On discrete boundary-value problems in a~quarter plane

V. B. Vasilev (Vasilyev) ; A. A. Khodyreva

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Classical and Modern Geometry» dedicated to the 100th anniversary of the birth of Professor Levon Sergeyevich Atanasyan (July 15, 1921—July 5, 1998). Moscow, November 1–4, 2021. Part 4, Tome 223 (2023), pp. 14-23.

Résumé

We consider a discrete elliptic pseudodifferential equation in a quadrant and the corresponding discrete boundary-value problem. Conditions for the solvability of the discrete boundary-value problem in discrete analogs of the Sobolev–Slobodetskii spaces are described. The discrete solution is compared with the solution of the corresponding continuum boundary-value problem depending on the discretization parameter.

Détail

Keywords: discrete pseudodifferential operator, two-dimensional periodic Riemann problem, periodic wave factorization, analyticity, solvability.

Bibliographie

[1] Vasilev A. V., Vasilev V. B., “Periodicheskaya zadacha Rimana i diskretnye uravneniya v svertkakh”, Differ. uravn., 51:5 (2015), 642–649 | DOI | Zbl

[2] Vasilev V. B., Multiplikatory integralov Fure, psevdodifferentsialnye uravneniya, volnovaya faktorizatsiya, kraevye zadachi, KomKniga, M., 2010

[3] Vasilev V. B., Tarasova O. A., “O diskretnykh kraevykh zadachakh i ikh approksimatsionnykh svoistvakh”, Itogi nauki i tekhn. Sovr. mat. prilozh. Temat. obz., 174 (2020), 12–19 | DOI

[4] Vladimirov V. S., Obobschennye funktsii v matematicheskoi fizike, Nauka, M., 1979 | MR

[5] Gokhberg I. Ts., Krupnik N. Ya., Vvedenie v teoriyu odnomernykh singulyarnykh integralnykh operatorov, Shtiintsa, Kishinev, 1973

[6] Rempel Sh., Shultse B.-V., Teoriya indeksa ellipticheskikh kraevykh zadach, Mir, M., 1986 | MR

[7] Ryabenkii V. S., Metod raznostnykh potentsialov i ego prilozheniya, Fizmatlit, M., 2002

[8] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1989

[9] Frank L. S., “Prostranstva setochnykh funktsii”, Mat. sb., 86:2 (1971), 187–233 | Zbl

[10] Khermander L., Analiz lineinykh differentsialnykh operatorov. T. 1-4, Mir, M., 1986

[11] Eskin G. I., Kraevye zadachi dlya ellipticheskikh psevdodifferentsialnykh uravnenii, Nauka, M., 1973 | MR

[12] Botchway L. N. A., Gaël Kibiti P., Ruzhansky M., “Difference equations and pseudo-differential operators on $\mathbb{Z}^n$”, J. Funct. Anal., 278:11 (2020), 108473 | DOI | MR | Zbl

[13] King F. W., Hilbert Transforms. Vols. 1, 2, Cambridge University Press, Cambridge, 2009 | MR

[14] Rabinovich V. S., “Wiener algebra of operators on the lattice $\mathbb{Z}^n$ depending on the small parameter $\mu>0$”, Complex Var. Elliptic Equations., 58:6 (2013), 751–766 | DOI | MR | Zbl

[15] Tarasova O. A., Vasilyev V. B., “To the theory of discrete boundary value problems”, Difference Differ. Equations Appl., 2 (2019), 17 | MR

[16] Vasilyev A. V., Vasilyev V. B., “Discrete singular operators and equations in a half-space”, Azerb. J. Math., 3:1 (2013), 81–93 | MR

[17] Vasilyev A. V., Vasilyev V. B., “Pseudo-differential operators and equations in a discrete half-space”, Math. Model. Anal., 23:3 (2018), 492–506 | DOI | MR | Zbl

[18] Vasilyev A. V., Vasilyev V. B., “On some discrete potential like operators”, Tatra Mt. Math. Publ., 71 (2018), 195–212 | MR | Zbl

[19] Vasilyev V. B., “The periodic Cauchy kernel, the periodic Bochner kernel, discrete pseudo-differential operators”, AIP Conf. Proc., 1863 (2017), 140014 | DOI

[20] Vasilyev V. B., “On discrete solutions for pseudo-differential equations”, AIP Conf. Proc., 2116 (2019), 040010 | DOI | MR