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Solution of integral equations in fractional Sobolev spaces

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Abstract

We consider linear integral equations and Urysohn equations with constant integration limits. Sufficient conditions are given for the solutions of these equations to be in Sobolev spacesW α2 (0,1), 0 ≤ α ≤ 2. Finite-difference schemes are constructed for approximate solution of the original equation by special averaging of the right-hand side kernel. The rate of convergence of the approximate solution to the averaged exact solution is shown to beO(hα|ln h|δ(1/2,α)+δ(3/2,α)).

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References

  1. B. A. Bel'tyukov, Some Methods of Approximate Solution of Linear and Nonlinear Integral Equations [in Russian], thesis, Irkutsk (1967).

  2. A. F. Verlan' and V. S. Sizikov, Methods of Solution of Integral Equations Using Computer Programs [in Russian], Naukova Dumka, Kiev (1978).

    Google Scholar 

  3. P. P. Zabreiko, A. I. Koshelev, M. A. Krasnosel'skii, et al., Integral Equations [in Russian], Nauka, Moscow (1968).

    Google Scholar 

  4. L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis [in Russian], Gostekhizdat, Moscow-Leningrad (1962).

    Google Scholar 

  5. R. D. Lazarov, V. L. Makarov, and A. A. Samarskii, “Application of exact difference schemes for construction and analysis of difference schemes on generalized solutions,” Matem. Sb.,117(159), No. 4, 469–480 (1982).

    Google Scholar 

  6. V. L. Makarov and A. A. Samarskii, “Application of exact difference schemes to estimate the convergence of the method of lines,” Zh. Vychisl. Matem. Mat. Fiz.,20, No. 2, 205–217 (1980).

    Google Scholar 

  7. S. G. Mikhlin and Kh. L. Smolitskii, Approximate Methods of Solution of Differential and Integral Equations [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  8. L. A. Oganesyan and L. A. Rukhovets, Variational-Difference Methods of Solution of Elliptic Equations [in Russian], Izd. Akad. Nauk Arm. SSSR, Erevan (1979).

    Google Scholar 

  9. V. A. Trenogin, Functional Analysis [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  10. F. Tricomi, Integral Equations, New York, Interscience (1957).

    Google Scholar 

  11. K. E. Atkinson and E. Kendall, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, Philadelphia, PA (1976).

    Google Scholar 

  12. L. M. Delves and J. Wals, Numerical Solution of Integral Equations, Clarendon Press, Oxford (1974).

    Google Scholar 

  13. M. A. Goldberg, “A survey of numerical methods for integral equations,” in: Solution Methods of Integral Equations, Theory and Applications, Academic Press, New York (1979), pp. 1–58.

    Google Scholar 

  14. B. Noble, “The numerical solution of nonlinear integral equations and related topics,” in: Nonlinear Integral Equations, P. M. Anselone (ed.), Univ. of Wisconsin Press, Madison (1964), pp. 215–318.

    Google Scholar 

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Authors and Affiliations

  1. Kiev University, USSR

    V. L. Makarov & G. S. Karkarashvili

  2. Mathematical Institute, Georgian Academy of Sciences, USSR

    V. L. Makarov & G. S. Karkarashvili

Authors
  1. V. L. Makarov
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  2. G. S. Karkarashvili
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Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 3–19, 1987.

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Makarov, V.L., Karkarashvili, G.S. Solution of integral equations in fractional Sobolev spaces. J Math Sci 66, 2125–2138 (1993). https://doi.org/10.1007/BF01098595

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  • DOI: https://doi.org/10.1007/BF01098595

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