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A continuous variant for Grünwald–Letnikov fractional derivatives

Author

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  • Néel, Marie-Christine
  • Abdennadher, Ali
  • Solofoniaina, Joelson

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.

Suggested Citation

  • Néel, Marie-Christine & Abdennadher, Ali & Solofoniaina, Joelson, 2008. "A continuous variant for Grünwald–Letnikov fractional derivatives," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(12), pages 2750-2760.
    • Handle: RePEc:eee:phsmap:v:387:y:2008:i:12:p:2750-2760
    DOI: 10.1016/j.physa.2008.01.090
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    References listed on IDEAS

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    1. Paradisi, Paolo & Cesari, Rita & Mainardi, Francesco & Tampieri, Francesco, 2001. "The fractional Fick's law for non-local transport processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 293(1), pages 130-142.
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    Cited by:

    1. Golder, J. & Joelson, M. & Néel, M.C., 2011. "Mass transport with sorption in porous media," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(10), pages 2181-2189.

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