الفهرس 
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Abstract
The aim of spectral methods is to approximate functions (solutions of differential equations) by means of truncated series of orthogonal polynomials. There are four well-known methods of spectral methods, namely, tau, collocation, Galerkin and Petrov-Galerkin methods. The choice of the suitable used spectral method suggested for solving the given equation depends certainly on the type of the differential equation and also on the type of the boundary conditions governed by it. The choice of test functions distinguishes between the four versions of spectral schemes.
The purpose of this thesis is twofold, the numerical approaches based on tau and collocation techniques in conjunction with operational matrices of some orthogonal polynomials/functions, for solving several kinds of fractional differential equations (FDEs) on finite and infinite domains are introduced. The employment of operational matrices for solving different kinds of differential equations is considered as a common technique. There are several studies in this respect. This approach has two main advantages, the first is its simplicity, and the second is the accuracy of the approximate solutions resulted from their uses.
Concerning the first topic, we present the construction of the shifted Legendre operational matrix (SLOM), shifted Chebyshev operational matrix (SCOM), shifted Jacobi operational matrix (SJOM), Laguerre operational matrix (LOM), modified generalized Laguerre operational matrix (MGLOM) and Bernstein operational matrix (BOM) of fractional derivatives and integrals that are employed with the tau method to provide efficient numerical schemes for solving linear FDEs on finite and infinite interval. Moreover, we present the construction of generalized Laguerre operational matrix (GLOM) and fractional-order generalized Laguerre operational matrix (FGLOM) of fractional derivatives and integrals which are employed with tau methods to provide two efficient numerical schemes for solving linear FDEs on a semi-infinite interval.
The second topic of this thesis concerns the spectral collocation method for solving the FDEs on finite and infinite domains. We introduce a Bernstein operational matrix (BOM) of fractional derivatives with collocation method for solving nonlinear FDEs. Also, we present the shifted Jacobi collocation (SJC) and the modified generalized Laguerre collocation (MGLC) methods, for solving fractional initial and boundary value problem of fractional order ν > 0 with nonlinear terms, in which the nonlinear FDE is collocated at the N zeros of the orthogonal functions. We also aim to propose a fractional-order generalized Laguerre collocation (FGLC) method, for solving fractional initial value problems of fractional order ν (0 < ν < 1) with nonlinear terms, in which the nonlinear FDE is collocated at the N zeros of the new function which is defined on the interval Λ = (0,∞). We extend the application of the FGLC method based on these functions to solve a system of FDEs with fractional orders less than 1.
Moreover, we derive three-term recurrence relations to efficiently calculate the variable-order fractional integrals and derivatives of the modified generalized Laguerre polynomials, which lead to the corresponding fractional differentiation matrices that will be used to construct the collocation methods. Besides, using the modified generalized Laguerre polynomials as the basis functions, we develop Laguerre-Gauss collocation methods to solve fractional differential equations of variable and constant orders on a semi-infinite domain. Finally, we propose a new efficient spectral collocation method for solving a time fractional sub-diffusion equation on a semi-infinite domain. The main advantage of the proposed approach is that a spectral method is implemented for both time and space discretizations, which allows us to present a new efficient algorithm for solving time fractional sub-diffusion equations.
The obtained numerical results are tabulated and displayed graphically whenever possible. These results show that our proposed algorithms of solutions are reliable and accurate. Comparisons with previously obtained results by other researchers or exact known solutions are made throughout the context whenever available.
To the best of our knowledge, the formulae and algorithms stated and proved in Chapters 2 up to 5 are completely new. The Programs used in this thesis are performed using the PC machine, with Intel(R) Core(TM) i7 CPU 2.00 GHz, 8.00 GB of RAM, and the symbolic computation software Mathematica Version 9.0 has been also used.