Construction of polynomial solutions to the Dirichlet problem for the polyharmonic equation in a ball

V. V. Karachik

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 7, pp. 1149-1170

Résumé

An algorithm is proposed for the analytical construction of a polynomial solution to Dirichlet problem for an inhomogeneous polyharmonic equation with a polynomial right-hand side and polynomial boundary data in the unit ball.

Bibliographie

[1] Karachik V. V., “Postroenie polinomialnykh reshenii nekotorykh kraevykh zadach dlya uravneniya Puassona”, Zh. vychisl. matem. i matem. fiz., 51:9 (2011), 1674–1694

[2] Karachik V. V., Antropova N. A., “O polinomialnykh resheniyakh zadachi Dirikhle dlya bigarmonicheskogo uravneniya v share”, Sib. zh. industr. matem., 15:2(50) (2012), 86–98

[3] Karachik V. V., Antropova N. A., “Polinomialnye resheniya zadachi Dirikhle dlya bigarmonicheskogo uravneniya v share”, Differents. ur-niya, 49:2 (2013), 250–254

[4] Karachik V. V., “Polinomialnye resheniya zadachi Dirikhle dlya 3-garmonicheskogo uravneniya v share”, Zh. Sib. feder. un-ta. Matem. i fiz., 5:4 (2012), 527–546

[5] Vladimirov B. C., Sbornik zadach po uravneniyam matematicheskoi fiziki, Fizmatlit, M., 2001

[6] Karachik V. V., “On one set of orthogonal harmonic polynomials”, Proc. Amer. Math. Soc., 126:12 (1998), 3513–3519 | DOI

[7] Almansi E., “Sull'integrazione dell'equazione differenziale $\Delta^{2n}u=0$”, Ann. Mat. Rura Appl., 1899, no. (3)2, 1–51 | DOI

[8] Karachik V. V., “Ob odnom razlozhenii tipa Almansi”, Matem. zametki, 83:3 (2008), 370–380 | DOI

[9] Nicolescu N., “Probléme de l'analyticité par rapportá un opérateur linéaire”, Studia Math., 16 (1958), 353–363

[10] Karachik V. V., “Primenenie formuly Almansi k postroeniyu polinomialnykh reshenii zadachi Dirikhle dlya uravneniya vtorogo poryadka”, Izv. vyssh. uchebn. zavedenii. Matematika, 2012, no. 6, 24–35

[11] Karachik V. V., Antropova N. A., “O reshenii neodnorodnogo poligarmonicheskogo uravneniya i neodnorodnogo uravneniya Gelmgoltsa”, Differents. ur-niya, 46:3 (2010), 384–395

[12] Kalmenov T. Sh., Suragan D., “O novom metode postroeniya funktsii Grina zadachi Dirikhle dlya poligarmonicheskogo uravneniya”, Differents. ur-niya, 48:3 (2012), 441–445

[13] Kanguzhin B. E., Koshanov B. D., “Neobkhodimye i dostatochnye usloviya razreshimosti kraevykh zadach dlya neodnorodnogo poligarmonicheskogo uravneniya v share”, Ufimskii matem. zh., 2:2 (2010), 41–52

[14] Karachik V. V., Turmetov B. Kh., Bekaeva A., “Solvability conditions of the Neumann boundary value problem for the biharmonic equation in the unit ball”, Internat. J. Pure and Appl. Mathem., 81:3 (2012), 487–495

[15] Karachik V. V., “On an arithmetical triangle”, J. Math. Sci., 195:5 (2013), 665–675 | DOI

[16] Karachik V. V., “Ob usloviyakh razreshimosti zadachi Neimana dlya poligarmonicheskogo uravneniya v edinichnom share”, Sib. zh. industr. matem., 16:4 (2013), 61–74

[17] Turmetov V., Ashurov R., “On solvability of the Neumann boundary value problem for a non-homogeneous polyharmonic equation in a ball”, Boundary Value Problems, 162 (2013)

[18] Karachik V. V., “Ob odnoi zadache dlya poligarmonicheskogo uravneniya v share”, Sibirsk. matem. zh., 32:5 (1991), 51–58

[19] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, Nauka, M., 1966

[20] Karachik V. V., “Ob odnom predstavlenii analiticheskikh funktsii garmonicheskimi”, Matem. trudy, 10:2 (2007), 142–162

[21] Karachik V. V., “Metod postroeniya reshenii lineinykh obyknovennykh differentsialnykh uravnenii s postoyannymi koeffitsientami”, Zh. vychisl. matem. i matem. fiz., 52:2 (2012), 237–252

[22] Karachik V. V., “O svoistve srednego dlya poligarmonicheskikh funktsii v share”, Matem. trudy, 16:2 (2013), 69–88