الفهرس 
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Abstract
Summary : The aim of the thesis is to discuss the effect of nonlocality in boundary conditions on the eigenvalues of some finite difference approaches. The thesis is organized in four chapters as follows : Chapter one : This chapter presents a history introduction about the origin and development of the use of non-local conditions in many applications and natural phenomena was followed by a review of some previous studies of research related to the subject of the thesis, followed by a detailed presentation of some of the mathematical concepts and equations used in this thesis and followed by a summary of the objectives of the research submitted to the thesis. Chapter two : Three new forms of multipoint non-local BCs were proposed for elliptic PDE. These conditions work as generalizations to the already known ones. We generate the system of equations of the finite difference scheme and the boundary conditions were implemented in it. The considered non-local BCs that were inside the domain produce a shift in the tridiagonal elements which appear in the matrix for elliptic problems with classical conditions, we end up with general relations that describe not only the considered problems. As well as these conditions contain several parameters that can describe the different cases of boundary conditions including Dirichlet, Neumann and mixed conditions. The effect of these parameters on the difference eigenvalue problem for an elliptic PDE was investigated subject to the proposed conditions. Chapter three : we consider an elliptic interface problem with nonlocal conditions. As far as we know, this is the first attempt to tackle the difference eigenvalue problem such problems. Due to the non-locality in boundary conditions and interface conditions, the matrix of eigenvalue difference scheme is non-symmetric. Thus, it may have zero, positive, negative, or complex eigenvalues. The system of equations of the finite difference scheme is obtained and the boundary conditions and inter-face conditions were implemented in it too. Chapter four : Concluding remarks and suggested future work are given.