Volume 9, no. 4Pages 40 - 52

On One Mathematical Model Described by Boundary Value Problem for the Biharmonic Equation

V.V. Karachik, B.T. Torebek
In this paper mathematical model described by a generalized third boundary value problem for the homogeneous biharmonic equation in the unit ball with boundary operators up to the third order containing normal derivatives and Laplacian is investigated. Particular cases of the considered mathematical model are the classical models described by Dirichlet, Riquier, and Robin problems, and the Steklov spectral problem, as well as many other mathematical models generated by these boundary conditions. Two existence theorems for the solution of the problem are proved. Existence conditions are obtained in the form of orthogonality on the boundary of some linear combination of boundary functions to homogeneous harmonic polynomials of a particular order. The obtained results are illustrated by some special cases of the general problem.
Full text
mathematical model; biharmonic equation; boundary value problems; Laplace operator.
1. Andersson L-E., Elfving T., Golub G.H. Solution of Biharmonic Equations with Application to Radar Imaging. Journal of Computational and Applied Mathematics, 1998, vol. 94, no. 2, pp. 153-180. DOI: 10.1016/S0377-0427(98)00079-X
2. Lai M.-C., Liu H.-C. Fast Direct Solver for the Biharmonic Equation on a Disk and Its Application to Incompressible Flows. Applied Mathematics and Computation, 2005, vol. 164, no. 3, pp. 679-695. DOI: 10.1016/j.amc.2004.04.064
3. Ehrlich L.N., Gupta M.M. Some Difference Schemes for the Biharmonic Equation. SIAM Journal on Numerical Analysis, 1975, vol. 12, no. 5, pp. 773-790. DOI: 10.1137/0712058
4. Almansi E. Sull'integrazione dell'equazione differenziale $Delta^{2n} u =0$. Annali di Matematica Pura ed Applicata, 1899, vol. 2, no. 3, pp. 1-51. DOI: 10.1007/BF02419286
5. Boggio T. Sulle funzioni di green d'ordinem. Rendiconti del Circolo Matematico di Palermo, 1905, pp. 97-135.
6. Love A.E.H. Biharmonic Analysis, Especially in a Rectangle, and Its Application to the Theory of Elasticity. Journal London Mathematical Society, 1928. vol. 3, pp. 144-156. DOI: 10.1112/jlms/s1-3.2.144
7. Zaremba S. Sur l'integration de l'equation biharmonique. Bulletin International de l'Academie des Sciences de Cracovie, 1908, pp. 1-29.
8. Karachik V.V. Construction of Polynomial Solutions to the Dirichlet Problem for the Polyharmonic Equation in a Ball. Computational Mathematics and Mathematical Physics, 2014, vol. 54, no. 7, pp. 1122-1143. DOI: 10.1134/S0965542514070070
9. Karachik V.V. Normalized System of Functions with Respect to the Laplace Operator and Its Applications. Journal of Mathematical Analysis and Applications, 2003, vol. 287, no. 2, pp. 577-592. DOI: 10.1016/S0022-247X(03)00583-3
10. Karachik V.V., Turmetov B.Kh., Bekaeva A. Solvability Conditions of the Neumann Boundary Value Problem for the Biharmonic Equation in the Unit Ball. International Journal of Pure and Applied Mathematics, 2012. vol. 81, no. 3, pp. 487-495.
11. Karachik V.V. Solvability Conditions for the Neumann Problem for the Homogeneous Polyharmonic Equation. Differential Equations, 2014, vol. 50, no. 11, pp. 1449-1456. DOI: 10.1134/S0012266114110032
12. Karachik V.V. On Solvability Conditions for the Neumann Problem for a Polyharmonic Equation in the Unit Ball. Journal of Applied and Industrial Mathematics, 2014, vol. 8, no. 1, pp. 63-75. DOI: 10.1134/S1990478914010074
13. Gazzola F., Sweers G. On Positivity for the Biharmonic Operator under Steklov Boundary Conditions. Archive for Rational Mechanics and Analysis, 2008, vol. 188, pp. 399-427. DOI: 10.1007/s00205-007-0090-4
14. Karachik V.V., Sadybekov M.A., Torebek B.T. Uniqueness of Solutions to Boundary-Value Problems for the Biharmonic Equation in a Ball. Electronic Journal of Differential Equations, 2015, vol. 2015, no. 244, pp. 1-9.
15. Karachik V.V. Construction of Polynomial Solutions to Some Boundary Value Problems for Poisson's Equation. Computational Mathematics and Mathematical Physics, 2011. vol. 51, no. 9, pp. 1567-1587. DOI: 10.1134/S0965542511090120
16. Karachik V.V. A Problem for the Polyharmonic Equation in the Sphere. Siberian Mathematical Journal, 1991. vol. 32, no. 5, pp. 767-774. DOI: 10.1007/BF00971175
17. Karachik V.V. On the Mean Value Property for Polyharmonic Functions in the Ball. Siberian Advances in Mathematics, 2014, vol. 24, no. 3, pp. 169-182. DOI: 10.3103/S1055134414030031
18. Karachik V.V., Torebek B.T. On Uniqueness and Correct Solvability of the Biharmonic Boundary Value Problem. AIP Conference Proceedings, 2016, vol. 1759, 020045, 4 p. DOI: 10.1063/1.4959659