الفهرس 
Chapter 1Only 14 pages are availabe for public view
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Abstract
Partial differential and partial integro-differential equations (PIDEs) are considered one of the most prevalent equations in many engineering, physical and chemical fields. Partial differential equations (PDEs) play a major role in the modeling of many processes. However, in many evolutionary problems, the history of the phenomena under investigation is of relevance and must be incorporated into the mathematical model. Classical PDEs models cannot reproduce this property, and they fail to provide a reliable description of such processes. As a solution to overcome this drawback, PDEs have been replaced by PIDEs. The theoretical and experimental research that has been conducted in recent years shows that mathematical models based on PIDEs, which consider this memory effect, are more accurate than the traditional PDE models. Approximate solutions to these equations are of great importance due to the limited availability of their exact solutions especially on complex domains. There are several methods used to obtain approximate solutions to these equations including finite element methods (FEMs). Researchers grew more interest in FEMs as a result of the progress in computer programs that greatly facilitated complicated mathematical operations required for applying these methods. In this thesis, we introduce a more realistic and more accurate representation of PDEs and PIDEs by generalizing the space partial derivative to Riesz fractional derivative and perform numerical simulations that increase the accuracy of FEM approximations to space-fractional partial differential and partial integro-differential equations. We proposed an adaptive refinement scheme for improving the accuracy when solving problems with Riesz fractional derivatives over nonuniform meshes using FEM. Thus, we deduced the fractional derivatives of FEM bases on nonuniform mesh. This scheme depends on the FEM solution and a recovery-based error estimator to adaptively refine the mesh so that better accuracy is achieved with the minimum number of degrees of freedom. The recovery technique used is a gradient recovery technique called the polynomial preserving recovery (PPR). The error analysis and stability conditions for the proposed scheme have been explored.