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Fractional diffusion equation for transport phenomena in random media

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  • Giona, Massimiliano
  • Eduardo Roman, H.

Abstract

A differential equation for diffusion in isotropic and homogeneous fractal structures is derived within the context of fractional calculus. It generalizes the fractional diffusion equation valid in Euclidean systems. The asymptotic behavior of the probability density function is obtained exactly and coincides with the accepted asymptotic form obtained using scaling argument and exact enumeration calculations on large percolation clusters at criticality. The asymptotic frequency dependence of the scattering function is derived exactly from the present approach, which can be studied by X-ray and neutron scattering experiments on fractals.

Suggested Citation

  • Giona, Massimiliano & Eduardo Roman, H., 1992. "Fractional diffusion equation for transport phenomena in random media," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 185(1), pages 87-97.
    • Handle: RePEc:eee:phsmap:v:185:y:1992:i:1:p:87-97
    DOI: 10.1016/0378-4371(92)90441-R
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    References listed on IDEAS

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    1. Joseph Aharony & Sasson Bar†Yosef, 1987. "Tests of the impact of LIFO adoption on stockholders: A stochastic dominance approach," Contemporary Accounting Research, John Wiley & Sons, vol. 3(2), pages 430-444, March.
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    Cited by:

    1. Lina Song, 2018. "A Semianalytical Solution of the Fractional Derivative Model and Its Application in Financial Market," Complexity, Hindawi, vol. 2018, pages 1-10, April.
    2. Duan, Jun-Sheng & Wang, Zhong & Liu, Yu-Lu & Qiu, Xiang, 2013. "Eigenvalue problems for fractional ordinary differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 46(C), pages 46-53.
    3. M. Rezaei & A. R. Yazdanian & A. Ashrafi & S. M. Mahmoudi, 2022. "Numerically Pricing Nonlinear Time-Fractional Black–Scholes Equation with Time-Dependent Parameters Under Transaction Costs," Computational Economics, Springer;Society for Computational Economics, vol. 60(1), pages 243-280, June.
    4. Vyacheslav Svetukhin, 2021. "Nucleation Controlled by Non-Fickian Fractional Diffusion," Mathematics, MDPI, vol. 9(7), pages 1-11, March.
    5. Osman, S.A. & Langlands, T.A.M., 2019. "An implicit Keller Box numerical scheme for the solution of fractional subdiffusion equations," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 609-626.
    6. Gorenflo, Rudolf & Mainardi, Francesco & Moretti, Daniele & Pagnini, Gianni & Paradisi, Paolo, 2002. "Fractional diffusion: probability distributions and random walk models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 305(1), pages 106-112.
    7. Liang, Jin-Rong & Ren, Fu-Yao, 1998. "Hausdorff dimensions of random net fractals," Stochastic Processes and their Applications, Elsevier, vol. 74(2), pages 235-250, June.
    8. Abdelkawy, M.A. & Alyami, S.A., 2021. "Legendre-Chebyshev spectral collocation method for two-dimensional nonlinear reaction-diffusion equation with Riesz space-fractional," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).

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