813 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
Stable L\'{e}vy diffusion and related model fitting
A fractional advection-dispersion equation (fADE) has been advocated for
heavy-tailed flows where the usual Brownian diffusion models fail. A stochastic
differential equation (SDE) driven by a stable L\'{e}vy process gives a forward
equation that matches the space-fractional advection-dispersion equation and
thus gives the stochastic framework of particle tracking for heavy-tailed
flows. For constant advection and dispersion coefficient functions, the
solution to such SDE itself is a stable process and can be derived easily by
least square parameter fitting from the observed flow concentration data.
However, in a more generalized scenario, a closed form for the solution to a
stable SDE may not exist. We propose a numerical method for solving/generating
a stable SDE in a general set-up. The method incorporates a discretized finite
volume scheme with the characteristic line to solve the fADE or the forward
equation for the Markov process that solves the stable SDE. Then we use a
numerical scheme to generate the solution to the governing SDE using the fADE
solution. Also, often the functional form of the advection or dispersion
coefficients are not known for a given plume concentration data to start with.
We use a Levenberg--Marquardt (L-M) regularization method to estimate advection
and dispersion coefficient function from the observed data (we present the case
for a linear advection) and proceed with the SDE solution construction
described above.Comment: Published at https://doi.org/10.15559/18-VMSTA106 in the Modern
Stochastics: Theory and Applications (https://vmsta.org/) by VTeX
(http://www.vtex.lt/
The convergence analysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity
© 2018, The Author(s). In this paper, we focus on the convergence analysis and error estimation for the unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity. By introducing a double iterative technique, in the case of the nonlinearity with singularity at time and space variables, the unique positive solution to the problem is established. Then, from the developed iterative technique, the sequences converging uniformly to the unique solution are formulated, and the estimates of the error and the convergence rate are derived
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