الفهرس 
Elliptic Partial Differential Equations in the PlaneOnly 14 pages are availabe for public view
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Abstract
Thesis title: “On the Numerical Treatment of Elliptic System of Partial
Differential Equations”.
The thesis introduces the numerical treatment of elliptic system of partial differential equations in the plane using the finite difference method. Two types of grids (square - triangular) are considered. The structure of the resultant algebraic system depends on the grid as well as the labeling of the grid points (the natural, the electronic, the RBG, and the spiral). Different forms of iterative methods (iterative methods without relaxation parameters, iterative methods with only one parameter, and iterative methods with two parameters) for solving linear algebraic systems are discussed.
This thesis consists of five chapters, Arabic summary, and English summary.
Chapter One: Elliptic Partial Differential Equations in the Plane
A system of two partial differential equations in the plane is studied. Transformation of systems of partial differential equations into canonical forms which contain the smallest possible number of parameters (only two parameters instead of twelve) is obtained. Classification of systems of partial differential equations in the plane is given. The finite difference method is used to transform differential equations into algebraic systems. The algebraic structures of the resultant algebraic systems as well as the grid labeling are established. The square grid is discussed with three labeling techniques (the natural, the electronic, and the red-black order) in connection with the standard Poission’s problem.
Chapter Two: Variants of Successive Overrelaxation Techniques
Iterative methods for solving linear algebraic systems are established. Jacobi and Gauss Seidel methods are classified as iterative techniques without relaxation factors. SOR and KSOR methods are classified as iterative techniques with only one relaxation factor. Different forms of the modified successive overrelaxation methods (MSOR, MKSOR, MKSOR1, and MKSOR2) are introduced as iterative techniques with two relaxation factors. Iteration matrices and functional eigenvalue relations are given. Because of the fixed values of the spectral radii of Jacobi and Gauss Seidel iteration matrices, they are used to compare the convergence speeds with other methods and for the selection of relaxation parameters.
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Chapter Three: Elliptic Systems Strongly Coupled Through the Mixed
Derivative Term
Elliptic systems strongly coupled through the mixed derivative term are considered. Ordering of unknowns in connection with different labeling of grid points in the square grid is given. selection of relaxation parameters which gives the smallest value of the spectral radius of the iteration matrix of different methods (SOR, KSOR, MSOR, MKSOR, MKSOR1, and MKSOR2) is established.
Chapter Four: Elliptic Systems Coupled Through the First Order
Derivative Term
Elliptic systems coupled through the term of first order derivative with the effect of decoupling are studied. Ordering of unknowns in connection with different labeling of grid points in the square grid is given. selection of relaxation parameters which gives the smallest value of the spectral radius of the iteration matrix for different methods (SOR, KSOR, MSOR, MKSOR, MKSOR1, and MKSOR2) is established. Reduction of the system to two uncoupled equations is considered. Application to a realistic physical problem is discussed.
Chapter Five: Boundary Value Problems on Triangular Domains
A treatment for the boundary value problem over a triangular grid is introduced similar to the treatment introduced by Young for the square grid. The effect of different labeling techniques (the natural, the electronic, the spiral, and red, black and green) on the structure of algebraic system is established. Elliptic equation with mixed derivative term is considered, a parameter r is introduced which enables one to obtain the results and the algebraic structures of elliptic problems without the mixed derivative term. Also, the system introduced in chapter four is discussed over the triangular grid.
It is worth to mention that:
- All calculations were done by the use Mathematica 8.0.
- The results of chapter five are published in the Journal of Applied and
Computational Mathematics.
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