الفهرس 
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Abstract
Fractional calculus is one of the most interdisciplinary fields of applied mathematics which has attracted the attention of many researchers. The memory and the non local property of the fractional calculus make it more accurate than the classical (integer-order) calculus for mathematical modeling of many various phenomena in science and engineering. So far there have been several fundamental works on the fractional derivatives and fractional differential equations. The approximate solutions of the fractional differential equations have a growing interest to study the behavior of these equations while finding the exact analytical solutions of such equations are almost impossible.
The main objective of this thesis is to develop new spectral tau methods based on the shifted Gegenbauer polynomials for the numerical treatment of a wide classes of fractional linear and nonlinear differential equations with either constant or variable coefficients in multi dimensions. The aforementioned fractional differential equations are considered with three different types of fractional derivatives, namely the constant, the distributed and the variable order fractional derivatives. Accordingly, the proposed methods are implemented on important classes of linear and nonlinear multi-dimensional mathematical models with the types of fractional derivatives to illustrate the applicability, the validity and the performance of these methods. These mathematical models contain the equations of telegraph, diffusion-wave, diffusion-wave with damping, wave and Klein-Gordon. The proposed fractional mathematical models play an important role in describing several phenomena in diverse applied fields, for example, blood flow under magnetic and vibration mode in a porous tube with thermochemical.
properties, propagation, and transmission of electrical signals, enhancing denoising of image structure of elasticmanifolds, bimolecular reactions in homogeneous media, anomalous diffusion, the neutron transport process inside the core of a nuclear reactor and removes the hiatuses in the characterization of neutron motions exists in the classical integer-order models.
In this dissertation, the spectral tau method based on the SGPs has been developed for solving a class of multi-dimensional linear and nonlinear TFPDEs with constant and variable coefficients. The proposed TFPDEs represent several equations of important applications in diverse fields, such as, the TFTE, the TFDWE, the TFDWE with damping, the TFWE and the TFKGE. The presented mathematical models have been defined with three different fractional derivatives, these derivatives include the COFD, DOFD and VOFD in the Caputo type. The spectral tau method presents high precision of its approximations because it does not require
a discretization of the given differential operator, a process which often alters the behavior of solutions. It takes the full advantage of the nonlocal nature of the presented three kinds fractional differential operators. Also, it has an exponential rate of convergence in both time and space.
To apply the proposed technique, we successfully derived novel OMs of the COFDs, DOFDs and VOFDs of SGPs in Caputo type, these OMs of fractional derivatives played significant role for solving the different kinds of the proposed TFPDEs. Also, the OMs of the multiplication of the Kronecker product of the space vectors and space-time vectors in multi-dimensional have been constructed effectively and in a simple way, these OMs played an essential role for completely applying the spectral tau method in the presence of the variable coefficients, the nonlinear terms and the variable-order functions included in the OM of VOFDs.
The spectral tau method and aforementioned OMs have been combined together. As a result, the equation under study turned into a linear or nonlinear algebraic system of equations depending on the linearity or nonlinearity of the proposed mathematical model, which are easier to solve. During our study, we found that the proposed spectral tau technique yielded an exponential rate of convergence with a small number of SGPs when the solution is smooth which is in agreement with the presented theoretical analysis in the L2ωα- norm. While, for non-smooth solutions, smooth transformations have been used in terms of SGPs to avoid the convergence deteriorated. Also the computational
techniques for all considered mathematical models were computationally less expensive and easy to implement. They shined best when high accuracy is sought in regular domains with smooth solutions. Also, they took full advantage of the nonlocal nature of constant-order, distributed-order and variable-order fractional differential operators. Our results confirmed that the nonlocal numerical methods are best suited to discretize the constant-order, distributed-order and variableorder-
TFPDEs as they naturally take the global behavior of the solution into account.
The proposed computational techniques presented high accurate solutions
at various shifted Gegenbauer parameters α for the different types of the fractional derivatives. It has been shown through the comparisons made during this thesis between the proposed computational techniques with many other methods that the proposed computational techniques gave more accurate numerical solutions. Also, our results gave an indication of the applicability of the proposed spectral tau method to linear and nonlinear models with constant-order, distributed-order and variable-order fractional differential operators that appear in engineering and
science. The proposed spectral tau methods have a relatively high computational time for the 2D nonlinear problems when increasing the basis functions due to the resultant large number of nonlinear systems of algebraic equations. Also, they cannot apply to the problems with irregular domains.