A fourth-order accurate difference Dirichlet problem for the approximate solution of Laplace\u2019s equation with integral boundary condition

Adiguzel Dosiyev,Rifat Reis

doi:10.1186/s13662-019-2282-2

Abstract

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) belongs to the Hölder calsses C^{4,lambda }, 0<lambda <1, on the closed solution domain, the uniform estimate of the error of the approximate solution is of order O(h^{4}), where h is the mesh step. Numerical experiments are given to support analysis made.

Highlights

Different finite-difference problems as approximations of the nonlocal problems with integral boundary condition have been studied by many authors
In [4], the radial basis function collocation technique is used to find an approximate solution of an elliptic equation with nonlocal integral boundary condition
We propose and justify a new constructive method to solve a system of nonlocal 9-point finite-difference problem for the Laplace equation with the integral boundary condition

Summary

Introduction

Different finite-difference problems as approximations of the nonlocal problems with integral boundary condition have been studied by many authors (see [1,2,3,4,5] and references given therein). We propose and justify a new constructive method to solve a system of nonlocal 9-point finite-difference problem for the Laplace equation with the integral boundary condition. The solution of this nonlocal difference problem is defined as a solution of the 9-point Dirichlet problem by constructing approximate values of the solution on the side where the integral condition was given. Other nonlocal boundary value problems are stated and developed in numerous papers (see [9,10,11,12,13,14,15,16,17,18,19,20] and references therein)

Nonlocal boundary value problem Let

Nonlocal finite-difference problem and its reduction to the Dirichlet problem

Numerical experiments Let

Conclusion