On the numerical solution for nonlinear elliptic equations with variable weight coefficients in an integral boundary conditions

Regimantas Čiupaila,Mifodijus Sapagovas,Kristina Pupalaigė

doi:10.15388/namc.2021.26.23929

Abstract

In the paper the two-dimensional elliptic equation with integral boundary conditions is solved by finite difference method. The main aim of the paper is to investigate the conditions for the convergence of the iterative methods for the solution of system of nonlinear difference equations. With this purpose, we investigated the structure of the spectrum of the difference eigenvalue problem. Some sufficient conditions are proposed such that the real parts of all eigenvalues of the corresponding difference eigenvalue problem are positive. The proof of convergence of iterative method is based on the properties of the M-matrices not requiring the symmetry or diagonal dominance of the matrices. The theoretical statements are supported by the results of the numerical experiment.

Highlights

Introduction and problem formulationIt is important to emphasize that in most cases of elliptic equations with nonlocal conditions in the form of (2), (3) the matrix of difference problem under some assumption to functions α(x) and β(x) has the properties appropriate for the M-matrices [22, 29]
Introduction and problem formulationIn this paper, we will consider the nonlinear elliptic equation ∂2u ∂2u∂x2 + ∂y2 = f (x, y, u), (x, y) ∈ Ω = {0 < x < 1, 0 < y < 1}, (1)with integral boundary conditions u(0, y) = α(x)u(x, y) dx + μ1(y), 0 y 1, (2)0 1 u(1, y) = β(x)u(x, y) dx + μ2(y), 0 y 1, (3)and Dirichlet boundary conditions at the points of the remaining two sides of the rectangle Ω u(x, 0) = μ3(x), u(x, 1) = μ4(x), 0 x 1. (4)The boundary value problems for elliptic equations with nonlocal conditions as some elementary generalization of classical boundary value problems were formulated in [5,8]
We write down the system of difference equations (13)–(16) in the matrix form

Summary

Introduction and problem formulation

It is important to emphasize that in most cases of elliptic equations with nonlocal conditions in the form of (2), (3) the matrix of difference problem under some assumption to functions α(x) and β(x) has the properties appropriate for the M-matrices [22, 29]. Based on this property, it is possible to prove the convergence of many iterative methods [22].

A difference problem

An eigenvalue problem of matrix A

Iterative methods

Conditions under which the matrix of difference problem is an M-matrix

Remarks and generalization