On Solvability of the Neumann Boundary Value Problem for Non-homogeneous Biharmonic Equation
Batirkhan Turmetov *
Department of Matematics, Kazakh - Turkish University named after A. Yasawi, Sattarkhanova st. 29, Turkestan, 161200, Kazakhstan.
Ravshan Ashurov
Institute of Mathematics National University of Uzbekistan, Hodjaeva st., 29, Tashkent, 100125, Uzbekistan.
*Author to whom correspondence should be addressed.
Abstract
In this work we investigate the Neumann boundary value problem in the unit ball for a nonhomogeneous biharmonic equation. It is well known, that even for the Poisson equation this problem does not have a solution for an arbitrary smooth right hand side and boundary functions; it follows from the Green formula, that these given functions should satisfy a condition called the solvability condition. In the present paper these solvability conditions are found in an explicit form for the natural generalization of the Neumann problem for the non-homogeneous biharmonic equation. The method used is new for these type of problems. We first reduce this problem to the Dirichlet problem, then use the Green function of the Dirichlet problem recently found by T. Sh. Kal’menov and D. Suragan.
Keywords: Non-homogeneous biharmonic equation, the Neumann problem, the necessary and sufficient conditions for solvability, the Dirichlet problem, the Green function.