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Analytical solution of the space–time fractional hyperdiffusion equation

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  • Tawfik, Ashraf M.
  • Fichtner, Horst
  • Elhanbaly, A.
  • Schlickeiser, Reinhard

Abstract

The so-called fractional hyperdiffusion equation is presented to develop a fractional derivative model of the transport of energetic particles. The fractional hyperdiffusion equation is defined in terms of Caputo and Riesz fractional derivatives for time and space, respectively. The solution is obtained by using the Laplace–Fourier transforms and given in terms of the M-Wright and Fox’s H functions. Profiles of particle densities are illustrated for different values of space-fractional order.

Suggested Citation

  • Tawfik, Ashraf M. & Fichtner, Horst & Elhanbaly, A. & Schlickeiser, Reinhard, 2018. "Analytical solution of the space–time fractional hyperdiffusion equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 510(C), pages 178-187.
    • Handle: RePEc:eee:phsmap:v:510:y:2018:i:c:p:178-187
    DOI: 10.1016/j.physa.2018.07.002
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    References listed on IDEAS

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    1. Pagnini, Gianni, 2014. "Short note on the emergence of fractional kinetics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 409(C), pages 29-34.
    2. Rocca, M.C. & Plastino, A.R. & Plastino, A. & Ferri, G.L. & de Paoli, A., 2016. "General solution of a fractional diffusion–advection equation for solar cosmic-ray transport," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 447(C), pages 402-410.
    3. 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.
    4. Sun, HongGuang & Chen, Wen & Li, Changpin & Chen, YangQuan, 2010. "Fractional differential models for anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(14), pages 2719-2724.
    5. 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.
    6. H. J. Haubold & A. M. Mathai & R. K. Saxena, 2011. "Mittag-Leffler Functions and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-51, May.
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