Iterative Methods and Fuzzy Algebraic Systems
Mahmoud AbdElmonem Farid;
Abstract
With scientific development featured mathematical models for most of natural laws. Since mathematical models contain natural variables and parameters that are determined by some measurements. And Since the measured values are often not precise, some mathematical methods have emerged that take into account the inaccuracy of the recorded measurements, the intervals arithmetic and related equations have many results and applications in physics and biological sciences. With the advent of the Fuzzy concepts, some types of equations (differential equations, integral equations, algebraic equations) contains fuzzy parameters have appeared. In this thesis, we focus on solving systems of algebraic equations, which contain some fuzzy-properties, because of the importance of this type of equation. Most of the differential or integral equations - in stages solution - are converted into a system of linear equations.
The thesis contains four chapters, a summary in Arabic, one in English and a bibliography.
Chapter One:
In this chapter, some basic concepts and definitions were introduced to solve equations systems using stationary iterative methods such as Jacob, Gauss Seidel, and Successive over Relaxation, KSOR, We also introduced Accelerated Overrelaxation method,which can be considered as a generalization of iterative methods Mentioned above and discussed some of the functional relations of these methods as we discussed some fuzzy concepts and intervals arithmetic.
Chapter Two:
In this chapter,we also introduced the KAOR method, which contains two parameters and we studied the values of these parameters at which the method can be convergent, indicated that this method can be considered as a generalization of some methods of Jacobi method and KSOR method. We also studied the convergence of the KAOR with some types of matrices, We compared the spectral radius of the KAOR and AOR methods and showed that the selection of the relaxation and acceleration parameters in the KAOR was less sensitive around the optimal values than in the AOR method and we illustrated this with numerical examples.
Chapter Three:
In this chapter we studied systems of algebraic fuzzy systems of equations containing fuzzy coefficients in the right-hand side only. We focused on the triangular fuzzy coefficients and its parametric form. We studied how to solve these fuzzy systems of equations using the embedding concept. We studied
The thesis contains four chapters, a summary in Arabic, one in English and a bibliography.
Chapter One:
In this chapter, some basic concepts and definitions were introduced to solve equations systems using stationary iterative methods such as Jacob, Gauss Seidel, and Successive over Relaxation, KSOR, We also introduced Accelerated Overrelaxation method,which can be considered as a generalization of iterative methods Mentioned above and discussed some of the functional relations of these methods as we discussed some fuzzy concepts and intervals arithmetic.
Chapter Two:
In this chapter,we also introduced the KAOR method, which contains two parameters and we studied the values of these parameters at which the method can be convergent, indicated that this method can be considered as a generalization of some methods of Jacobi method and KSOR method. We also studied the convergence of the KAOR with some types of matrices, We compared the spectral radius of the KAOR and AOR methods and showed that the selection of the relaxation and acceleration parameters in the KAOR was less sensitive around the optimal values than in the AOR method and we illustrated this with numerical examples.
Chapter Three:
In this chapter we studied systems of algebraic fuzzy systems of equations containing fuzzy coefficients in the right-hand side only. We focused on the triangular fuzzy coefficients and its parametric form. We studied how to solve these fuzzy systems of equations using the embedding concept. We studied
Other data
| Title | Iterative Methods and Fuzzy Algebraic Systems | Other Titles | الطرق التكرارية وأنظمة المعادلات الجبرية الفازية | Authors | Mahmoud AbdElmonem Farid | Issue Date | 2017 |
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