About the nonlocal contact problem for the second order differential equation with constant coefficients in one-dimensional and two-dimensional cases

Author: Giorgi Iliaevi
Keywords: nonlocal, nonlocal contact, contact problem
Annotation:

This master's thesis addresses the numerical solutions of non-local contact problems for both ordinary and partial differential equations. In the first part of the thesis, we explore non-local contact problems for non-homogeneous ordinary differential equations (ODEs) of the second order with constant coefficients, considering both two-point and multi-point non-local contact conditions. Analytical solutions for these problems are constructed, and an iterative process is developed which reduces the solution of the non-local problem to a sequence of classical boundary value problems for second-order ODEs. The numerical methods for their solution are well-established. The iterative process is constructed, and the finite difference method is employed for the numerical solution. 3 Computational experiments are conducted under various conditions for the two-point contact problem. The second part of the thesis focuses on non-local contact problems for the two dimensional linear elliptic equation with constant coefficients, specifically under two-point non local contact boundary conditions within a rectangular domain. The existence and uniqueness of the regular solution are proved. An iterative method is considered and investigated for solving the stated problem. Similar to the ODE case, this problem also follows a sequence of classical problems, with the finite difference method utilized for numerical solutions. To outline future research directions, the classical Dirichlet problem for the elliptic equation is studied. Also, several non-local contact problems for circular arcs are stated.

Lecture files:

არალოკალური საკონტაქტო ამოცანის შესახებ [ka]
ანოტაცია, Abstract [ka]
ანოტაცია - ძველი ვერსია [ka]