الفهرس 
1 FundamentalsOnly 14 pages are availabe for public view
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Abstract
Many physical phenomena can be described by fractional-orders differential equations. The solutions to most fractional-order differential equations have no convenient explicit forms and hence the numerical solution of such equations is a subject of fundamental importance. In this work, we are concerned with how to find explicit approximations for functions defined by fractional-order differential equations using spectral methods. In particular, we consider linear and nonlinear fractional-order differential equations. One of the main objectives of this thesis is to introduce and develop new algorithms based on the shifted Chebyshev- and shifted Jacobi-tau approximations for solving linear multi-term fractional-order differential equations. Moreover, we aim to introduce and construct efficient algorithms based on the shifted Chebyshev- and shifted Jacobi-collocation approximations for solving nonlinear multi-term fractional-order differential equations. We demonstrate the advantage of using the collocation approximations, by noting that the nonlinear fractional-order differential equations are reduced to nonlinear algebraic equations. We also aim to introduce and develop new algorithms based on the shifted chebyshevtau approximations in combination with the shifted Jacobi operational matrix of fractional derivatives for solving initial and boundary value problems of fractional-orders with constant coefficients. As an interesting generalization, we give efficient algorithms based on the shifted Jacobi collocation approximations for the nonlinear initial and boundary value problems of fractional-orders which are based on shifted Jacobi operational matrix of fractional derivatives. The expansion for the function (solution), in terms of Jacobi polynomials P(α,β)
n (x) (n ≥ 0, α > −1, β > −1), enables one to get the sought- for Jacobi-tau approximation for any possible values of the real parameters α and β. That is, instead of developing approximation results for each particular pair of indexes (α, β), it would be very useful to carry out a study on Jacobi polynomials P(α,β) n (x) with general indexes which can then be directly applied to other applications. In particular, the four important (symmetric Jacobi) special cases of ultraspherical polynomials α = β and each is replaced by (α− 1 2 ), Chebyshev
polynomials of the first kind (α = β = −1/2), Legendre polynomials (α = β = 0)
and Chebyshev polynomials of the second kind (α = β = 1/2) are considered; the two
(nonsymmetric Jacobi) special cases of Chebyshev polynomials of the third and fourth
kinds corresponding to (α = −β = ±1
2 ) are also noted. The obtained results show that
the linear systems based on the expansion in Chebyshev polynomials of the first kind
(α = β = −1/2) are not always better than the expansion in other Jacobi polynomials.
In Chapter 1, we give a general introduction to fractional calculus, and spectral methods
and their advantages over the standard finite-difference and finite-element methods.
We also clarify the differences between the three most commonly used spectral methods,
namely, the Galerkin, Collocation and tau methods. A brief account of orthogonal
polynomials, their properties and expansion of functions in terms of them are given.
Some general properties and important relations concerned with the Jacobi polynomials
P(α,β)
n (x) are considered.
In Chapter 2, we state and prove a new formula expressing explicitly the derivatives
of shifted Chebyshev polynomials of any degree and for any fractional-order in terms of
shifted Chebyshev polynomials themselves. We develop also a direct solution technique for
solving the linear multi-order fractional differential equations (FDEs) with constant coefficients
using a spectral tau method. The spatial approximation with its fractional-order
derivatives (described in the Caputo sense) are based on shifted Chebyshev polynomials
TL,n(x) with x ∈ (0,L), L > 0 and n is the polynomial degree. We presented a shifted
Chebyshev collocation method with shifted Chebyshev-Gauss points used as collocation
nodes for solving nonlinear multi-order fractional initial value problems. Several numerical
examples are considered aiming to demonstrate the validity and applicability of the
proposed techniques and to compare with the existing results.
In Chapter 3, we are concerned with a linear and nonlinear multi-term fractional
differential equations. The shifted Chebyshev operational matrix (COM) of fractional
derivatives is derived and used together with spectral methods for solving FDEs. Our
approach is based on the shifted Chebyshev tau and collocation methods. The proposed
method is applied to solve two types of FDEs, linear and nonlinear subject to initial
or boundary conditions, and the exact solutions are obtained for some tested problems Numerical results with comparisons are given to confirm the reliability of the proposed
method for some FDEs.
In Chapter 4, we state and prove a new formula expressing explicitly the derivatives of
shifted Jacobi polynomials of any degree and for any fractional-order in terms of shifted
Jacobi polynomials themselves. We develop a direct solution technique for solving linear
multi-order FDEs with constant coefficients using a spectral tau approximation. Some
efficient direct solvers for the same equations with variable coefficients using quadrature
shifted Jacobi tau (Q-SJT) approximation is introduced. We also present a shifted Jacobi
collocation method for solving nonlinear multi-order fractional initial value problems. We
present the advantages of using our proposed techniques and compare them with several
other methods. Some numerical experiments are exhibited to solve FDEs including linear
and nonlinear terms.
Finally in Chapter 5, we derive the shifted Jacobi operational matrix (JOM) of fractional
derivatives which applied together with spectral tau method for numerical solution
of general linear multi-term FDEs. A new approach implementing shifted Jacobi operational
matrix in combination with the shifted Jacobi collocation technique is introduced
for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind
this approach is that it reduces such problems to those of solving a system of algebraic
equations which greatly simplifying the problem. The proposed methods are applied for
solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions,
and the exact solutions are obtained for some tested problems. Special attention is given
to the comparison of the numerical results obtained by the new algorithm with those
found by other known methods.
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) 2 Duo CPU 2.00 GHz, 2.00 GB of RAM,
and the symbolic computation software Mathematica Version 6.0 has been also used.