A Second Order Finite Difference Approximation for the Fractional Diffusion Equation
TL;DR: In this paper, it was shown that the finite difference approximation obtained from the Grünwald-Letnikov formulation, often claimed to be of first order accuracy, is in fact a second order approximation of the fractional derivative at a point away from the grid points.
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Abstract: Abstract—We consider an approximation of one-dimensional fractional diffusion equation. We claim and show that the finite difference approximation obtained from the Grünwald-Letnikov formulation, often claimed to be of first order accuracy, is in fact a second order approximation of the fractional derivative at a point away from the grid points. We use this fact to device a second order accurate finite difference approximation for the fractional diffusion equation. The proposed method is also shown to be unconditionally stable. By this approach, we treat three cases of difference approximations in a unified setting. The results obtained are justified by numerical examples.
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Citations
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TL;DR: In this paper, the authors developed practical numerical methods to solve one dimensional fractional advection-dispersion equations with variable coefficients on a finite domain and demonstrated the practical application of these results is illustrated by modeling a radial flow problem.
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TL;DR: In this article, a transport equation that uses fractional-order dispersion derivatives has fundamental solutions that are Le´vy's a-stable densities, which represent plumesthat spread proportional to time 1/a, have heavy tails, and incorporate any degree of skewness.
Discretized fractional calculus
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