الفهرس | Only 14 pages are availabe for public view |
Abstract Summary Mathematical models for most of natural laws appeared due to sci- enti c development. As a result of human performance the mea- sured parameters included in the mathematical methods su¤er from the inaccuracy of the recorded measurements. As a remedy for this inaccuracy interval and fuzzy mathematics appear in many of the mathematical models which represent many biological and physical phenomenons. The fuzzy parameters appeared in many mathemati- cal models (di¤erential equations, integral equations, algebraic equa- tions). In this thesis we focused on linear algebraic systems which contain fuzzy parameters. Linear algebraic systems appear in the numerical treatment of di¤erential equations and integral equations The thesis contains four chapters, Arabic summary, English sum- mary and a list of references. Chapter One: In this chapter, some basic concepts and de nitions necessary to solve linear systems are introduced. Stationary iterative methods such as Jacobi, Gauss Seidel, and Successive over Relaxation, KSOR are considered. As a generalization of the previous iterative methods the Accelerated Overrelaxation method (AOR) is presented. Some fuzzy concepts and intervals arithmetic are studied. 5 6 Chapter Two: The KAOR method is introduced, which contains two relaxation parameters. The values of the relaxation parameters at which the method can be convergent are studied. Moreover some selected val- ues of the relaxation parameters are considered to illustrate that the KAOR method is a generalization of some methods like Jacobi method and KSOR method. We also studied the convergence of the KAOR with some types of matrices, We compared the spectral ra- dius of the KAOR and AOR methods and showed that the selection of the relaxation 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 fuzzy algebraic systems of equations containing fuzzy coe¢ cients in the right-hand side only. We focused on the triangular fuzzy coe¢ cients and its parametric form. The basic idea is to convert the fuzzy system of equations into one or two algebraic systems by using the embedding concept in two di¤erent techniques. In the rst technique, the fuzzy system of order nn was converted into a classical system of order 2n 2n and the solutions were studied and displayed. In the second technique, the fuzzy system is converted to two classic systems of type n n. We have applied this to numerical examples. 6 Summary Chapter Four: In this chapter, we applied the iterative methods presented in the rst and second chapters to fuzzy systems of equations. Also, we considered algebraic system of special construction which resulting from the numerical treatment of boundary value problems with fuzzy boundary conditions, results obtained are in a good value and the subject is promising. The thesis contains tables and gures to show and compare the results and all calculations were implemented using the calculation software designed using MATLAB. It should be noted that the content of the second chapter is pub- lished as paper entitled On the Accelerated Overrelaxation Method in the Pure and Applied Mathematics Journal, Fourth Edition, pages 26-31, 2015. The contents of chapter four will be prepared for publi- cation. |