A Resistor Network for the Approximate Solution of the Laplace Equation

1947 ◽  
Vol 18 (10) ◽  
pp. 798-799 ◽  
Author(s):  
David C. dePackh
Keyword(s):  
Approximate Solution ◽  
Laplace Equation ◽  
Resistor Network

On an ill-posed boundary value problem for the Laplace equation in a circular cylinder

2021 ◽  
pp. 35-43
Author(s):  
Evgeniy B. Laneev ◽  
Dmitriy Yu. Bykov ◽  
Anastasia V. Zubarenko ◽  
Olga N. Kulikova ◽  
Darya A. Morozova ◽  
...  
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Circular Cylinder ◽  
Boundary Value Problem ◽  
Laplace Equation ◽  
Boundary Value ◽  
Stable Solution ◽  
Medical Diagnostics ◽  
Cauchy Data ◽  
Ill Posed

In this paper, we consider a mixed problem for the Laplace equation in a region in a circular cylinder. On the lateral surface of a cylidrical region, the homogeneous boundary conditions of the first kind are given. The cylindrical area is bounded on one side by an arbitrary surface on which the Cauchy conditions are set, i.e. a function and its normal derivative are given. The other border of the cylindrical area is free. This problem is ill-posed, and to construct its approximate solution in the case of Cauchy data known with some error it is necessary to use regularizing algorithms. In this paper, the problem is reduced to a Fredholm integral equation of the first kind. Based on the solution of the integral equation, an explicit representation of the exact solution of the problem is obtained in the form of a Fourier series with the eigenfunctions of the first boundary value problem for the Laplace equation in a circle. A stable solution of the integral equation is obtained by the Tikhonov regularization method. The extremal of the Tikhonov functional is considered as an approximate solution. Based on this solution, an approximate solution of the problem in the whole is constructed. The theorem on convergence of the approximate solution of the problem to the exact one as the error in the Cauchy data tends to zero and the regularization parameter is matched with the error in the data is given. The results can be used for mathematical processing of thermal imaging data in medical diagnostics.

Combined Adaptive Meshfree Method for Modeling Potential Physical Fields

2021 ◽  
Vol 64 (1) ◽  
pp. 5-12
Author(s):  
Alexander Tkachev ◽  
◽  
Dmitry Chernoivan ◽  
Keyword(s):  
Monte Carlo ◽  
Approximate Solution ◽  
Monte Carlo Method ◽  
Laplace Equation ◽  
Meshfree Method ◽  
Fundamental Solutions ◽  
Computational Domain ◽  
The Best Approximation ◽  
Physical Fields ◽  
Two Stages

A procedure for numerical solution of the Dirichlet problem for the Laplace equation, which is often re-duced to modeling potential physical fields in homogeneous media, is described. The approximate solution is proposed to be found using a combined meshfree Monte Carlo method and fundamental solutions, which is implemented in two stages. At the first stage, the element of the best approximation in the linear shell of the fundamental solutions of the Laplace equation is determined. At the second stage, the solution is refined using the potential values found by the Monte Carlo method at individual points in the computational domain. Algorithms are given for finding the defining parameters of both methods used to reduce the error. The procedure for evaluating the accuracy of the found approximate solution of the problem is described. An example of calculating the potential distribution in the angular zone under specified boundary conditions using the combined meshfree method is given. The accuracy of the approximate solution is estimated by comparing it with the exact solution. It is shown that the use of the meshfree method leads to a decrease in the error without a significant increase in the computational resources required for its implementation

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.

Approximate Solution of One Dynamic Problem of Geoinformatics

2010 ◽  
Vol 42 (5) ◽  
pp. 1-11 ◽  
Author(s):  
Vladimir M. Bulavatskiy ◽  
Vasiliy V. Skopetsky
Keyword(s):  
Approximate Solution ◽  
Dynamic Problem

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.

Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

2008 ◽  
Vol 8 (2) ◽  
pp. 143-154 ◽  
Author(s):  
P. KARCZMAREK
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Half Plane ◽  
Singular Integral ◽  
Trigonometric Polynomials ◽  
Cauchy Kernel

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

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