الفهرس 
Basic ConceptsOnly 14 pages are availabe for public view
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Abstract
Differential equations are considered one of the most important and oldest methods used in building mathematical models for real problems. The methods for solving differential equations vary between analytical and numerical methods. Because of the development in numerical methods and techniques, as well as the systems used in numerical computations, the numerical treatment of equations and systems of differential equations is one of the basic topics. In this thesis, we focus on taking advantage of the memory and hereditary effects of the mathematical models as in differential equations of fractional orders, or in the properties of the method used as in iterative methods, or in improving the results of the same method used as in multigrid methods.
We use a technique to combine two iterative methods for solving systems of linear equations in one method by defining an updated iteration matrix to the composite of the two iterative methods. In addition, different algorithms (three) were used to be able to use the conjugate gradient method for solving non-symmetric systems of equations that result from the Poisson and Helmholtz equations in their fractional order. The Two-Grid and V-Cycle methods were used to solve a system of two elliptic differential equations.
This thesis consists of four chapters, Abstract, Arabic summary, and English summary.
Chapter One: Basic Concepts
In this chapter, we provide a brief overview of some basic definitions about matrices (symmetric, positive definite, spectral radius, and condition number). We present an introduction about stationary iterative methods (Jacobi, Gauss-Seidel, SOR, and KSOR methods). We consider refinement of stationary iterative methods (refinement of Jacobi, refinement of Gauss-Seidel, and refinement of SOR methods). We introduce a brief overview of fractional calculus (gamma function, beta function, Mittag-Leffler function, fractional integration, Riemann-Liouville fractional derivative, Caputo fractional derivative, and shifted Grünwald-Letnikov fractional derivative). We introduce the descant methods (steepest descant, and conjugate gradient methods) for solving linear system of algebraic equations and the multigrid methods (brief history for multigrid methods, model problems, and elements of multigrid) for solving elliptic partial differential equations.
Chapter Two: Composite Refinement of Stationary Iterative Techniques
In this chapter, we introduce a refinement of the KSOR (RKSOR) iterative method [32], composite refinement Jacobi Gauss-Seidel (RJGS) [33], composite refinement Gauss-Seidel Jacobi (RGSJ) [33], composite refinement Jacobi-SOR (RJSOR) [34], composite refinement SOR-Jacobi (RSORJ) [34], composite refinement GS-SOR (RGSSOR) [34] and composite refinement SOR-GS (RSORGS) [34] for solving linear system of algebraic equations. The efficient performance of the new techniques is illustrated theoretically and confirmed through the numerical examples. We prove that the new seven methods (RKSOR, RJGS, RGSJ, RJSOR, RSORJ, RGSSOR and RSORGS) are more efficient compared to the classical methods in terms of the rate of convergence and the number of iterations used to obtain the same approximate solution.
Chapter Three: Conjugate Gradient Methods for Partial Differential Equations
We present a preconditioned conjugate gradient method for solving non-symmetric linear systems. We consider three algorithms for converting non-symmetric linear systems to symmetric. We apply the three algorithms for solving linear system of algebraic equations, which arise from discretization of fractional order Poisson’s equation and fractional order Helmholtz’s equation. We compare these algorithms with SOR and KSOR methods. The optimum value of the relaxation parameter is sensitive in the SOR method than in the KSOR method.
Chapter Four: Multigrid Methods for Elliptic Problems
We introduce two methods of multigrid (Two-Grid and V-Cycle) for solving elliptic problems of partial differential equations. We use the SOR and KSOR methods as pre-smoothing and post-smoothing for relaxing the resultant algebraic linear system from elliptic problems. The choice of the appropriate values of the relaxation parameter is sensitive in the SOR method than in the KSOR method. We apply the Two-Grid and V-Cycle methods for solving magneto-hydrodynamic (MHD) elliptic system and fractional order Poisson’s equation.
It is worth to mentioning that:
All calculations were implemented using Mathematica 12.2.0 and MATLAB 7.9.0 (R2009b).
The results of chapter two are published in the three journals:
1. Sh. A. Meligy, I. K. Youssef, A Refinement of the KSOR Iterative Method, International Journal of Mathematics and Computer Science, Vol. 17, No. 3, pp. 1193-1199, 2022.
2. Sh. A. Meligy, I. K. Youssef, Composite Refinement Techniques for Solving Linear Systems, Journal of Mathematical and Computational Science, Vol. 12, ID 145, 2022.
3. Sh. A. Meligy, I. K. Youssef, Relaxation Parameters and Composite Refinement Techniques, Results in Applied Mathematics, Vol. 15, Article No. 100282, 2022.