الفهرس 
FundamentalsOnly 14 pages are availabe for public view
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Abstract
The main objectives of this thesis can be summarized in the following points:
A survey study on special polynomials such as; generalized Fibonacci and Lucas polynomials.
A theoretical study on fractional di erential equations subject to initial and boundary conditions.
A comprehensive study on spectral methods, and in particular tau, col-location and Galerkin methods.
Finding explicit formulae for the operational matrices of derivatives for certain basis functions that satisfy the homogeneous initial and boundary conditions, on the problems under investigation.
Implementing spectral algorithms for handling some certain types of dif-ferential equations involving some speci c problems such as; fractional Bagley-Torvik equation and one-dimensional telegraph type equation..
Comparing our algorithms with some other algorithms to show the ac-curacy and the applicability of the proposed methods.
The thesis consists of four chapters as follows:
Chapter 1
The purpose of this chapter is to introduce a brief introduction to the spectral methods and their advantages over the other standard methods. In
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addition, an overview on fractional calculus and the celebrated sequences of Fibonacci and Lucas polynomials.
Chapter 2
The main objectives of this chapter can be summarized in the following points:
Deriving operational matrices for integer and fractional derivatives of the generalized Fibonacci polynomials.
Constructing and developing algorithm for solving multi-term fractional-order di erential equation by using spectral collocation method.
Presenting some numerical results to investigate the applicability and accuracy of the scheme.
The results of this chapter are:
published in International Journal of Applied and Computational Mathematics with speci cations:
A. G. Atta, G. M. Moatimid and Y. H. Youssri, Generalized Fibonacci Oper-ational Collocation Approach for Fractional Initial Value Problems, Interna-tional Journal of Applied and Computational Mathematics, 5, (2019), 11 pages.
Chapter 3
The purpose of this chapter is twofold:
Presenting an algorithm for solving fractional Bagley-Trovic equation by using spectral tau method.
Reducing the solution of the equation with its homogeneous boundary conditions into a system of algebraic equations, then solving it.
Discussion of the convergence and error analysis of the proposed method.
The results of this chapter are:
accepted in Progress in Fractional Di erentiation and Applications, vol. 5, (2019).
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Chapter 4
The purpose of this chapter is twofold:
Presenting an algorithm for solving linear one-dimensional telegraph type equation by using spectral tau method.
Using generalized Lucas polynomials as basis functions to solve linear one-dimensional telegraph type equation.
Discussion of the convergence and error analysis of the proposed method.
The results of this chapter are:
submitted for publication and still under review.