Dirichlet Problem For Biharmonic Equation Research Articles

ЗАДАЧА НЕЙМАНА ДЛЯ НЕЛОКАЛЬНОГО БИГАРМОНИЧЕСКОГО УРАВНЕНИЯ

The solvability conditions for a class of boundary value problems for a nonlocal biharmonic equation in the unit ball with the Neumann conditions on the boundary are studied. The nonlocality of the equation is generated by some orthogonal matrix. The presence and uniqueness of a solution to the proposed Neumann boundary condition is examined, and an integral representation of the solution to the Dirichlet problem in terms of the Green's function for the biharmonic equation in the unit ball is obtained. First, some auxiliary statements are established: the Green's function of the Dirichlet problem for the biharmonic equation in the unit ball is given, the representation of the solution to the Dirichlet problem in terms of this Green's function is written, and the values of the integrals of the functions perturbed by the orthogonal matrix are found. Then a theorem for the solution to the auxiliary Dirichlet problem for a nonlocal biharmonic equation in the unit ball is proved. The solution to this problem is written using the Green's function of the Dirichlet problem for the regular biharmonic equation. An example of solving a simple problem for a nonlocal biharmonic equation is given. Next, we formulate a theorem on necessary and sufficient conditions for the solvability of the Neumann boundary condition for a nonlocal biharmonic equation. The main theorem is proved based on two lemmas, with the help of which it is possible to transform the solvability conditions of the Neumann boundary condition to a simpler form. The solution to the Neumann boundary condition is presented through the solution to the auxiliary Dirichlet problem.

Presentation of solution of the Dirichlet problem for biharmonic equation in the unit ball through the Green function

Presentation of solution of the Dirichlet problem for biharmonic equation in the unit ball through the Green function

Green's function of Dirichlet problem for biharmonic equation in the ball

ABSTRACTAn explicit representation of the Green's function of the Dirichlet problem for the biharmonic equation in the unit ball is given. Expansion of the constructed Green's function in the complete system of homogeneous harmonic polynomials that are orthogonal on the unit sphere is obtained. Solution of the homogeneous Dirichlet problem with a simple right-hand side of the biharmonic equation is found.

ОБ ОДНОМ ПРЕДСТАВЛЕНИИ ФУНКЦИИ ГРИНА ЗАДАЧИ ДИРИХЛЕ ДЛЯ БИГАРМОНИЧЕСКОГО УРАВНЕНИЯ В ШАРЕ

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

Polynomial solutions of the Dirichlet problem for the biharmonic equation in the ball

We find a polynomial solution of the Dirichlet problem for the inhomogeneous biharmonic equation with polynomial right-hand side and polynomial boundary data in the unit ball. To this end, we use the closed-form representation of harmonic functions in the Almansi formula.

A multiunit ADI scheme for biharmonic equation with accelerated convergence

We consider the problem of acceleration of the Alternative Directions Implicit (ADI) scheme for Dirichlet problem for biharmonic equation. The second Douglas scheme is used as the main vehicle and two full time steps are organised in a single iteration unit in which the explicit operators are arranged differently for the second step. Using an a priori estimate for the spectral radius of the operator, we show that there exists an optimal value for the acceleration parameter. An algorithm is devised implementing the scheme and the optimal range of the parameter is verified through numerical experiments. One iteration unit speeds up the convergence from two to three times in comparison with the standard ADI scheme. To obtain more significant acceleration for the cases when the standard ADI scheme has extremely slow convergence, a generalised multiunit scheme is constructed by introducing another acceleration parameter and treating two consecutive iteration units as one basic element.

SOLVING THE MULTIDIMENSIONAL DIFFERENCE BIHARMONIC EQUATION BY THE MONTE CARLO METHOD

We construct and justify new weighted Monte Carlo methods for estimation of a solution to the Dirichlet problem for the multidimensional difference biharmonic equation by modeling a “random walk by a grid.” Vector versions of our algorithms extend to the difference metaharmonic equations, with the shape of unbiasedness conditions of estimators preserved together with boundedness of their variances. In this connection, we construct a simple algorithm for estimation of the first eigenvalue of the multidimensional difference Laplace operator. Moreover, we construct special algorithms of a “random walk by a grid” which under certain conditions allow us to estimate solutions of the Dirichlet problem for the biharmonic equation with a weak nonlinearity as well as solutions to problems with mixed boundary conditions, the Neumann condition inclusively.

Some Difference Schemes for the Biharmonic Equation

The Dirichlet problem for biharmonic equation in a rectangular region is considered. The method of splitting is used and two classes of finite difference approximations are defined. Two semi-iterative procedures are considered for obtaining the solution of the resulting coupled system of algebraic equations. It is shown that the rate of convergence of the iterative procedures depends upon the choice of the difference approximation. Estimates for optimum iteration parameters are given and several comparisons are made. An attempt is made to unify the ideas on the splitting technique for solving the first biharmonic boundary value problem.