الفهرس 
Theory and Basic Concepts
Stationary Iterative MethodsOnly 14 pages are availabe for public view
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Abstract
Thesis Title: On The Performance of Some Numerical Methods for Elliptic Systems
The thesis focuses on the study of the systems of elliptic equations in the plane starting with the standard model of the Poisson equation and discusses two methods of generalization. In the first method, the Poisson equation of fractional order is studied. In the second method, the elliptic system is studied in general with a focus on the system consisting of two equations. The thesis is also subjected to the biharmonic equation of order four, which is transformed into elliptic system of two second order partial differential equations. Also a system of two coupled Magneto hydrodynamics, MHD, equations has been transformed to uncoupled equations, which is produced when studying the stable flow of fluid through the pipes with optional conductivity walls in the presence of a magnetic field perpendicular to the direction of flow. The thesis is subjected to two methods of numerical solution in the first method; the finite difference method is used with emphasis on implicit alternating direction method. In the second method, the weighted residual method is used using Haar functions.
Since the methods used lead to the solution of a system of linear equations, the properties of the resulting linear equations as well as some methods of obtaining these equations have been studied efficiently. It is also shown that the resulting matrix in the case of fractional order Poisson equation is close to Poisson equation in the standard form, as the fractional differential is closer to the standard image.
Key words: Poisson’s Equation, Fractional Poisson’s Equation, Finite Difference, Wavelet, Haar Wavelet, Alternating directional.