Abstract
We study a general discrete boundary value problem in Sobolev–Slobodetskii spaces in a plane quadrant and reduce it to a system of integral equations. We show a solvability of the system for a small size of discreteness starting from a solvability of its continuous analogue.
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References
Taylor, M.: Pseudodifferential operators. Princeton University Press, Princeton (1981)
Treves, F.: Introduction to pseudodifferential operators and Fourier integral op- erators. Springer, New York (1980)
Eskin, G.: Boundary value problems for elliptic pseudodifferential equations. AMS, Providence (1981)
Vasilyev, V.B.: Operators and equations: discrete and continuous. J. Math. Sci. 257(1), 17–26 (2021)
Edwards, R.: Fourier series. A modern introduction. Springer-Verlag, Heidelberg (1982)
Samarskii, A.: The theory of difference schemes. CRC Press, Boca Raton (2001)
Ryaben’kii, V.: Method of difference potentials and its applications. Springer- Verlag, Berlin-Heidelberg (2002)
Vasilyev, A.V., Vasilyev, V.B., Tarasova, O.A.: Discrete boundary value problems as approximate constructions. Lobachevskii J. Math. 43(6), 1446–1457 (2022)
Kozak, A.V.: On a projection method for solving operator equations in Banach space. Sov. Math. Dokl. 14, 1159–1162 (1973)
Vasil’ev, V.B.: Wave factorization of elliptic symbols: Theory and applications. Kluwer Academic Publishers, Dordrecht-Boston-London (2000)
Mashinets, A.A., Vasilyev, A.V., Vasilyev, V.B.: On discrete Neumann problem in a quadrant. Lobachevskii J. Math 44(3), 1011–1021 (2023)
Vasilyev, A.V., Vasilyev, V.B.: Pseudo-differential operators and equations in a discrete half-space. Math. Model. Anal. 23(3), 492–506 (2018)
Vasilyev, V. B.: Discreteness, periodicity, holomorphy, and factorization, Integral Methods in Science and Engineering (New York) In: Constanda, C., Dalla Riva, M., Lamberti, P.D., Musolino, P. (eds.) Theoretical technique, vol. 1, Birkhauser, pp. 315–324 (2017)
Vasilyev, V. B.: On discrete boundary value problems. In: Kal’menov, T., adybekov, M. (eds.) Proceedings of the International Conference on Functional Analysis in Interdisciplinary Applications" (FAIA2017) (Melville), AIP Conf. Proc., vol. 1880, AIP Publishing, p. 050010 (2017)
Vasilyev, V. B.: The periodic Cauchy kernel, the periodic Bochner kernel, discrete pseudo-differential operators. In: Simos, T., Tsitouras, C. (eds.) Proceedings of the International Conference on Numerical Analysis and Applications (ICNAAM-2016) (Melville), AIP Conf. Proc., vol. 1863, AIP Publishing, p. 140014 (2017)
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Vasilyev, V., Vasilyev, A. & Mashinets, A. On a general discrete boundary value problem for an elliptic pseudo-differential equation in a quadrant. Bol. Soc. Mat. Mex. 29, 53 (2023). https://doi.org/10.1007/s40590-023-00527-x
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DOI: https://doi.org/10.1007/s40590-023-00527-x
Keywords
- Elliptic symbol
- Invertibility
- Digital pseudo-differential operator
- Discrete equation
- Periodic wave factorization
- System of integral equation