Geodesic
Numerical integration of a biharmonic equation in square field
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 1 (2013), pp. 52-62 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The finite-difference algorithm of the numerical solution of a boundary-value problem for a biharmonic equation with a known right-hand member and zero conditions on required function and its normal gradient on boundary of a square field which was found on ideas of a relaxation method is suggested. The comparative analysis of the results obtained has confirmed efficiency of a computing circuit. Bibliogr. 20. Il. 2. Tabl. 2.
Keywords: biharmonic equation, relaxation method, finite-difference scheme. 
@article{VSPUI_2013_1_a5,
     author = {V. I. Ryazhskikh and M. I. Slyusarev and M. I. Popov},
     title = {Numerical integration of a biharmonic equation in square field},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {52--62},
     year = {2013},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2013_1_a5/}
}
TY  - JOUR
AU  - V. I. Ryazhskikh
AU  - M. I. Slyusarev
AU  - M. I. Popov
TI  - Numerical integration of a biharmonic equation in square field
JO  - Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
PY  - 2013
SP  - 52
EP  - 62
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VSPUI_2013_1_a5/
LA  - ru
ID  - VSPUI_2013_1_a5
ER  - 
%0 Journal Article
%A V. I. Ryazhskikh
%A M. I. Slyusarev
%A M. I. Popov
%T Numerical integration of a biharmonic equation in square field
%J Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
%D 2013
%P 52-62
%N 1
%U http://geodesic.mathdoc.fr/item/VSPUI_2013_1_a5/
%G ru
%F VSPUI_2013_1_a5
V. I. Ryazhskikh; M. I. Slyusarev; M. I. Popov. Numerical integration of a biharmonic equation in square field. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 1 (2013), pp. 52-62. http://geodesic.mathdoc.fr/item/VSPUI_2013_1_a5/

[1] Timoshenko S. P., Vojnovskij-Kriger S., Plastiny i obolochki (Plates and envelopes), Nauka, Moscow, 1966, 636 pp.

[2] Happel J., Brenner H., Low Reynolds numbers hydrodymics

[3] Happel' Dzh., Brenner G., Gidrodinamika pri malyh chislah Rejnol'dsa (Low Reynolds numbers hydrodymics), Per. s angl. V. S. Bermana, G. V. Markova, ed. Ju. A. Buevich, Mir, Moscow, 1976, 630 pp.

[4] Postnov V. A., Rostovcev D. M., Suslov V. P., Kochanov Ju. P., Stroitel'naja mehanika korablja i teorija uprugosti (Structural mechanics of the ship and elastic theory), V 2 t., v. 2, Sudostroenie, Leningrad, 1987, 416 pp.

[5] Sljusarev M. I., Chertov E. Ju., Rjazhskih V. I., “Analiticheskoe reshenie pervoj testovoj zadachi svobodnoj konvekcii dlja konduktivno-laminarnogo rezhima (Analytical solution the first test problem of the free convection for conductive and laminar mode)”, Vestn. Voronezh. gos. tehn. un-ta, 6:7 (2010), 165–167

[6] Tihonov A. N., Samarskij A. A., Uravnenija matematicheskoj fiziki (Equations of mathematical physics), Nauka, Moscow, 1977, 742 pp.

[7] Bubnov I. G., “Naprjazhenija v obshivke sudov ot davlenija vody (Tension in a covering of vessels from water pressure)”, Morskoj sbornik, 312:10 (1902), 119–138

[8] Chehov V. N., Pan A. V., “Ob uluchshenii shodimosti rjadov dlja bigarmonicheskoj zadachi v prjamougol'nike (About an acceleration by powering of ranks for the biharmonic problems in a rectangle)”, Dinamicheskie sistemy, 2008, no. 25, 135–144 | Zbl

[9] Suslov V. P., Kachanov Ju. P., Spiharenko V. N., Stroitel'naja mehanika korablja na osnove teorii uprugosti (Structural mechanics of the ship on the basis of the elastic theory), Sudostroenie, Leningrad, 1972, 720 pp.

[10] Selvadurai A. P., Partial differention equations in mechanics, v. 2, Springer, New York, 2004, 698 pp. | MR

[11] Sljusarev M. I., Chertov E. Ju., Rjazhskih V. I., Boger A. A., “Konduktivno-laminarnaja estestvennaja konvekcija n'jutonovskoj teplovydeljajushhej zhidkosti v kvadratnoj kaverne s postojannoj temperaturoj stenok (Conductive and laminar free convection of the Newtonian heat-generating liquid in a square cavity from a constant temperature of walls)”, Vestn. Voronezh. gos. un-ta. Ser. Fizika. Matematika, 2011, no. 1, 214–218

[12] Kobel'kov G. M., “O svedenii kraevoj zadachi dlja bigarmonicheskogo uravnenija k zadache tipa Stoksa (About reduction of a boundary value problem for a biharmonic equation to a problem of type of Stokes)”, Dokl. AN SSSR, 283:1 (1985), 539–542 | MR

[13] Fletcher C. A. J., Computational techniquies for fluid dynamics | MR

[14] Fletcher K., Vychislitel'nye metody v dinamike zhidkostej (Computational techniquies for fluid dynamics), V 2 t. Per. s angl. A. I. Derzhavinoj, v. 1, ed. V. I. Shidlovskii, Mir, Moscow, 1991, 504 pp. | MR

[15] Algazin S. D., Chislennye algoritmy klassicheskoj matematicheskoj fiziki (Numerical algorithms of classical mathematical physics), Dialog-MIFI, Moscow, 2010, 240 pp.

[16] Galanin M. F., Milyutin D. S., Savenkov E. B., Development, research and application of a finite superelements method for solution biharmonical equation, Preprint Inst. Appl. Math., the Russian Academy of science, M., 2005, 24 pp.

[17] Marchuk G. I., Metody vychislitel'noj matematiki (Methods of calculus mathematics), Nauka, Moscow, 1980, 535 pp. | MR

[18] Anderson D. A., Tannehil J. C., Pletcher R. H., Computational fluid mechanics and heat transfer | MR | Zbl

[19] Anderson D., Tannehil Dzh., Pletcher R., Vychislitel'naja mehanika i teploobmen (Computational fluid mechanics and heat transfer), V 2 t. Per. s angl. S. V. Senina, E. Ju. Shal'mana, v. 1, ed. G. L. Podvidz, Mir, Moscow, 1990, 384 pp. | MR | Zbl

[20] Samarskij A. A., Vvedenie v teoriju raznostnyh shem (Introduction in the theory of difference schemes), Nauka, Moscow, 1971, 552 pp. | MR | Zbl

[21] Trenogin A. A., Funkcional'nyj analiz (Functional analysis), Fizmatlit, Moscow, 2002, 488 pp.

[22] Ahiezer N. I., Glazman I. M., Teorija linejnyh operatorov v gil'bertovom prostranstve (The theory of the linear operators in Hilbert space), Nauka, Moscow, 1966, 544 pp. | MR | Zbl

[23] Filippov A. F., Rjaben'kij V. S., Ob ustojchivosti raznostnyh uravnenij (About stability of the difference equations), Gos. izd-vo teor.-tehn. lit., Moscow, 1956, 171 pp. | MR