Ukrainian Mathematical Journal

, Volume 49, Issue 2, pp 269–280 | Cite as

Nonlinear nonlocal problems for a parabolic equation in a two-dimensional domain

  • Yu. A. Mitropol’skii
  • A. A. Berezovskii
  • M. Kh. Shkhanukov-Lafishev


We establish the convergence of the Rothe method for a parabolic equation with nonlocal boundary conditions and obtain an a priori estimate for the constructed difference scheme in the grid norm on a ball. We prove that the suggested iterative process for the solution of the posed problem converges in the small.


Parabolic Equation Integral Identity Ukrainian Academy Nonlocal Condition Nonlocal Boundary Condition 
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  1. 1.
    M. I. Vishik, “On general boundary-value problems,” Tr. Mosk. Mat. Obshch., 1, 186–246 (1952).Google Scholar
  2. 2.
    A. N. Tikhonov, “On functional Volterra-type equations and their applications to some problems in mathematical physics,” Vestn. Mosk. Univ., 8, No. 1, 1–25 (1938).MathSciNetGoogle Scholar
  3. 3.
    Yu. A. Surinov, “Integral equations of heat radiation and methods for the analysis of radiation heat exchange in systems of ‘gray’ bodies separated by diathermic media,” Izv. Akad. Nauk SSSR, No. 7, 981–1002 (1948).Google Scholar
  4. 4.
    A. A. Berezovskii, Nonlinear Boundary-Value Problems for Heat-Radiating Bodies [in Russian], Naukova Dumka, Kiev (1968).Google Scholar
  5. 5.
    L. M. Berezovskaya, “Temperature field of a nonsymmetrically heated thin cylindrical shell,” in: Theoretical and Applied Problems of Differential Equations and Algebra [in Russian]. Naukova Dumka, Kiev (1978), pp. 25–32.Google Scholar
  6. 6.
    O. A. Ladyzhenskaya, Boundary-Value Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973).Google Scholar
  7. 7.
    A. A. Samarskii, “Homogeneous difference schemes with nonuniform grids for parabolic equations,” Zh. Vychisl. Mat. Mat. Fiz., 3, No. 2, 266–298 (1963).MathSciNetGoogle Scholar
  8. 8.
    I. V. Fryazinov, “On difference schemes for the Poisson equation in polar, cylindrical, and spherical coordinates,” Zh. Vychisl. Mat. Mat. Fiz., 11, No. 5, 1219–1228 (1971).Google Scholar
  9. 9.
    D. G. Gordeziani, Methods for the Numerical Analysis of One Class of Nonlocal Boundary-Value Problems [in Russian], Preprint, Vekua Institute of Applied Mathematics, Georgian Academy of Sciences, Tbilisi (1981).Google Scholar
  10. 10.
    V. B. Andreev, “A method for the numerical solution of the third boundary-value problem for a parabolic equation in a p-dimensional parallelepiped,” in: Computational Methods and Programming [in Russian], Moscow University, Moscow (1967), pp. 64–75.Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • Yu. A. Mitropol’skii
    • 1
  • A. A. Berezovskii
    • 2
  • M. Kh. Shkhanukov-Lafishev
    • 3
  1. 1.Academician. Institute of MathematicsUkrainian Academy of SciencesKiev
  2. 2.Institute of MathematicsUkrainian Academy of SciencesKiev
  3. 3.Kabardino-Balkarian UniversityNalchik

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