4 research outputs found
Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method
In this work we present a new numerical method for the solution of the distributed order time fractional diffusion equation. The method is based on the approximation of the solution by a double Chebyshev truncated series, and the subsequent collocation of the resulting discretised system of equations at suitable collocation points. An error analysis is provided and a comparison with other methods used in the solution of this type of equation is also performed
Fractional Calculus and the Future of Science
Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding
Numerical approximations of fractional differential equations: a Chebyshev pseudo-spectral approach.
Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.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