6,274 research outputs found
On an explicit finite difference method for fractional diffusion equations
A numerical method to solve the fractional diffusion equation, which could
also be easily extended to many other fractional dynamics equations, is
considered. These fractional equations have been proposed in order to describe
anomalous transport characterized by non-Markovian kinetics and the breakdown
of Fick's law. In this paper we combine the forward time centered space (FTCS)
method, well known for the numerical integration of ordinary diffusion
equations, with the Grunwald-Letnikov definition of the fractional derivative
operator to obtain an explicit fractional FTCS scheme for solving the
fractional diffusion equation. The resulting method is amenable to a stability
analysis a la von Neumann. We show that the analytical stability bounds are in
excellent agreement with numerical tests. Comparison between exact analytical
solutions and numerical predictions are made.Comment: 22 pages, 6 figure
A comparison between numerical solutions to fractional differential equations: Adams-type predictor-corrector and multi-step generalized differential transform method
In this note, two numerical methods of solving fractional differential
equations (FDEs) are briefly described, namely predictor-corrector approach of
Adams-Bashforth-Moulton type and multi-step generalized differential transform
method (MSGDTM), and then a demonstrating example is given to compare the
results of the methods. It is shown that the MSGDTM, which is an enhancement of
the generalized differential transform method, neglects the effect of non-local
structure of fractional differentiation operators and fails to accurately solve
the FDEs over large domains.Comment: 12 pages, 2 figure
Local discontinuous Galerkin methods for fractional ordinary differential equations
This paper discusses the upwinded local discontinuous Galerkin methods for
the one-term/multi-term fractional ordinary differential equations (FODEs). The
natural upwind choice of the numerical fluxes for the initial value problem for
FODEs ensures stability of the methods. The solution can be computed element by
element with optimal order of convergence in the norm and
superconvergence of order at the downwind point of each
element. Here is the degree of the approximation polynomial used in an
element and () represents the order of the one-term
FODEs. A generalization of this includes problems with classic 'th-term
FODEs, yielding superconvergence order at downwind point as
. The underlying mechanism of the
superconvergence is discussed and the analysis confirmed through examples,
including a discussion of how to use the scheme as an efficient way to evaluate
the generalized Mittag-Leffler function and solutions to more generalized
FODE's.Comment: 17 pages, 7 figure
Numerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control
This chapter presents some numerical methods to solve problems in the
fractional calculus of variations and fractional optimal control. Although
there are plenty of methods available in the literature, we concentrate mainly
on approximating the fractional problem either by discretizing the fractional
term or expanding the fractional derivatives as a series involving integer
order derivatives. The former method, as a subclass of direct methods in the
theory of calculus of variations, uses finite differences, Grunwald-Letnikov
definition in this case, to discretize the fractional term. Any quadrature rule
for integration, regarding the desired accuracy, is then used to discretize the
whole problem including constraints. The final task in this method is to solve
a static optimization problem to reach approximated values of the unknown
functions on some mesh points.
The latter method, however, approximates fractional problems by classical
ones in which only derivatives of integer order are present. Precisely, two
continuous approximations for fractional derivatives by series involving
ordinary derivatives are introduced. Local upper bounds for truncation errors
are provided and, through some test functions, the accuracy of the
approximations are justified. Then we substitute the fractional term in the
original problem with these series and transform the fractional problem to an
ordinary one. Hereafter, we use indirect methods of classical theory, e.g.
Euler-Lagrange equations, to solve the approximated problem. The methods are
mainly developed through some concrete examples which either have obvious
solutions or the solution is computed using the fractional Euler-Lagrange
equation.Comment: This is a preprint of a paper whose final and definite form appeared
in: Chapter V, Fractional Calculus in Analysis, Dynamics and Optimal Control
(Editor: Jacky Cresson), Series: Mathematics Research Developments, Nova
Science Publishers, New York, 2014. (See
http://www.novapublishers.com/catalog/product_info.php?products_id=46851).
Consists of 39 page
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