514 research outputs found
Differential quadrature method for space-fractional diffusion equations on 2D irregular domains
In mathematical physics, the space-fractional diffusion equations are of
particular interest in the studies of physical phenomena modelled by L\'{e}vy
processes, which are sometimes called super-diffusion equations. In this
article, we develop the differential quadrature (DQ) methods for solving the 2D
space-fractional diffusion equations on irregular domains. The methods in
presence reduce the original equation into a set of ordinary differential
equations (ODEs) by introducing valid DQ formulations to fractional directional
derivatives based on the functional values at scattered nodal points on problem
domain. The required weighted coefficients are calculated by using radial basis
functions (RBFs) as trial functions, and the resultant ODEs are discretized by
the Crank-Nicolson scheme. The main advantages of our methods lie in their
flexibility and applicability to arbitrary domains. A series of illustrated
examples are finally provided to support these points.Comment: 25 pages, 25 figures, 7 table
Matrix approach to discrete fractional calculus II: partial fractional differential equations
A new method that enables easy and convenient discretization of partial
differential equations with derivatives of arbitrary real order (so-called
fractional derivatives) and delays is presented and illustrated on numerical
solution of various types of fractional diffusion equation. The suggested
method is the development of Podlubny's matrix approach (Fractional Calculus
and Applied Analysis, vol. 3, no. 4, 2000, 359--386). Four examples of
numerical solution of fractional diffusion equation with various combinations
of time/space fractional derivatives (integer/integer, fractional/integer,
integer/fractional, and fractional/fractional) with respect to time and to the
spatial variable are provided in order to illustrate how simple and general is
the suggested approach. The fifth example illustrates that the method can be
equally simply used for fractional differential equations with delays. A set of
MATLAB routines for the implementation of the method as well as sample code
used to solve the examples have been developed.Comment: 33 pages, 12 figure
A New Grünwald-Letnikov Derivative Derived from a Second-Order Scheme
A novel derivation of a second-order accurate Grünwald-Letnikov-type approximation to the fractional derivative of a function is presented. This scheme is shown to be second-order accurate under certain modifications to account for poor accuracy in approximating the asymptotic behavior near the lower limit of differentiation. Some example functions are chosen and numerical results are presented to illustrate the efficacy of this new method over some other popular choices for discretizing fractional derivatives
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