Adiguzel Dosiyev

Eastern Mediterranean University · Department of Mathematics

A pointwise error estimation of the form O(ρh8),h is the mesh size, for the approximate solution of the Dirichlet problem for Laplace’s equation on a rectangular domain is obtained as a result of three-stage (9-point, 5-point and 5-point) finite difference method; here ρ=ρ(x,y) is the distance from the current grid point (x,y)∈Πh to the boundary of...

A three stage (9-point, 5-point and 5-point) difference method for solving the Dirichlet problem for Laplace's equation on a rectangle is proposed and justified. It is proved that the proposed difference solution converges uniformly to the exact solution of order O(h⁸ |ln h|), h is the mesh size, when the boundary functions are from C9,1. Numerical...

A uniform estimation of order O(h⁴), for the convergence of the finite difference solution for the general second order elliptic equation with nonlocal integral boundary condition is obtained where h is the mesh step.

A new constructive method for the finite-difference solution of the Laplace equation with the integral boundary condition is proposed and justified. In this method, the approximate solution of the given problem is defined as a sequence of 9-point solutions of the local Dirichlet problems. It is proved that when the exact solution u(x,y)\documentcla...

We present and justify finite difference schemes with the 14-point averaging operator for the second derivatives of the solution of the Dirichlet problem for Laplace?s equations on a rectangular parallelepiped. The boundary functions ?j on the faces ?j,j = 1,2,..., 6 of the parallelepiped are supposed to have fifth derivatives belonging to the H?ld...

We consider the multipoint nonlocal boundary value problem for the two-dimensional Laplace equation in a rectangular domain. The solution of this problem is defined as a 9-point finite difference solution, with the fourth order gluing operator of the local Dirichlet boundary value problem, by constructing a special method to find a function as the...

The boundary functions φj of the Dirichlet problem, on the faces Γj, j = 1, 2, …, 6 of the parallelepiped R are supposed to have seventh derivatives satisfying the Hölder condition and on the edges their second, fourth and sixth order derivatives satisfy the compatibility conditions which result from the Laplace equation. For the error uh-u of the...

A 14-point difference operator is used to construct finite difference problems for the approxi- mation of the solution, and the first order derivatives of the Dirichlet problem for Laplace’s equations in a rectangular parallelepiped. The boundary functions φj on the faces Γj, j = 1, 2, …, 6 of the parallelepiped are supposed to have pth order deriv...

A new method for the solution of a nonlocal boundary value problem with integral boundary condition for Laplace's equation on a rectangular domain is proposed and justified. The solution of the given problem is defined as a solution of the Dirichlet problem by constructing the approximate value of the unknown boundary function on the side of the re...

A pointwise error estimation of the form 0(ρh ⁸ ),h is the mesh size, for the approximate solution of the Dirichlet problem for Laplace's equation on a rectangular domain is obtained as a result of three stage (9-point, 5-point and 5-point) finite difference method; here ρ = ρ(x,y) is the distance from the current grid point ( x,y, ) ε Π h to the b...

The solution of the Dirichlet problem for Laplace's equation on a special polygon is harmonically extended to a sector with the center at the singular vertex. This is followed by an integral representation of the extended function in this sector, which is approximated by the mid-point rule. By using the extension properties for the approximate valu...

O(h⁸) order (h is the mesh size) of accurate three-stage difference method on a square grid for the approximate solution of the Dirichlet problem for Laplace’s equation on a rectangle is proposed and justified without taking more than 9 nodes of the grid. At the first stage, by using the 9-point scheme the sum of the pure fourth derivatives of the...

An interpolation operator is proposed using the cubic grid solution of order O(h ⁴), h is the mesh size, of the Dirichlet problem for Laplace’s equation in a rectangular paralellepiped. It is proved that when the boundary functions on the faces of the rectangular parallelepiped are from the Hölder classes C 4,λ , λ ∈ (0, 1), and their second and fo...

In a rectangular domain, we consider the 5-point approximate solution of the multilevel nonlocal boundary value problem for Laplace’s equation. By constructing the approximate value of the unknown boundary function on the side of the rectangle where the nonlocal condition was given, the solution of the multilevel nonlocal problem is defined as a so...

We propose and justify difference schemes for the approximation of the first and pure second derivatives of a solution of the Dirichlet problem in a rectangular parallelepiped. The boundary values on the faces of the parallelepiped are supposed to have six derivatives satisfying the Hölder condition, to be continuous on the edges, and to have secon...

The hexagonal grid version of the block-grid method, which is a difference-analytical method, has been applied for the solution of Laplace’s equation with Dirichlet boundary conditions, in a special type of polygon with corner singularities. It has been justified that in this polygon, when the boundary functions away from the singular corners are f...

In this paper, we discuss an approximation of the first and pure second order derivatives for the solution of the Dirichlet problem on a rectangular domain. The boundary values on the sides of the rectangle are supposed to have the sixth derivatives satisfying the Holder condition. On the vertices, besides the continuity condition, the compatibilit...

A highly accurate approximation for the coefficients of the series expansion of the solution of Laplace’s equation around the singular vertex, which are called the generalized stress intensity factors (GSIFs), is obtained by One-Block Method. The method is demonstrated for the slit problem, which has a strong singularity, and for the popular proble...

The combined block-grid method is developed for the highly accurate approximation of the pure second-order derivatives for the solution of Laplace’s equation on a staircase polygon. By approximating the pure derivatives with respect to one of the variables on an artificial boundary around the reentry vertices, the approximation problem of this deri...

The fourth order matching operator on the hexagonal grid is constructed. Its application to the interpolation problem of the numerical solution obtained by hexagonal grid approximation of Laplace’s equation on a rectangular domain is investigated. Furthermore, the constructed matching operator is applied to justify a hexagonal version of the combin...

In a rectangular domain, we consider the Bitsadze-Samarskii nonlocal boundary value problem for the two-dimensional Poisson equation. The solution of this problem is defined as a solution of the local Dirichlet boundary value problem, by constructing a special method to find a function as the boundary value on the side of the rectangle, where the n...

The integral representations of the solution around the vertices of the interior reentered angles (on the “singular” parts) are approximated by the composite midpoint rule when the boundary functions are from These approximations are connected with the 9-point approximation of Laplace's equation on each rectangular grid on the “nonsingular” part of...

The highly accurate block-grid method for solving Laplace’s boundary value problems on polygons is developed for nonanalytic boundary conditions of the first kind. The quadrature approximation of the integral representations of the exact solution around each reentrant corner(“singular” part) are combined with the 9-point finite difference equations...

In this paper, a homogeneous scheme with 26-point averaging operator for the solution of Dirichlet problem for Laplace’s equation on rectangular parallelepiped is analyzed. It is proved that the order of convergence is O(h 4), where h is the mesh step, when the boundary functions are from C 3, 1, and the compatibility condition, which results from...

Two new properties of the 9-point finite difference solution of the Laplace equation are obtained, when the boundary functions are given from C 5,1. It is shown that the maximum error is of order $${O\,\left(h^6\,(|{\rm ln}\,h| + 1)\right)}$$, and this order cannot be obtained for the class of boundary functions from C 5,λ, 0 < λ < 1. These propert...

An extremely accurate, exponentially convergent solution is presented for both symmetric and non-symmetric Laplacian problems on L-shaped domains by using one-block version of the block method (BM). A simple and highly accurate formula for computing the stress intensity factor is given. Comparisons with various results in the literature are include...

The block-grid method (see Dosiyev, 2004) for the solution of the Dirichlet problem on polygons, when a boundary function on each side of the boundary is given from C2,λ, 0<λ<1, is analized. In the integral represetations around each singular vertex, which are combined with the uniform grids on "nonsingular" part the boundary conditions are taken i...

The finite difference solution of the Dirichlet problem on rectangles when a boundary function is given from C_{1,1} is analized. It is shown that the maximum error for 9-point approximation is of order O(h²(|ln h|+1)) as 5-point approximation. This order can be improved up to O(h²) when the 9-point approximation in the grids which are in h distanc...

The finite difference solution of the Dirichlet problem on rectangles when a boundary function is given from C1,1 is analyzed. It is shown that the maximum error for a nine-point approximation is of the order of O(h2(|ln h|+1)) as a five-point approximation. This order can be improved up to O(h2) when the nine-point approximation in the grids which...

An extremely accurate solution is obtained for the cracked-beam problem by one-block version of the block method (BM). The obtained numerical results demonstrate the exponential convergence of the BM with respect to the number of quadrature nodes. A simple and high accurate formula to compute the stress intensity factor is given. The comparisons wi...

A sixth-order accurate composite grid method for solving a mixed boundary value problem for Laplace's equation on staircase polygons (the polygons may have polygonal cuts and be multiply connected) is constructed and justified. The O(h 6) order of accuracy for the number of nodes 0(h -2lnh -1) is obtained by using 9-point scheme on exponentially co...

High accurate difference-analytical method of solving the mixed boundary value problem for Laplace’s equation on graduated polygons (which can have broken sections and be multiply connected) is described and justified. The uniform estimate for the error of the approximate solution is of order O(h 4), where h is the mesh step, for the errors of deri...

A high accurate difference-analytical method is introduced for the solution of the mixed boundary value problem for Laplace's equation on graduated polygons. The polygon can have broken sections and be multiply connected. The uniform estimate of the error of the approximate solution is of order O(h 6), whereas it is of order O(h 6/r jp-λj) for the...

Ad ifference-analytical method of solving the mixed boundary value prob- lem for Laplace's equation on polygons (which can have broken sections and be multiply connected) is described and justified. The uniform estimate for the error of the approximate solution is of order O ¡ h2 ¢ ,w hereh is the mesh step, for the errors of the derivatives of ord...

The author considers a finite difference scheme of 9 points on square grids to approximate the Laplace equation on rectangular region with Dirichlet boundary conditions. The case when the boundary functions are in C 5,1 is discussed. The necessary differential properties of the exact solution are also given. Two properties for the finite difference...