scholarly journals Study of approximate solution to integral equation associated with mixed boundary value problem for Laplace equation

2021 ◽  
Vol 13 (1) ◽  
pp. 85-97
Author(s):  
Elnur Hasan ogly Khalilov ◽  
Mehpara Nusrat kyzy Bakhshaliyeva
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Laplace Equation ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Value Problem

scholarly journals On a method for approximate solution of a mixed boundary value problem for an elliptic equation

2021 ◽  
Vol 23 (1) ◽  
pp. 58-71
Author(s):  
Mahmut E. Fairuzov ◽  
Fedor V. Lubyshev
Keyword(s):  
Boundary Condition ◽  
Approximate Solution ◽  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Variable Coefficients ◽  
Natural Boundary Condition ◽  
Mixed Boundary Value Problem ◽  
Integration Region

A mixed boundary value problem for an elliptic equation of divergent type with variable coefficients is considered. It is assumed that the integration region is a rectangle, and the boundary of the integration region is the union of two disjoint pieces. The Dirichlet boundary condition is set on the first piece, and the Neumann boundary condition is set on the other one. The given problem is a problem with a discontinuous boundary condition. Such problems with mixed conditions at the boundary are most often encountered in practice in process modeling, and the methods for solving them are of considerable interest. This work is related to the paper [1] and complements it. It is focused on the approbation of the results established in [1] on the convergence of approximations of the original mixed boundary value problem with the main boundary condition of the third boundary value problem already with the natural boundary condition. On the basis of the results obtained in this paper and in [1], computational experiments on the approximate solution of model mixed boundary value problems are carried out.

scholarly journals Electrostatic potential calculation by application of walk-on-spheres method

2018 ◽  
pp. 152-160
Author(s):  
Liudmila Vladimirova ◽  
Irina Rubtsova ◽  
Nikolai Edamenko
Keyword(s):  
Integral Equation ◽  
Problem Solving ◽  
Boundary Value Problem ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Value Problem ◽  
Equation Solving ◽  
Electrostatic Potential Calculation ◽  
Walk On Spheres Algorithm ◽  
Boundary Form

The paper is devoted to mixed boundary-value problem solving for Laplace equation with the use of walk-on-spheres algorithm. The problem under study is reduced to finding a solution of integral equation with the kernel nonzero only at some sphere in the domain considered. Ulam-Neumann scheme is applied for integral equation solving; the appropriate Markov chain is introduced. The required solution value at a certain point of the domain is approximated by the expected value of special statistics defined on Markov paths. The algorithm presented guarantees the average Markov trajectory length to be finite and allows one to take into account boundary conditions on required solution derivative and to avoid Markov paths ending in the neighborhood of the boundaries where solution values are not given. The method is applied for calculation of electric potential in the injector of linear accelerator. The purpose of the work is to verify the applicability and effectiveness of walk-on-spheres method for mixed boundary-value problem solving with complicated boundary form and thus to demonstrate the suitability of Monte Carlo methods for electromagnetic fields simulation in beam forming systems. The numerical experiments performed confirm the simplicity and convenience of this method application for the problem considered.

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.

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