Method N.\,M.~Korobova approximate solution of the Dirichlet problem

A. V. Rodionov

Čebyševskij sbornik, Tome 15 (2014) no. 3, pp. 48-85.

Résumé

The paper discusses the generalization of the method embodiments N. M. Korobov approximate solution of the Dirichlet problem for equations of the form $$Q\left(\frac{\partial }{\partial x_1},\ldots,\frac{\partial }{\partial x_s}\right)u(\mathbf{x})=f(\mathbf{x}),$$ where the functions $u(\mathbf{x}),f(\mathbf{x}),\varphi(\mathbf{x})$ belongs to the class of functions $E_s^\alpha$ in case of using generalized Parallelepipedal nets $M(\Lambda)$ integral lattices $\Lambda$. Particular attention is paid to the class of differential operators, consisting of operators $Q\left(\frac{\partial }{\partial x_1},\ldots,\frac{\partial }{\partial x_s}\right)$ with zero kernel. The importance of this class of operators due to the fact that up to a constant solution of differential equations with partial derivatives for these operators is uniquely determined. An example of such an operator is the Laplace operator. In the work, an approximate solution of the Dirichlet problem for partial differential equations using arbitrary generalized parallelepiped mesh $M(\Lambda)$ integer lattice $\Lambda$ for a certain class of periodic functions and shown that by using an infinite sequence of nested grids is generalized parallelepipedal nets sufficiently fast convergence of the approximate solutions to the function $u(\mathbf{x})$. Bibliography: 15 titles.

Détail

Keywords: parallelepiped nets, partial differential equations, the Dirichlet problem.

Bibliographie

[1] Rodionov A. V., “Number theoretic metods for solving partial differential equations”, Algebra and Number Theory: Modern Problems and Application, Proceedings XII Iinternational Conference, dedicated to 80-th anniversary of Professor V. N. Latyshev (2014, Tula), 159–161

[2] Dobrovolskaya L. P., Dobrovolskii N. M., Simonov A. S., “O pogreshnosti priblizhennogo integrirovaniya po modifitsirovannym setkam”, Chebyshevskii sbornik, 9:1(25) (2008), 185–223 | MR | Zbl

[3] Dobrovolskii M. N., “Ob optimalnykh koeffitsientakh kombinirovannykh setok”, Chebyshevskii sbornik, 5:1(9) (2004), 95–121 | MR | Zbl

[4] Dobrovolskii M. N., “Otsenki summ po giperbolicheskomu krestu”, Izv. TulGU. Ser. Matematika. Mekhanika. Informatika, 9:1 (2003), 82–90 | MR

[5] Dobrovolskii N. M., Giperbolicheskaya dzeta funktsiya reshetok, Dep. v VINITI 24.08.84, No 6090-84

[6] Dobrovolskii N. M., Manokhin E. V., “Banakhovy prostranstva periodicheskikh funktsii”, Izv. TulGU. Ser. Mekhanika. Matematika. Informatika, 4:3 (1998), 56–67 | MR | Zbl

[7] Dobrovolskii M. N., Dobrovolskii N. M., Kiseleva O. V., “O proizvedenii obobschennykh parallelepipedalnykh setok tselochislennykh reshetok”, Chebyshevskii sbornik, 3:2(4) (2002), 43–59 | MR

[8] Dobrovolskii N. M., Esayan A. R., Andreeva O. V., Zaitseva N. V., “Mnogomernaya teoretiko-chislovaya Fure interpolyatsiya”, Chebyshevskii sbornik, 5:1(9) (2004), 122–143 | MR | Zbl

[9] Korobov N. M., Teoretiko-chislovye metody v priblizhennom analize, Fizmatgiz, M., 1963 | MR | Zbl

[10] Korobov N. M., Teoretiko-chislovye metody v priblizhennom analize, 2-e izd., MTsNMO, M., 2004 | MR

[11] Rodionov A. V., Chuprin S. Yu., “O giperbolicheskikh parametrakh reshetki lineinogo sravneniya”, Izvestiya TulGU. Estestvennye nauki, 2014, no. 1-1, 50–63 | MR

[12] Rodionov A. V., “O metode V. S. Ryabenkogo–N. M. Korobova priblizhennogo resheniya uravnenii s chastnymi proizvodnymi”, Chebyshevskii sbornik, 10:3(31) (2009), 110–136

[13] Rodionov A. V., “Reshenie differentsialnykh uravnenii v chastnykh proizvodnykh metodom V. S. Ryabenkogo”, Izvestiya Saratovskogo universiteta, 13:4(2) (2013), 120–124

[14] Rodionov A. V., “Teoretiko-chislovye metody resheniya differentsialnykh uravnenii v chastnykh proizvodnykh”, Algebra i teoriya chisel: sovremennye problemy i prilozheniya, Materialy XII Mezhdunar. konf., posvyaschennoi 80-letiyu prof. Viktora Nikolaevicha Latysheva (Tula, 2014), 297–300

[15] Ryabenkii V. S., “Ob odnom sposobe polucheniya raznostnykh skhem i ob ispolzovanii teoretiko-chislovykh setok dlya resheniya zadachi Koshi metodom konechnykh raznostei”, Tr. mat. in-ta im. V. A. Steklova, 60, 1961, 232–237 | MR