Voir la notice de l'article provenant de la source Math-Net.Ru
@article{MZM_2017_102_1_a4, author = {V. V. Karachik and B. T. Torebek}, title = {On the {Dirichlet--Riquier} {Problem} for {Biharmonic} {Equations}}, journal = {Matemati\v{c}eskie zametki}, pages = {39--51}, publisher = {mathdoc}, volume = {102}, number = {1}, year = {2017}, language = {ru}, url = {http://geodesic.mathdoc.fr/item/MZM_2017_102_1_a4/} }
V. V. Karachik; B. T. Torebek. On the Dirichlet--Riquier Problem for Biharmonic Equations. Matematičeskie zametki, Tome 102 (2017) no. 1, pp. 39-51. http://geodesic.mathdoc.fr/item/MZM_2017_102_1_a4/
[1] L.-E. Andersson, T. Elfving, G. H. Golub, “Solution of biharmonic equations with application to radar imaging”, J. Comput. Appl. Math., 94:2 (1998), 153–180 | DOI | MR | Zbl
[2] M.-C. Lai, H.-C. Liu, “Fast direct solver for the biharmonic equation on a disk and its application to incompressible flows”, Appl. Math. Comput., 164:3 (2005), 679–695 | MR | Zbl
[3] L. N. Ehrlich, M. M. Gupta, “Some difference schemes for the biharmonic equation”, SIAM J. Numer. Anal., 12:5 (1975), 773–790 | DOI | MR | Zbl
[4] A. H. A. Ali, K. R. Raslan, “Variational iteration method for solving biharmonic equations”, Phys. Lett. A, 370:5-6 (2007), 441–448 | DOI | MR | Zbl
[5] P. Bjorstad, “Fast numerical solution of the biharmonic Dirichlet problem on rectangles”, SIAM J. Numer. Anal., 20:1 (1983), 59–71 | DOI | MR | Zbl
[6] Q. A. Dang, “Iterative method for solving the Neumann boundary problem for biharmonic type equation”, J. Comput. Appl. Math., 196:2 (2006), 634–643 | DOI | MR | Zbl
[7] Q. A. Dang, X. T. Mai, “Iterative method for solving a problem with mixed boundary conditions for biharmonic equation arising in fracture mechanics”, Bol. Soc. Parana. Mat. (3), 31:1 (2013), 65–78 | MR
[8] E. Almansi, “Sull'integrazione dell'equazione differenziale $\Delta^{2n}u=0$”, Ann. Mat. Pura Appl. (3), 2:1 (1899), 1–51 | DOI | Zbl
[9] T. Boggio, “Sulle funzioni di Green d'ordine $m$”, Rend. Circ. Mat. Palermo, 20 (1905), 97–135 | DOI | Zbl
[10] A. E. H. Love, “Biharmonic analysis, especially in a rectangle, and its application to the theory of elasticity”, J. London Math. Soc., 3:2 (1928), 144–156 | DOI | MR | Zbl
[11] S. Zaremba, “Sur l'integration de l'equation biharmonique”, Bull. Acad. Sci. Cracovie, 1908, 1–29 | Zbl
[12] C. Miranda, “Formule di maggiorazione e teorema di esistenza per le funzioni biarmoniche de due variabili”, Giorn. Mat. Battaglini (4), 2 (78) (1948), 97–118 | MR | Zbl
[13] D. D. Joseph, L. Sturgers, “The convergence of biorthogonal series for biharmonic and Stokes flow edge problems. II”, SIAM J. Appl. Math., 34:1 (1978), 7–26 | DOI | MR | Zbl
[14] V. V. Karachik, “Ob odnom razlozhenii tipa Almansi”, Matem. zametki, 83:3 (2008), 370–380 | DOI | MR | Zbl
[15] V. V. Karachik, N. A. Antropova, “Polinomialnye resheniya zadachi Dirikhle dlya bigarmonicheskogo uravneniya v share”, Differents. uravneniya, 49:2 (2013), 250–254 | MR | Zbl
[16] V. V. Karachik, “Postroenie polinomialnykh reshenii zadachi Dirikhle dlya poligarmonicheskogo uravneniya v share”, Zh. vychisl. matem. i matem. fiz., 54:7 (2014), 1149–1170 | DOI | MR | Zbl
[17] V. V. Karachik, “Normalized system of functions with respect to the Laplace operator and its applications”, J. Math. Anal. Appl., 287:2 (2003), 577–592 | DOI | MR | Zbl
[18] O. A. Matevosyan, “O resheniyakh zadachi Neimana dlya bigarmonicheskogo uravneniya v neogranichennykh oblastyakh”, Matem. zametki, 98:6 (2015), 944–947 | DOI | MR | Zbl
[19] G. C. Verchota, “The biharmonic Neumann problem in Lipschitz domains”, Acta Math., 194:2 (2005), 217–279 | DOI | MR | Zbl
[20] A. Gómez-Polanco, J. M. Guevara-Jordan, B. Molina, “A mimetic iterative scheme for solving biharmonic equations”, Math. Comput. Modelling, 57:9-10 (2013), 2132–2139 | DOI | MR | Zbl
[21] F. Gazzola, H.-Ch. Grunau, G. Sweers, Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Math., 1991, Springer-Verlag, Berlin, 2010 | DOI | MR | Zbl
[22] B. D. Koshanov, “About solvability of boundary value problems for the nonhomogeneous polyharmonic equation in a ball”, Adv. Pure Appl. Math., 4:4 (2013), 351–373 | DOI | MR | Zbl
[23] A. S. Berdyshev, A. Cabada, B. Kh. Turmetov, “On solvability of a boundary value problem for a nonhomogeneous biharmonic equation with a boundary operator of a fractional order”, Acta Math. Sci. Ser. B Engl. Ed., 34:6 (2014), 1695–1706 | DOI | MR | Zbl
[24] V. V. Karachik, B. Kh. Turmetov, B. T. Torebek, “O nekotorykh integro-differentsialnykh operatorakh v klasse garmonicheskikh funktsii i ikh primenenii”, Matem. tr., 14:1 (2011), 99–125 | MR | Zbl
[25] V. V. Karachik, M. A. Sadybekov, B. T. Torebek, “Uniqueness of solutions to boundary-value problems for the biharmonic equation in a ball”, Electron. J. Differential Equations, 2015:244 (2015), 9 pp. | MR | Zbl
[26] S. L. Sobolev, Vvedenie v teoriyu kubaturnykh formul, Nauka, M., 1974 | MR | Zbl
[27] J. Pipher, G. Verchota, “Area integral estimates for the biharmonic operator in Lipschitz domains”, Trans. Amer. Math. Soc., 327:2 (1991), 903–917 | DOI | MR | Zbl
[28] D. Buoso, L. Provenzano, “A few shape optimization results for a biharmonic Steklov problem”, J. Differential Equations, 259:5 (2015), 1778–1818 | DOI | MR | Zbl
[29] V. V. Karachik, “Postroenie polinomialnykh reshenii nekotorykh kraevykh zadach dlya uravneniya Puassona”, Zh. vychisl. matem. i matem. fiz., 51:9 (2011), 1674–1694 | MR | Zbl
[30] V. V. Karachik, “Ob odnoi zadache dlya poligarmonicheskogo uravneniya v share”, Sib. matem. zhurn., 32:5 (1991), 51–58 | MR | Zbl