scholarly journals On the high order convergence of the difference solution of Laplace’s equation in a rectangular parallelepiped

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 893-901 ◽  
Author(s):  
Adiguzel Dosiyev ◽  
Ahlam Abdussalam
Keyword(s):  
Approximate Solution ◽  
Dirichlet Problem ◽  
Laplace Equation ◽  
Grid Point ◽  
Order Convergence ◽  
Compatibility Conditions ◽  
Second Derivatives ◽  
Boundary Functions ◽  
High Order Convergence ◽  
The Difference

The boundary functions ?j of the Dirichlet problem, on the faces ?j, j = 1,2,..., 6 of the parallelepiped R are supposed to have seventh derivatives satisfying the H?lder 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 approximate solution uh at each grid point (x1,x2,x3), a pointwise estmation O(?h6) is obtained, where ?= ?(x1,x2,x3) is the distance from the current grid point to the boundary of R; h is the grid step. The solution of difference problems constructed for the approximate values of the first and pure second derivatives converge with orders O(h6 ?ln h?) and O(h5+?), 0 < ? < 1, respectivly.

scholarly journals On the difference method for approximation of second order derivatives of a solution of Laplace’s equation in a rectangular parallelepiped

Filomat ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 633-643
Author(s):  
Adiguzel Dosiyev ◽  
Hediye Sarikaya
Keyword(s):  
Difference Schemes ◽  
Rectangular Parallelepiped ◽  
Finite Difference Schemes ◽  
Compatibility Conditions ◽  
Averaging Operator ◽  
Second Derivatives ◽  
Boundary Functions ◽  
The Difference ◽  
Derivatives Of ◽  
Theoretical Results

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?lder classes C5?, 0 < ? < 1. On the edges, the boundary functions as a whole are continuous, and their second and fourth order derivatives satisfy the compatibility conditions which result from the Laplace equation. It is proved that the proposed difference schemes for the approximation of the pure and mixed second derivatives converge uniformly with order O(h3+?), 0 < ? < 1 and O(h3), respectively. Numerical experiments are illustrated to support the theoretical results.

scholarly journals A highly accurate corrected scheme in solving the Laplace's equation on a rectangle

2018 ◽  
Vol 22 ◽  
pp. 01015
Author(s):  
Adıgüzel A. Dosiyev ◽  
Hediye Sarıkaya
Keyword(s):  
Approximate Solution ◽  
Dirichlet Problem ◽  
Finite Difference Method ◽  
Grid Point ◽  
Mesh Size ◽  
Rectangular Domain ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Pointwise Error ◽  
Corrected Scheme

A pointwise error estimation of the form 0(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,) ε Πh to the boundary of the rectangle Π.

scholarly journals The Dirichlet Problem for the 2D Laplace Equation in a Domain with Cracks without Compatibility Conditions at Tips of the Cracks

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
P. A. Krutitskii
Keyword(s):  
Dirichlet Problem ◽  
Classical Solution ◽  
Laplace Equation ◽  
Boundary Data ◽  
Exterior Domains ◽  
Compatibility Conditions ◽  
Closed Curves ◽  
Asymptotic Formulae ◽  
Well Posed ◽  
Solution Gradient

We study the Dirichlet problem for the 2D Laplace equation in a domain bounded by smooth closed curves and smooth cracks. In the formulation of the problem, we do not require compatibility conditions for Dirichlet's boundary data at the tips of the cracks. However, if boundary data satisfies the compatibility conditions at the tips of the cracks, then this is a particular case of our problem. The cases of both interior and exterior domains are considered. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integral representation for a solution is obtained. It is shown that weak solution of the problem does not typically exist, though the classical solution exists. The asymptotic formulae for singularities of a solution gradient at the tips of the cracks are presented.

scholarly journals 14-point difference operator for the approximation of the first derivatives of a solution of Laplace’s equation in a rectangular parallelepiped

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 791-800 ◽  
Author(s):  
Adiguzel Dosiyev ◽  
Hediye Sarikaya
Keyword(s):  
Exact Solution ◽  
Difference Operator ◽  
Grid Point ◽  
Rectangular Parallelepiped ◽  
Constant Independent ◽  
Compatibility Conditions ◽  
First Order ◽  
First Derivative ◽  
Boundary Functions ◽  
Derivatives Of

A 14-point difference operator is used to construct finite difference problems for the approximation of the solution, and the first order derivatives of the Dirichlet problem for Laplace?s equations in a rectangular parallelepiped. The boundary functions ?j on the faces ?j, j = 1,2,...,6 of the parallelepiped are supposed to have pth order derivatives satisfying the H?lder condition, i.e., ?j ? Cp,?(?j), 0 < ? < 1, where p = {4,5}. On the edges, the boundary functions as a whole are continuous, and their second and fourth order derivatives satisfy the compatibility conditions which result from the Laplace equation. For the error uh - u of the approximate solution uh at each grid point (x1,x2,x3), ?uh-u?? c?p-4(x1,x2,x3)h4 is obtained, where u is the exact solution, ? = ? (x1, x2,x3) is the distance from the current grid point to the boundary of the parallelepiped, h is the grid step, and c is a constant independent of ? and h. It is proved that when ?j ? Cp,?, 0 < ? < 1, the proposed difference scheme for the approximation of the first derivative converges uniformly with order O(hp-1), p ? {4,5}.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

The Dirichlet problem for the Laplace equation in a starlike domain of a Riemann surface

2008 ◽  
Vol 49 (1-4) ◽  
pp. 299-313 ◽  
Author(s):  
Pierpaolo Natalini ◽  
Roberto Patrizi ◽  
Paolo E. Ricci
Keyword(s):  
Dirichlet Problem ◽  
Riemann Surface ◽  
Laplace Equation ◽  
Starlike Domain
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