Numerical approximations of fractional differential equations: a Chebyshev pseudo-spectral approach.
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Date
2020
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Abstract
This study lies at the interface of fractional calculus and numerical methods. Recent
studies suggest that fractional differential and integral operators are well suited to model
physical phenomena with intrinsic memory retention and anomalous behaviour. The global
property of fractional operators presents difficulties in fnding either closed-form solutions
or accurate numerical solutions to fractional differential equations. In rare cases, when
analytical solutions are available, they often exist only in terms of complex integrals and
special functions, or as infinite series. Similarly, obtaining an accurate numerical solution
to arbitrary order differential equation is often computationally demanding. Fractional
operators are non-local, and so it is practicable that when approximating fractional
operators, non-local methods should be preferred. One such non-local method is the
spectral method. In this thesis, we solve problems that arise in the
ow of non-Newtonian
fluids modelled with fractional differential operators. The recurrent theme in this thesis
is the development, testing and presentation of tractable, accurate and computationally
efficient numerical schemes for various classes of fractional differential equations. The
numerical schemes are built around the pseudo{spectral collocation method and shifted
Chebyshev polynomials of the first kind. The literature shows that pseudo-spectral
methods converge geometrically, are accurate and computationally efficient. The objective
of this thesis is to show, among other results, that these features are true when the method
is applied to a variety of fractional differential equations. A survey of the literature
shows that many studies in which pseudo-spectral methods are used to numerically
approximate the solutions of fractional differential equations often to do this by expanding
the solution in terms of certain orthogonal polynomials and then simultaneously solving
for the coefficients of expansion. In this study, however, the orthogonality condition of
the Chebyshev polynomials of the first kind and the Chebyshev-Gauss-Lobatto quadrature
are used to numerically find the coefficients of the series expansions. This approach is
then applied to solve various fractional differential equations, which include, but are not
limited to time{space fractional differential equations, two{sided fractional differential
equations and distributed order differential equations. A theoretical framework is provided
for the convergence of the numerical schemes of each of the aforementioned classes of
fractional differential equations. The overall results, which include theoretical analysis
and numerical simulations, demonstrate that the numerical method performs well in
comparison to existing studies and is appropriate for any class of arbitrary order differential
equations. The schemes are easy to implement and computationally efficient.
Description
Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.