On representation of Green’s function of the Dirichlet problem for biharmonic equation in a ball. (Russian. English summary) Zbl 1459.31003
Summary: Elementary solution of a biharmonic equation is introduced in analogy to the known elementary solution of the Laplace equation. Relation of this elementary solution with the elementary solution of the Laplace equation gets determined. Depending on dimensionality of space in which a boundary problem is being under research, a symmetric function of two variables gets determined in explicit form through the introduced elementary solution. Then it gets proved that this function possesses properties of Green’s function of the Dirichlet problem for biharmonic equation in a unit ball. Two cases when space dimensionality equals two and when space dimensionality is more than two are being researched separately. Analogous to Green’s function of the Dirichlet problem for Poisson’s equation in a ball, there is expansion of Green’s function of the Dirichlet problem for biharmonic equation in a ball in the full, orthogonal-at-the-unit-sphere, system of homogenous harmonic polynominals. This is to be done in case when space dimensionality is more than four. Using the obtained expansion of Green’s function, integral gets calculated by a ball with the kernel out of Green’s function from a homogenous harmonic polynominal multiplied by the positive degree of norm of the independent variable. The obtained results get complied with the previously known results in this sphere.
MSC:
| 31B30 | Biharmonic and polyharmonic equations and functions in higher dimensions |
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 35J30 | Higher-order elliptic equations |
References:
| [1] | Bitsadze A. V., Equations of Mathematical Physics, Nauka Publ., M., 1982, 336 pp. (in Russ.) · Zbl 0554.35001 |
| [2] | Y. Wang, L. Ye, “Biharmonic Green function and biharmonic Neumann function in a sector”, Complex Variables and Elliptic Equations, 58:1 (2013), 7-22 · Zbl 1266.30026 · doi:10.1080/17476933.2010.551199 |
| [3] | Y. Wang, “Tri-harmonic boundary value problems in a sector”, Complex Variables and Elliptic Equations, 59:5 (2014), 732-749 · Zbl 1295.31012 · doi:10.1080/17476933.2012.759566 |
| [4] | E. Constantin, N.H. Pavel, “Green function of the Laplacian for the Neumann problem in \(\mathbb{R}_+^n\)”, Libertas Mathematica, 30 (2010), 57-69 · Zbl 1210.35038 |
| [5] | H. Begehr, T. Vaitekhovich, “Modified harmonic Robin function”, Complex Variables and Elliptic Equations, 58:4 (2013), 483-496 · Zbl 1272.31003 · doi:10.1080/17476933.2011.625092 |
| [6] | M.A. Sadybekov, B.T. Torebek, B.Kh. Turmetov, “On an explicit form of the Green function of the third boundary value problem for the Poisson equation in a circle”, AIP Conf. Proc., 1611 (2015), 255-260 · doi:10.1063/1.4893843 |
| [7] | M.A. Sadybekov, B.T. Torebek, B.Kh. Turmetov, “On an explicit form of the Green function of the Robin problem for the Laplace operator in a circle”, Advances in Pure and Applied Mathematics, 6:3 (2015), 163-172 · Zbl 1317.35025 · doi:10.1515/apam-2015-0003 |
| [8] | Kal’menov T. Sh., Koshanov B. D., Nemchenko M. Yu., “Green function representation in the Dirichlet problem for polyharmonic equations in a ball”, Dokl. Math., 78:1 (2008), 528-530 · Zbl 1220.31011 · doi:10.1134/S1064562408040169 |
| [9] | Kal’menov T. Sh., Suragan D., “On a new method for constructing the Green Function of the Dirichlet problem for the polyharmonic equation”, Differ. Equ., 48:3 (2012), 441-445 · Zbl 1247.31003 · doi:10.1134/S0012266112030160 |
| [10] | Karachik V. V., Antropova N. A., “On polynomial solutions to the Dirichlet problem for a biharmonic equation in a ball”, Sibirskii Zhurnal Industrial’noi Matematiki, 15:2 (2012), 86-98 (in Russ.) · Zbl 1324.35017 |
| [11] | Karachik V. V., “Green Function of the Dirichlet Boundary Value Problem for Polyharmonic Equation in a Ball Under Polynomial Data”, Izvestiya of Saratov University. New Series. Series Mathematics. Mechanics. Informatics, 14:4(2) (2014), 550-558 (in Russ.) · Zbl 1310.35084 |
| [12] | Karachik V. V., Turmetov B. Kh., Matematicheskie Trudy, 2018, no. 1, 17-34 (in Russ.) · Zbl 1440.11103 |
| [13] | Sadybekov M. A., Torebek B. T., Turmetov B. K., “Representation of the Green”s function of the exterior Neumann problem for the Laplace operator”, Siberian Mathematical Journal, 58:1 (2017), 153-158 · Zbl 1371.35044 · doi:10.1134/S0037446617010190 |
| [14] | M.A. Sadybekov, B.T. Torebek, B.Kh. Turmetov, “Representation of Green”s function of the Neumann problem for a multi-dimensional ball”, Complex Variables and Elliptic Equations, 61:1 (2016), 104-123 · Zbl 1336.35127 · doi:10.1080/17476933.2015.1064402 |
| [15] | V.V. Karachik, “On one set of orthogonal harmonic polynomials”, Proceedings of American Mathematical Society, 126:12 (1998), 3513-3519 · Zbl 0916.33007 · doi:10.1090/S0002-9939-98-05019-9 |
| [16] | Bateman H., Erdelyi A., Higher transcendental functions, v. 2, McGraw-Hill, New York-London, 1953, 396 pp. · Zbl 0143.29202 |
| [17] | Karachik V. V., “Construction of Polynomial Solutions to the Dirichlet Problem for the Polyharmonic Equation in a Ball”, Computational Mathematics and Mathematical Physics, 54:7 (2014), 1122-1143 · Zbl 1313.35093 · doi:10.1134/S0965542514070070 |
| [18] | Vladimirov V. S., Equations of mathematical physics, Nauka Publ., M., 1981, 512 pp. (in Russ.) |
| [19] | Karachik V. V., “Solution of the Dirichlet problem with polynomial data for the polyharmonic equation in a ball”, Differential Equations, 51:8 (2015), 1033-1042 · Zbl 1331.35118 · doi:10.1134/S0012266115080078 |
| [20] | Karachik V. V., “On an expansion of Almansi type”, Mathematical Notes, 83 (2008), 335-344 · Zbl 1152.35351 · doi:10.1134/S000143460803005X |
| [21] | Karachik V. V., “Construction of polynomial solutions to some boundary value problems for Poisson”s equation”, Computational Mathematics and Mathematical Physics, 51:9 (2011), 1567-1587 · Zbl 1274.31002 · doi:10.1134/S0965542511090120 |
| [22] | Karachik V. V., “A Neumann-type problem for the biharmonic equation”, Siberian Advances in Mathematics, 27:2 (2017), 103-118 · Zbl 1399.35186 · doi:10.3103/S105513441702002X |
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