A continuous variant for Grünwald–Letnikov fractional derivatives

Marie-Christine Néel, Ali Abdennadher and Joelson Solofoniaina

Physica A: Statistical Mechanics and its Applications, 2008, vol. 387, issue 12, 2750-2760

Abstract: 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.

Keywords: Transport processes; Random media; Random walks; Integro-differential equations (search for similar items in EconPapers)
Date: 2008
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:387:y:2008:i:12:p:2750-2760

DOI: 10.1016/j.physa.2008.01.090

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