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Generalized fractional diffusion equation with arbitrary time varying diffusivity

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  • Tawfik, Ashraf M.
  • Abdelhamid, Hamdi M.

Abstract

Anomalous diffusion processes in many complex systems are frequently described by the diffusion equation with a time-dependent diffusion coefficient. This paper introduces an exact solution to the broad classes of the fractional diffusion equation with the arbitrary time-dependent diffusion coefficient by using the Laplace-Fourier technique. The Riesz fractional derivative serves to replace the Laplacian operator, while the new regularized Caputo-type fractional derivative is employed instead of the time derivative. We examine our results by introducing and analyzing the most three common cases that represent diffusivity that varying with time. Our calculation shows exact matching with the probability distribution function and mean square displacement illustrated in the literature.

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  • Tawfik, Ashraf M. & Abdelhamid, Hamdi M., 2021. "Generalized fractional diffusion equation with arbitrary time varying diffusivity," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    • Handle: RePEc:eee:apmaco:v:410:y:2021:i:c:s0096300321005385
    DOI: 10.1016/j.amc.2021.126449
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    References listed on IDEAS

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    1. Dassios, Ioannis K. & Baleanu, Dumitru I., 2018. "Caputo and related fractional derivatives in singular systems," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 591-606.
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    3. Qi, Haitao & Jiang, Xiaoyun, 2011. "Solutions of the space-time fractional Cattaneo diffusion equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(11), pages 1876-1883.
    4. Jaume Masoliver & Katja Lindenberg, 2017. "Continuous time persistent random walk: a review and some generalizations," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 90(6), pages 1-13, June.
    5. Tomovski, Živorad & Sandev, Trifce & Metzler, Ralf & Dubbeldam, Johan, 2012. "Generalized space–time fractional diffusion equation with composite fractional time derivative," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(8), pages 2527-2542.
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    1. dos Santos, M.A.F. & Menon, L. & Cius, D., 2022. "Superstatistical approach of the anomalous exponent for scaled Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).

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