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 Uspekhi Mat. Nauk, 1971, Volume 26, Issue 3(159), Pages 105–160 (Mi umn5199)

The method of intrinsic boundary conditions in the theory of difference boundary value problems

V. S. Ryaben'kii

Abstract: In this article we consider general systems of difference equations with constant coefficients in arbitrary many-dimensional network domains. We define the boundary of a network domain in a certain way and give a formula that expresses the values of a solution at each point of the network domain in terms of its values at points of the boundary. We use this formula to derive necessary and sufficient conditions, which we call 'intrinsic boundary conditions', for a vector function given on the boundary of a network domain to be extendable everywhere in this domain to a solution. This formula allows us to appreciate that it is natural to consider a general boundary value problem for the systems in question.
The method we suggest for investigating and calculating solutions of difference boundary value problems consists in going from the original problem to the problem on the boundary that arises when we consider a combination of given and intrinsic boundary conditions. We present results obtained by the method of intrinsic boundary conditions. In the main they relate to non-stationary problems in simple and composite domains and have various degrees of effectiveness.
We refer in the paper to other methods only to appreciate the position of the new method among those already available; it interacts with them and supplements them.

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English version:
Russian Mathematical Surveys, 1971, 26:3, 117–176

Bibliographic databases:

UDC: 517.9
MSC: 45Exx, 39B12, 35Sxx, 35J25

Citation: V. S. Ryaben'kii, “The method of intrinsic boundary conditions in the theory of difference boundary value problems”, Uspekhi Mat. Nauk, 26:3(159) (1971), 105–160; Russian Math. Surveys, 26:3 (1971), 117–176

Citation in format AMSBIB
\Bibitem{Rya71} \by V.~S.~Ryaben'kii \paper The method of intrinsic boundary conditions in the theory of difference boundary value problems \jour Uspekhi Mat. Nauk \yr 1971 \vol 26 \issue 3(159) \pages 105--160 \mathnet{http://mi.mathnet.ru/umn5199} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=303149} \zmath{https://zbmath.org/?q=an:0257.35076} \transl \jour Russian Math. Surveys \yr 1971 \vol 26 \issue 3 \pages 117--176 \crossref{https://doi.org/10.1070/RM1971v026n03ABEH003836} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. V. Gulin, A. A. Samarskii, “On some results and problems of the stability theory of difference schemes”, Math. USSR-Sb., 28:3 (1976), 263–290
2. D. G. Vasil'ev, “Two-term asymptotics of the spectrum of a boundary-value problem under an interior reflection of general form”, Funct. Anal. Appl., 18:4 (1984), 267–277
3. I. L. Sofronov, “Nondegeneracy of the equations related to the method of difference potentials”, Funct. Anal. Appl., 18:4 (1984), 347–349
4. V. S. Ryaben'kii, “Boundary equations with projections”, Russian Math. Surveys, 40:2 (1985), 147–183
5. V. S. Ryaben'kii, “Difference Potentials Method and its Applications”, Math Nachr, 177:1 (1996), 251
6. V. S. Ryaben'kiǐ, “Potentials for general linear systems of difference equations on abstract grids”, Comput. Math. Math. Phys., 36:4 (1996), 451–456
7. A. O. Rodnikov, B. A. Samokish, “Finite difference method in the problem of diffraction of a plane acoustic wave in a half-plane with a cut”, Comput. Math. Math. Phys., 49:12 (2009), 2117–2134
8. V. S. Ryaben'kii, “Difference potentials analogous to Cauchy integrals”, Russian Math. Surveys, 67:3 (2012), 541–567
9. Ryabenkii V.S., “Potentsialy dlya abstraktnykh raznostnykh skhem”, Preprinty IPM im. M.V. Keldysha, 2012, no. 10, 1–30
10. V. S. Ryabenkii, “Potentsialy dlya abstraktnykh raznostnykh skhem”, Preprinty IPM im. M. V. Keldysha, 2012, 010, 30 pp.
11. V. S. Ryabenkii, “Aktivnaya zaschita akusticheskogo polya zhelatelnykh istochnikov ot vneshnego shuma v realnom vremeni”, Preprinty IPM im. M. V. Keldysha, 2016, 027, 21 pp.
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