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On a discrete boundary value problem in a quarter-plane

Annotation

We study the solvability of a discrete analogue of a model pseudo-differential equation in a quarter-plane in discrete Sobolev–Slobodetskii spaces. Using a concept of periodic wave factorization for elliptic periodic symbol, we describe solvability conditions for the equation and for a certain boundary value problem related to this equation. In particular, for certain values of the index of periodic wave factorization, a formula for a general solution of the model discrete pseudo-differential equation is obtained, there are some arbitrary functions in the formula. For their unique determination, we introduce certain additional conditions such as a discrete analogues of integral conditions on angle sides. The existence and uniqueness theorem for the stated boundary value problem is proved and a priori estimates for the solution are obtained. A comparison between discrete and continuous solutions for a special choice of discrete objects is also given.

Keywords

elliptic symbol, invertibility, digital pseudo-differential operator, discrete equation, periodic wave factorization

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DOI

10.20310/2686-9667-2023-28-142-169-181

UDC

517.95+517.983

Pages

169-181

References

[1] A.A. Samarskii, The Theory of Difference Schemes, CRC Press, Boca Raton, 2001. [2] V.S. Ryaben’kii, Method of Difference Potentials and its Applications, Springer–Verlag, Berlin–Heidelberg, 2002. [3] A. Vasilyev, V. Vasilyev, “Pseudo-differential operators and equations in a discrete half-space”, Mathematical Modelling and Analysis, 23:3 (2018), 492–506. [4] V.S. Vladimirov, Generalized Functions in Mathematical Physics, Mir Publ., Moscow, 1979. [5] A. Vasilyev, V. Vasilyev, “Discrete singular operators and equations in a half-space”, Azerbaijan Journal of Mathematics, 3:1 (2013), 84–93. [6] G.I. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Equations, AMS, Providence, 1981. [7] F.D. Gakhov, Boundary Value Problems, 3rd ed., Dover Publications, Mineola, 1981. [8] N.I. Muskhelishvili, Singular Integral Equations, 3rd ed., North Holland, Amsterdam, 1976. [9] V.B. Vasil’ev, Wave Factorization of Elliptic Symbols: Theory and Applications, Kluwer Academic Publ., Dordrecht–Boston–London, 2000. [10] V. Vasilyev, “The periodic Cauchy kernel, the periodic Bochner kernel, discrete pseudodifferential operators”, AIP Conference Proceedings, Proceedings of the International Conference on Numerical Analysis and Applications (ICNAAM-2016) (Rhodes, Greece, September 19–25), 1863, AIP Publishing, New York, 2017, 140014. [11] V. Vasilyev, “Discrete equations and periodic wave factorization”, AIP Conference Proceedings, Proceedings of the Third International Conference on Analysis and Applied Mathematics (ICAAM 2016) (Almaty, Kazakhstan, September 7–10), 1759, AIP Publishing, New York, 2016, 020126. [12] V. Vasilyev, “On discrete boundary value problems”, AIP Conference Proceedings, Proceedings of the International Conference “Functional Analysis in Interdisciplinary Applications” (FAIA2017) (Astana, Kazakhstan, October 2–5), 1880, AIP Publishing, New York, 2017, 050010. [13] V. B. Vasilyev, “Discreteness, periodicity, holomorphy, and factorization”, Integral Methods in Science and Engineering. V. 1: Theoretical Technique, eds. C. Constanda, M. Dalla Riva, P.D. Lamberti, P. Musolino, Springer International Publ., New York, 2017, 315–324. [14] V. B. Vasil’ev, “On Some new boundary-value problems in nonsmooth domains”, Journal of Mathematical Sciences, 173:2 (2011), 225–230.

Received

2023-04-19

Section of issue

Scientific articles

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