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The numerical solution of integral equations with weakly singular kernels

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Numerical Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1066))

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References

  1. Abdalkhani, J.: Collocation and Runge-Kutta-type methods for Volterra integral equations with weakly singular kernels, Ph. D. Thesis, Dalhousie University, Halifax, N.S. (Canada), 1982.

    MATH  Google Scholar 

  2. Atkinson, K., Graham, I., and Sloan, I.: Piecewise continuous collocation for integral equations, SIAM J. Numer. Anal., 20 (1983), 172–186.

    Article  MathSciNet  MATH  Google Scholar 

  3. Brunner, H.: A survey of recent advances in the numerical treatment of Volterra integral and integro-differential equations, J. Comp. Appl. Math., 8 (1982), 213–229.

    Article  MathSciNet  MATH  Google Scholar 

  4. Brunner, H.: Nonpolynomial spline collocation for Volterra equations with weakly singular kernels, SIAM J. Numer. Anal., 20 (1983).

    Google Scholar 

  5. Brunner, H.: Iterated collocation methods and their discretizations for Volterra integral equations, SIAM J. Numer. Anal. (to appear).

    Google Scholar 

  6. Brunner, H., and Graham, I.G.: Product integration for weakly singular Volterra integral equation (to appear).

    Google Scholar 

  7. Brunner, H., and Evans, M.D.: Piecewise polynomial collocation for Volterra-type integral equations of the second kind, J. Inst. Math. Appl., 20 (1977), 415–423.

    Article  MathSciNet  MATH  Google Scholar 

  8. Brunner, H., and Nørsett, S.P.: Superconvergence of collocation methods for Volterra and Abel integral equations of the second kind, Numer. Math., 36 (1981), 347–358.

    Article  MathSciNet  MATH  Google Scholar 

  9. Chandler, G.A.: Superconvergence of numerical methods to second kind integral equations, Ph. D. Thesis, Australian National University, Canberra, 1979.

    Google Scholar 

  10. Chandler, G.A.: Superconvergence for second kind integral equations, in: The Application and Numerical Solution of Integral Equations (R.S. Anderssen, F.R. de Hoog and M.A. Lukas, eds.), Sijthoff & Noordhoff, Alphen/Rijn (Netherlands), 1980. pp. 103–117.

    Chapter  Google Scholar 

  11. Chatelin, F., and Lebbar, R.: The iterated projection solution for the Fredholm integral equation of second kind, J. Austral. Math. Soc. Ser. B, 22 (1981), 439–451.

    Article  MathSciNet  MATH  Google Scholar 

  12. Garey, L.: The numerical solution of Volterra integral equations with singular kernels, BIT, 14 (1974), 33–39.

    Article  MathSciNet  MATH  Google Scholar 

  13. Graham, I.G.: The numerical solution of Fredholm integral equations of the second kind, Ph. D. Thesis, University of New South Wales, Sydney, 1980.

    Google Scholar 

  14. Graham, I.G.: Singularity expansions for the solutions of second kind Fredholm integral equations with weakly singular convolution kernels, J. Integral Equations, 4 (1982), 1–30.

    MathSciNet  MATH  Google Scholar 

  15. Graham, I.G.: Galerkin methods for second kind integral equations with singularities, Math. Comp., 39 (1982), 519–533.

    Article  MathSciNet  MATH  Google Scholar 

  16. Handelsman, R.A., and Olmstead, W.E.: Asymptotic solution to a class of nonlinear Volterra integral equations, SIAM J. Appl. Math., 22 (1972), 373–384.

    Article  MathSciNet  MATH  Google Scholar 

  17. de Hoog, F., and Weiss, R.: Asymptotic expansions for product integration, Math. Comp., 27 (1973), 295–306.

    Article  MathSciNet  MATH  Google Scholar 

  18. de Hoog, F.R., and Weiss, R.: High order methods for a class of Volterra integral equations with weakly singular kernels, SIAM J. Numer. Anal., 11 (1974), 1166–1180.

    Article  MathSciNet  MATH  Google Scholar 

  19. Kershaw, D.: Some results for Abel-Volterra integral equations of the second kind, in: Treatment of Integral Equations by Numerical Methods (C.T.H. Baker and G.F. Miller, eds.), Academic Press, London, 1982, pp. 273–282.

    Google Scholar 

  20. Linz, P.: Numerical methods for Volterra integral equations with singular kernels, SIAM J. Numer. Anal., 6 (1969), 365–374.

    Article  MathSciNet  MATH  Google Scholar 

  21. Logan, J.E.: The approximate solution of Volterra integral equations of the second kind, Ph. D. Thesis, University of Iowa, Iowa City, 1976.

    Google Scholar 

  22. Lubich, C.: Runge-Kutta theory for Volterra and Abel integral equations of the second kind, Preprint Nr. 154, Sonderforschungsbereich 123, University of Heidelberg, 1982.

    Google Scholar 

  23. Lyness, J.N., and Ninham, B.W.: Numerical quadrature and asymptotic expansions, Math. Comp., 21 (1967), 162–178.

    Article  MathSciNet  MATH  Google Scholar 

  24. McKee, S.: Generalized discrete Gronwall lemmas, Z. Angew. Math. Mech., 62 (1982), 429–434.

    Article  MathSciNet  MATH  Google Scholar 

  25. Miller, R.K., and Feldstein, A.: Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM J. Math. Anal., 2 (1971), 242–258.

    Article  MathSciNet  MATH  Google Scholar 

  26. Olmstead, W.E.: A nonlinear integral equation associated with gas absorption in a liquid, Z. Angew. Math. Phys., 28 (1977), 513–523.

    Article  MathSciNet  MATH  Google Scholar 

  27. Oulès, H.: Résolution numérique d’une équation intégrale singulière, Rev. Française Trait. Inform. (Chiffres), 7 (1964), 117–124.

    MATH  Google Scholar 

  28. Pitkäranta, J.: On the differential properties of solutions to Fredholm equations with weakly singular kernels, J. Inst. Math. Appl., 24 (1979), 109–119.

    Article  MathSciNet  MATH  Google Scholar 

  29. Rice, J.R.: On the degree of convergence of nonlinear spline approximation, in: Approximation with Special Emphasis on Spline Functions (I.J. Schoenberg, ed.), Academic Press, New York, 1969, pp. 349–365.

    Google Scholar 

  30. Richter, G.R.: On weakly singular Fredholm integral equations with displacement kernels, J. Math. Anal. Appl., 55 (1976), 32–42.

    Article  MathSciNet  MATH  Google Scholar 

  31. te Riele, H.J.J.: Collocation methods for weakly singular second-kind Volterra integral equations with non-smooth solution, IMA J. Numer. Anal., 2 (1982), 437–449.

    Article  MathSciNet  MATH  Google Scholar 

  32. Schneider, C.: Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind, Integral Equations Operator Theory, 2 (1979), 62–68.

    Article  MathSciNet  MATH  Google Scholar 

  33. Schneider, C.: Produktintegration mit nicht-äquidistanten Stützstellen, Numer. Math., 35 (1980), 35–43.

    Article  MathSciNet  MATH  Google Scholar 

  34. Schneider, C.: Product integration for weakly singular integral equations, Math. Comp., 36 (1981), 207–213.

    Article  MathSciNet  MATH  Google Scholar 

  35. Sloan, I.H.: Improvement by interation for compact operator equations, Math. Comp., 30 (1976), 758–764.

    Article  MathSciNet  MATH  Google Scholar 

  36. Sloan, I.H.: A review of numerical methods for Fredholm equations of the second kind, in: The Application and Numerical Solution of Integral Equations (R.S. Anderssen, F.R. de Hoog and M.A. Lukas, eds.), Sijthoff & Noordhoff, Alphen/Rijn (Netherlands), 1980, pp. 51–74.

    Chapter  Google Scholar 

  37. Vainikko, G., and Pedas, A.: The properties of solutions of weakly singular integral equations, J. Austral. Math. Soc. Ser. B, 22 (1981), 419–430.

    Article  MathSciNet  MATH  Google Scholar 

  38. Vainikko, G., and Uba, P.: A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel, J. Austral. Math. Soc. Ser. B, 22 (1981), 431–438.

    Article  MathSciNet  MATH  Google Scholar 

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David F. Griffiths

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© 1984 Springer-Verlag

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Brunner, H. (1984). The numerical solution of integral equations with weakly singular kernels. In: Griffiths, D.F. (eds) Numerical Analysis. Lecture Notes in Mathematics, vol 1066. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0099518

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  • DOI: https://doi.org/10.1007/BFb0099518

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