,et al.  On the Solution of an Integral Equation Arising in Potential Problems for Circular and Elliptic Disks , 1993, SIAM J. Appl. Math..

[6]  F. Penzel.  Sobolev space methods for the Laplace equation in the exterior of the disk , 1993 .

[7]  V. I. Fabrikant,et al.  Mixed Boundary Value Problems of Potential Theory and Their Applications in Engineering , 1991 .

[8]  Frank J. Rizzo,et al.  On boundary integral equations for crack problems , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[9]  E. Love,et al.  On a method of Fabrikant, Sankar and Swamy for a generalized potential problem , 1988 .

[10]  N. Ioakimidis.  Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity , 1982 .

[11]  T. Sankar,et al.  Asymmetric Contact Problem Including Wear for Nonhomogeneous Half Space , 1982 .

[12]  C. Kahane.  The solution of mildly singular integral equation of the first kind on a disk , 1981 .

[13]  R. W. Lardner,et al.  Penny-shaped cracks , 1975 .

[14]  P. Wolfe.  Eigenfunctions of the Integral Equation for the Potential of the Charged Disk , 1971 .

[15]  V. Panasyuk.  LIMITING EQUILIBRIUM OF BRITTLE SOLIDS WITH FRACTURES , 1969 .

[16]  A. Ramm,et al.  Justification of Fabrikant's method for solving mixed problems of potential theory , 1991 .

[17]  V. I. Fabrikant,et al.  Applications of Potential Theory in Mechanics: A Selection of New Results , 1989 .

[18]  T. Sankar,et al.  On contact problems in an inhomogeneous half-space , 1984 .

[19]  T. Sankar,et al.  On the generalized potential problem for a surface of revolution , 1984 .

[20]  H. Bueckner.  Field singularities and related integral representations , 1973 .