Ian Sloan’s Legacy in Integral Equation Methods

  • Thanh TranEmail author


In almost four decades, from the early 1970s until the first decade of this century, Ian Sloan has contributed immensely in the area of integral equation methods for elliptic boundary value problems. A search on MathSciNet with entries “Author=Sloan” and “Title=integral equation” reveals 44 papers. This review article sheds some lights on this historic path.



I am deeply indebted to Ian Sloan for his invaluable guides in the early years of my career. Happy 80th birthday, Ian.


  1. 1.
    Adhikari, S.K., Sloan, I.H.: Separable operator expansions for the t-matrix. Nucl. Phys. A 241, 429–442 (1975)CrossRefGoogle Scholar
  2. 2.
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  3. 3.
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  4. 4.
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  5. 5.
    Costabel, M., Stephan, E.P.: Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation. In: Mathematical Models and Methods in Mechanics. Banach Center Publications, vol. 15, pp. 175–251. Polish Scientific Publishers, Warszawa (1985)MathSciNetCrossRefGoogle Scholar
  6. 6.
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  8. 8.
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  9. 9.
    Kress, R.: Linear Integral Equations. Springer, New York (1999)CrossRefGoogle Scholar
  10. 10.
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  11. 11.
    Le Roux, M.N.: Equations intégrales pour le problème du potentiel électrique dans le plan. C. R. Acad. Sci. Paris Sér. A 278, 541–544 (1974)Google Scholar
  12. 12.
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  14. 14.
    Richter, G.R.: Numerical solution of integral equations of the first kind with nonsmooth kernels. SIAM J. Numer. Anal. 15(3), 511–522 (1978)MathSciNetCrossRefGoogle Scholar
  15. 15.
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  16. 16.
    Sloan, I.H.: The method of polarized orbitals for the elastic scattering of slow electrons by ionized helium and atomic hydrogen. Proc. R. Soc. A (London) 281, 151–163 (1964)Google Scholar
  17. 17.
    Sloan, I.H.: Convergence of degenerate-kernel methods. J. Aust. Math. Soc. Ser. B 19(4), 422–431 (1975/76)MathSciNetCrossRefGoogle Scholar
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    Sloan, I.H.: Improvement by iteration for compact operator equations. Math. Comput. 30(136), 758–764 (1976)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sloan, I.H.: Iterated Galerkin method for eigenvalue problems. SIAM J. Numer. Anal. 13(5), 753–760 (1976)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sloan, I.H.: Three-body collisions involving breakup. In: Devins, D. (ed.) Momentum Wave Functions, AIP Conference Proceedings, pp. 187–194 (1977)Google Scholar
  22. 22.
    Sloan, I.H.: A quadrature-based approach to improving the collocation method. Numer. Math. 54, 41–56 (1988)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Sloan, I.H.: Error analysis of boundary integral methods. Acta Numer. 1, 287–339 (1992)MathSciNetCrossRefGoogle Scholar
  24. 24.
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  25. 25.
    Sloan, I.H., Adhikari, S.K.: Method for Lippmann-Schwinger equations. Nucl. Phys. A 235, 352–360 (1974)CrossRefGoogle Scholar
  26. 26.
    Sloan, I.H., Moore, E.J.: Integral equation approach to electron-hydrogen collisions. J. Phys. B (Proc. Phys. Soc.) 1(3), 414–422 (1968)CrossRefGoogle Scholar
  27. 27.
    Sloan, I.H., Spence, A.: The Galerkin method for integral equations of the first kind with logarithmic kernel: theory. IMA J. Numer. Anal. 8(1), 105–122 (1988)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Sloan, I.H., Thomée, V.: Superconvergence of the Galerkin iterates for integral equations of the second kind. J. Integr. Equ. 9(1), 1–23 (1985)MathSciNetzbMATHGoogle Scholar
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    Sloan, I.H., Tran, T.: The tolerant qualocation method for variable-coefficient elliptic equations on curves. J. Integr. Eqn. Appl. 13, 73–98 (2001)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Sloan, I.H., Wendland, W.L.: A quadrature-based approach to improving the collocation method for splines of even degree. Z. Anal. Anwend. 8(4), 361–376 (1989)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Sloan, I.H., Wendland, W.L.: Qualocation methods for elliptic boundary integral equations. Numer. Math. 79, 451–483 (1998)MathSciNetCrossRefGoogle Scholar
  32. 32.
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  33. 33.
    Sloan, I.H., Burn, B.J., Datyner, N.: A new approach to the numerical solution of integral equations. J. Comput. Phys. 18, 92–105 (1975)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Tran, T., Sloan, I.H.: Tolerant qualocation – a qualocation method for boundary integral equations with reduced regularity requirement. J. Integr. Eqn. Appl. 10, 85–115 (1998)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Wahlbin, L.B.: Superconvergence in Galerkin Finite Element Methods. Lecture Notes in Mathematics, vol. 1605. Springer, Berlin/Heidelberg (1995)CrossRefGoogle Scholar
  36. 36.
    Wendland, W.L.: Boundary element methods and their asymptotic convergence. In: Theoretical Acoustics and Numerical Techniques. CISM Courses and Lectures, vol. 277, pp. 135–216. Springer, Vienna (1983)CrossRefGoogle Scholar
  37. 37.
    Yan, Y., Sloan, I.H.: On integral equations of the first kind with logarithmic kernels. J. Integr. Equ. Appl. 1(4), 549–579 (1988)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Yan, Y., Sloan, I.H.: Mesh grading for integral equations of the first kind with logarithmic kernel. SIAM J. Numer. Anal. 26(3), 574–587 (1989)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsThe University of New South WalesSydneyAustralia

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