A continuous variant for Grünwald–Letnikov fractional derivatives
The names of Grünwald and Letnikov are associated with discrete convolutions of mesh h, multiplied by h−α. When h tends to zero, the result tends to a Marchaud’s derivative (of the order of α) of the function to which the convolution is applied. The weights wkα of such discrete convolutions form well-defined sequences, proportional to k−α−1 near infinity, and all moments of integer order r<α are equal to zero, provided α is not an integer. We present a continuous variant of Grünwald–Letnikov formulas, with integrals instead of series. It involves a convolution kernel which mimics the above-mentioned features of Grünwald–Letnikov weights. A first application consists in computing the flux of particles spreading according to random walks with heavy-tailed jump distributions, possibly involving boundary conditions.
Year of publication: |
2008
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Authors: | Néel, Marie-Christine ; Abdennadher, Ali ; Solofoniaina, Joelson |
Published in: |
Physica A: Statistical Mechanics and its Applications. - Elsevier, ISSN 0378-4371. - Vol. 387.2008, 12, p. 2750-2760
|
Publisher: |
Elsevier |
Subject: | Transport processes | Random media | Random walks | Integro-differential equations |
Saved in:
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