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Generalized space–time fractional diffusion equation with composite fractional time derivative

Author

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  • Tomovski, Živorad
  • Sandev, Trifce
  • Metzler, Ralf
  • Dubbeldam, Johan

Abstract

We investigate the solution of space–time fractional diffusion equations with a generalized Riemann–Liouville time fractional derivative and Riesz–Feller space fractional derivative. The Laplace and Fourier transform methods are applied to solve the proposed fractional diffusion equation. The results are represented by using the Mittag-Leffler functions and the Fox H-function. Special cases of the initial and boundary conditions are considered. Numerical scheme and Grünwald–Letnikov approximation are also used to solve the space–time fractional diffusion equation. The fractional moments of the fundamental solution of the considered space–time fractional diffusion equation are obtained. Many known results are special cases of those obtained in this paper. We investigate also the solution of a space–time fractional diffusion equations with a singular term of the form δ(x)⋅t−βΓ(1−β)(β>0).

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:391:y:2012:i:8:p:2527-2542
    DOI: 10.1016/j.physa.2011.12.035
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    Citations

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    Cited by:

    1. Ram K. Saxena & Zivorad Tomovski & Trifce Sandev, 2015. "Analytical Solution of Generalized Space-Time Fractional Cable Equation," Mathematics, MDPI, vol. 3(2), pages 1-18, April.
    2. Tawfik, Ashraf M. & Abdelhamid, Hamdi M., 2021. "Generalized fractional diffusion equation with arbitrary time varying diffusivity," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    3. Emilia Bazhlekova & Ivan Bazhlekov, 2019. "Subordination Approach to Space-Time Fractional Diffusion," Mathematics, MDPI, vol. 7(5), pages 1-12, May.
    4. Maike A. F. dos Santos, 2019. "Mittag–Leffler Memory Kernel in Lévy Flights," Mathematics, MDPI, vol. 7(9), pages 1-13, August.
    5. 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.
    6. Sun, HongGuang & Hao, Xiaoxiao & Zhang, Yong & Baleanu, Dumitru, 2017. "Relaxation and diffusion models with non-singular kernels," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 590-596.

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