On Solvability of the Neumann Boundary Value Problem for Non-homogeneous Biharmonic Equation

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

Application of the Least Squares Method in Axisymmetric Biharmonic Problems

“…To expand the developed approach for a short cylindrical body, the height of which is less than its diameter, the interactions of strain-stressed states, caused by boundary data prescribed on the opposite end faces, should be accounted. To do that one can use four sequences (1) , (2) , (3) , and (4) of homogeneous solutions to construct the solution.…”

“…. }, V (3) ( , ), and V (4) ( , ) are biharmonic functions corresponding transcendental equation (32) roots { (3) , = 1, 2, . .…”

“…Various kinds of Neumann problems for (1) can also be considered [2,3]. In simplest cases these are the problems with boundary conditions (4), (5), or (6):…”

On solvability of a boundary value problem for an inhomogeneous polyharmonic equation with a fractional order boundary operator

“…The applications of boundary value problems for elliptic equations with fractional order boundary operators were considered in works [11]- [13]. The analogues of the Neumann problems for the biharmonic and polyharmonic equations in the case of the integer order boundary operators were studied in works [15]- [19], while boundary operators fractional in Riemann-Liouville, Caputo and Hadamard-Marchaud sense were addressed in [20]- [23].…”

On a Generalization of the Initial-Boundary Problem for the Vibrating String Equation

“…Boundary value problems for the Laplace, Poisson and Helmholtz equations with boundary conditions containing the higher order derivatives were studied in works by Bavrin [1], Karachik [2][3][4][5], Sokolovskii [6]. In the papers [7][8][9][10][11][12][13], boundary problems, including higher derivatives on the boundary, were studied for the Poisson, Helmholtz, and biharmonic equations. It should be noted that unlike our work, in the mentioned papers [1][2][3][4][5][6]14], the higher order derivative is given on the entire boundary.…”