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On digital approximations for pseudo-differential equations and related boundary value problems

Annotation

We consider a discrete version of pseudo-differential operators as a first stage for constructing approximate methods of solving pseudo-differential equations and their numerical realization. For this purpose we introduce the classes of periodic symbols and discrete operators, study a solvability of corresponding discrete equations and suggest some computations algorithms.

Keywords

digital pseudo-differential operator; discrete equation; approximate Solution

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DOI

10.20310/1810-0198-2018-23-122-216-227

UDC

517.98

Pages

216-227

References

1. Taylor M.E. Pseudodifferential Operators. Princeton, Princeton Univ. Press, 1981, 464 p. 2. Treves J.-F. Introduction to Pseudodifferential and Fourier Integral Operators. New York, Springer, 1980, 299 p. 3. Eskin G. Boundary Value Problems for Elliptic Pseudodifferential Equations. Providence, AMS, 1981, 375 p. 4. Gradshteyn I.S., Ryzhik I.M. Tables of Integrals, Series, and Products. New York, Academic Press, 2007, 1200 p. 5. Vasilyev V.B. O razreshimosti nekotorykh diskretnykh uravneniy [On solvability of some discrete equations]. Materialy Mezhdunarodnoy konferentsii «Sovremennye problemy fiziko-matematicheskikh nauk» [Proceedings of International Conference “Modern Problems of Physical and Mathematical Sciences”]. Orel, 2017, pp. 37-41. (In Russian). 6. Vasilyev V.B. O kraevykh zadachakh dlya diskretnogo laplasiana v poluprostranstve [On boundary value problems for the discrete Laplacian in the half-space]. Matematicheskiy forum (Itogi nauki. Yug Rossii) – Mathematical Forum (Results of Science. South of Russia), 2017, vol. 11, pp. 95-102. (In Russian). 7. Vasilyev A.V., Vasilyev V.B. Discrete singular operators and equations in a half-space. Azerb. J. Math., 2013, vol. 3, no. 1, pp. 84-93. 8. Vasilyev A.V., Vasilyev V.B. O razreshimosti nekotorykh diskretnykh uravneniy i svyazannykh s nimi otsenkakh diskretnykh operatorov [On the solvability of certain discrete equations and related estimates of discrete operators]. Doklady Akademii nauk – Proceedings of the Russian Academy of Sciences, 2015, vol. 464, no. 6, pp. 651-655. (In Russian). 9. Vasilyev A.V., Vasilyev V.B. Discrete singular integrals in a half-space. In: Mityushev V., Ruzhansky M. (eds.). Current Trends in Analysis and its Applications. Trends in Mathematics. Basel, Birkhäuser, 2015, pp. 663-670. 10. Vasilyev A.V., Vasilyev V.B. On a digital approximation for pseudo-differential operators. Proc. Appl. Math. Mech., 2017, vol. 17, issue 1, pp. 763-764. 11. Vasilyev V.B. Discrete equations and periodic wave factorization. AIP Conference Proceedings, 2016, vol. 1759, pp. 0200126-1-5. 12. Vasilyev V.B. The periodic Cauchy kernel, the periodic Bochner kernel, and discrete pseudodifferential operators. AIP Conference Proceedings, 2017, vol. 1863, pp. 140014-1-4. 13. Vasilyev V.B. Discrete operators in canonical domains. WSEAS Trans. Math., 2017, vol. 16, pp. 197-201.

Received

2018-03-27

Section of issue

Scientific articles

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