Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On the solution of integral equations with strongly singular kernels

Authors: A. C. Kaya and F. Erdogan
Journal: Quart. Appl. Math. 45 (1987), 105-122
MSC: Primary 45E99; Secondary 45L10, 65R20
DOI: https://doi.org/10.1090/qam/885173
MathSciNet review: 885173
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Abstract: In this paper some useful formulas are developed to evaluate integrals having a singularity of the form $ {\left( {t - x} \right)^{ - m}},m \ge 1$. Interpreting the integrals with strong singularities in the Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term $ {\left( {t - x} \right)^{ - m}}$, terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.

References [Enhancements On Off] (What's this?)

  • [1] J. Hadamard, Lectures on Cauchy's problem in linear partial differential equations, Yale University Press, 1923.
  • [2] L. Schwartz, Théorie des distributions, Vols. I and II, Hermann, 1957-1959
  • [3] K. W. Mangler, Improper integrals in theoretical aerodynamics, Royal Aircraft Establishment, Farnborough, Report No. 2424 (1951)
  • [4] H. R. Kutt, The numerical evaluation of principal value integrals by finite-part integration, Numer. Math. 24, 205-210 (1975) MR 0378366
  • [5] H. R. Kutt, Quadrature formulae for finite-part integrals, Special Report WISK 178, Pretoria, National Research Institute for Mathematical Sciences, 1975
  • [6] H. R. Kutt, On the numerical evaluation of finite-part integrals involving an algebraic singularity, Special Report WISK 179, Pretoria, National Research Institute for Mathematical Sciences, 1975
  • [7] N. I. Muskhelishvili, Singular integral equations, Noordhoff, Groningen, 1953 MR 0355494
  • [8] N. I. Ioakimidis, On the numerical evaluation of derivatives of Cauchy principal value integrals, Computing 27, 81-88 (1981) MR 623178
  • [9] C. T. H. Baker, The numerical treatment of integral equations, Clarendon Press, Oxford (1977) MR 0467215
  • [10] A. C. Kaya, Applications of integral equations with strong singularities in fracture mechanics, Ph. D. thesis, Lehigh University, 1984
  • [11] W. T. Koiter, Discussion of ``Rectangular tensile sheet with symmetrical edge cracks,'' by O. L. Bowie, J. Appl. Mech. 32, Trans. ASME 87, Series E, 237 (1965)
  • [12] F. G. Tricomi, On the finite Hilbert transformation, Quart. J. Math. Oxford Ser. (2) 2, 199-211 (1951) MR 0043258
  • [13] M. Abramowitz and I. A. Stegun, ed., Handbook of mathematical functions, Dover Publications, Inc., New York, 1965.
  • [14] I. S. Gradshtein and I. M. Ryzhik, Table of integrals, series and products, Academic Press, New York, 1965 MR 1773820

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DOI: https://doi.org/10.1090/qam/885173
Article copyright: © Copyright 1987 American Mathematical Society

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