Construction of Polynomial Solutions to the Dirichlet Boundary Value Problem for the 3-Harmonic Equation in the Unit Ball
Pure and Applied Mathematics Journal
Volume 1, Issue 1, December 2012, Pages: 1-9
Received: Dec. 1, 2012; Published: Dec. 30, 2012
Views 3280      Downloads 157
Authors
Valery V. Karachik, Department of Mathematical Analysis, South-Ural State University, Chelyabinsk, Russia
Sanjar Abdoulaev, Department of Computational Mathematics, South-Ural State University, Chelyabinsk, Russia
Article Tools
Follow on us
Abstract
Polynomial solution to the Dirichlet boundary value problem for the nonhomogeneous 3-harmonic equation in the unit ball with polynomial right-hand side and polynomial boundary data is constructed. Representation of the Green’s function of the Dirichlet boundary value problem in the unit ball in the case of polynomial data is found.
Keywords
3-Harmonic Equation, Almansi Decomposition, Harmonic Polynomials, Dirichlet Boundary Value Problem, Polynomial Solutions
To cite this article
Valery V. Karachik, Sanjar Abdoulaev, Construction of Polynomial Solutions to the Dirichlet Boundary Value Problem for the 3-Harmonic Equation in the Unit Ball, Pure and Applied Mathematics Journal. Vol. 1, No. 1, 2012, pp. 1-9. doi: 10.11648/j.pamj.20120101.11
References
[1]
E. Almansi. Sull'integrazione dell'equazione differenziale . Ann. Mat. Pura Appl., (3) 2 1899, pp.1-51.
[2]
V. Karachik. On an expansion of Almansi type. Mathematical Notes, 83:3-4, 2008, pp. 335-344.
[3]
N. Nicolescu. Probléme de lánalyticité par rapport á un opérateur linéaire. Studia Math., 16, 1958, pp. 353-363.
[4]
V. Karachik. Application of the Almansi formula for constructing polynomial solutions to the Dirichlet problem for a second-order equation. Russian Mathematics, vol. 56, Issue 6, June 2012, pp. 20-29.
[5]
V.S. Vladimirov i dr. Sbornik zadach po uravneniyam matematicheskoi fiziki, Fizmatlit, 2001 (in Russian).
[6]
V. Karachik. On one set of orthogonal harmonic polynomials. Proceedings of American Mathematical Society, 126:12, 1998, pp. 3513-3519.
[7]
H. Bateman and A. Erdélyi. Higher Transcendental Functions, vol. 2, New York, 1953.
[8]
V. Karachik, N. Antropova. On the solution of the inhomogeneous polyharmonic equation and the inhomogeneous Helmholtz equation. Differential Equations, 46:3, 2010, pp. 387-399.
[9]
V. Karachik. Construction of polynomial solutions to some boundary value problems for Poisson's equation. Computational Mathematics and Mathematical Physics, 51:9, 2011, pp. 1567-1587.
[10]
V. Karachik, N. Antropova. Construction of polynomial solutions to the Dirichlet problem for the biharmonic equations in a ball. Vestnik SUSU, seriya ``Mathematika. Mechanika. Phisika'', 32(249):5, 2011, pp. 39-50 (in Russian).
[11]
V. Karachik. On one representation of analytic functions by harmonic functions. Siberian Advances in Mathematics, 18:2, 2008, pp. 142-162.
[12]
D. Khavinson, H.S. Shapiro. Dirichlet's problem when the data is an entire function. Bull. London Math. Soc., 24, 1992, pp. 456-468.
[13]
H. Render. Real Bargmann spaces, Fischer decompositions and sets of uniqueness for polyharmonic functions. Duke Math. J., 142, 2008, pp. 313-352.
[14]
V. Karachik. A problem for the polyharmonic equation in the sphere. Siberian Mathematical Journal, 32:5, 2005, pp. 767-774.
[15]
V. Karachik. Method for constructing solutions of linear ordinary differential equations with constant coefficients. Computational Mathematics and Mathematical Physics, 52:2, 2012, pp. 219-234.
[16]
V. Karachik, B.Kh. Turmetov, A. Bekaeva. Solvability conditions of the Neumann boundary value problem for the biharmonic equation in the unit ball. International Journal of Pure and Applied Mathematics, vol. 81, No. 3, 2012, 487-495.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186