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Stable solvability of Padé difference schemes for parabolic equations in Hölder spaces

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Ukrainian Mathematical Journal Aims and scope

Abstract

We study Padé difference scheme for the approximate solution of the Cauchy problem for parabolic equations generated by the Padé fractions Rj,ℓ, of exponential approximation. We establish an estimate of the coerciveness of the difference schemes for j=ℓ−2,ℓ −1, or even j=ℓ in a smaller space than C0 α(E).

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References

  1. P. E. Sobolevskii and Khoang Van Lai, “Difference schemes of optimal type for the approximate solution of parabolic equations (Banach case),” Ukr. Mat. Zhurn.,33, No. 1, 39–46 (1981).

    Google Scholar 

  2. Kh. A. Alibekov and P. E. Sobolevskii, “A method of construction and study of Padé class schemes,” Differents. Uravnen. Primen.,32, 9–29 (1982).

    Google Scholar 

  3. R. Hersh and T. Kato, “High-accuracy stable difference schemes for well-posed initial value problem,” SIAM J. Numer. Anal.,16, No. 4, 670–682 (1979).

    Google Scholar 

  4. P. Brenner and V. Thomee, “On rational approximations of semigroups,” SIAM J. Numer. Anal.,16, No. 4, 683–694 (1979).

    Google Scholar 

  5. P. Brenner, M. Grouzeix, and V. Thomee, “Single step methods for inhomogeneous differential equations in Banach space,” PAIRO J. Numer. Anal.,16, No. 1, 5–26 (1982).

    Google Scholar 

  6. N. Yu. Bakaev, “Estimates of stability of difference schemes for a differential equation with constant operator. I,” in: Partial Differential Equations [in Russian], Nauka, Novosibirsk (1989), pp. 3–14.

    Google Scholar 

  7. S. G. Krein, Linear Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  8. P. E. Sobolevskii and M. F. Tiunchuk, “Difference method of approximate solution of quasilinear elliptic and parabolic equations,” Tr. Mat. F. Voronezh. Univ.,1, 82–106 (1970).

    Google Scholar 

  9. V. B. Andreev, “Stability of difference schemes for parabolic equations with respect to the initial data,” Zhurn. Vychisl. Mat. Mat. Fiz.,11, No. 6, 1462–1475 (1971).

    Google Scholar 

  10. A. G. Grif, “Stability in W2 2,1 of difference schemes for parabolic equations,” in: Studies on the Theory of Difference Schemes for Elliptic and Parabolic Equations [in Russian], Izdat. Mosk. Univ., Moscow (1973), pp. 88–112.

    Google Scholar 

  11. N. I. Ionkin and Yu. I. Mokin, “Parabolic difference schemes,” Zhurn. Vychisl. Mat. Mat. Fiz.,14, No. 2, 402–417 (1974).

    Google Scholar 

  12. P. E. Sobolevskii, “Coercive solvability of difference equations,” Dokl. Akad. Nauk SSSR,201, No. 5, 1063–1066 (1971).

    Google Scholar 

  13. P. E. Sobolevskii, “Theory of semigroups and stability of difference schemes,” in: Theory of Operators on Function Spaces [in Russian], Nauka, Novosibirsk (1977), pp. 304–337.

    Google Scholar 

  14. A. E. Polichka and P. E. Sobolevskii, “Stable solvability of difference parabolic equations in Bochner spaces,” Tr. Mosk. Mat. Obshch.,36, 29–57 (1978).

    Google Scholar 

  15. A. E. Polichka and P. E. Sobolevskii, “New Lp-estimates for difference parabolic problems,” Zhurn. Vychisl. Mat. Mat. Fiz.,3, No. 2, 266–298 (1963).

    Google Scholar 

  16. A. O. Ashyralyev and P. E. Sobolevskii, “Stable solvability of the Crank-Nicholson difference scheme for parabolic equations,” Izv. AN Turkm. SSR. Ser. Fiz.-Tekhn., Khim. Geol. Nauk., No. 6, 10–16 (1981).

    Google Scholar 

  17. A. O. Ashyralyev and P. E. Sobolevskii, “Coercive stability of Crank-Nicholson difference scheme in spaces,” in: Approximate Methods of Study of Differential Equations and their Application [in Russian], Kuibyshev. Univ., Kuibyshev (1982), pp. 16–24.

    Google Scholar 

  18. A. O. Ashyralyev, “A purely implicit second-order approximation difference scheme for parabolic equations,” Izv. AN Turkm. SSR. Ser. Fiz.-Tekhn., Khim., Geol. Nauk., No. 4, 3–13 (1987).

    Google Scholar 

  19. J. Baker and P. Graves-Morris, Padé Approximants [Russian translation], Mir, Moscow (1986).

    Google Scholar 

  20. A. O. Ashyralyev and P. E. Sobolevskii, “Difference schemes for parabolic equations,” in: Differential Equations and their Applications: Abstracts of Reports to the All-Union Conference, [in Russian], Ashkhabad (1985) pp. 39–40.

  21. P. E. Sobolevskii, “Coercivity inequalities for abstract parabolic equations,” Dokl. Akad. Nauk SSSR,157, No. 1, 52–56 (1964).

    Google Scholar 

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Authors and Affiliations

  1. Mathematics Institute of the Academy of Sciences of Ukraine, Kiev

    A. Ashyralyev

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  1. A. Ashyralyev
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1466–1476, November, 1992.

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Ashyralyev, A. Stable solvability of Padé difference schemes for parabolic equations in Hölder spaces. Ukr Math J 44, 1349–1358 (1992). https://doi.org/10.1007/BF01071507

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  • DOI: https://doi.org/10.1007/BF01071507

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