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October  2016, 21(8): 2509-2530. doi: 10.3934/dcdsb.2016058

Random walk's models for fractional diffusion equation

1. 

Laboratoire d'Ingénierie Mathématique, Université de Carthage, Ecole Polytechnique de Tunisie, BP 743, 2078 La Marsa, Tunisia

2. 

Laboratoire d'Ingénierie Mathématique, Université de Carthage, Ecole Polytechnique de Tunisie-Institut National des Sciences Appliquées et de Technologie, Centre Urbain Nord, BP 676 Cedex 1080 Charguia Tunis, Tunisia

Received  April 2015 Revised  May 2016 Published  September 2016

Fractional diffusion equations are used for mass spreading in inhomogeneous media. They are applied to model anomalous diffusion, where a cloud of particles spreads in a different manner than the classical diffusion equation predicts. Thus, they involve fractional derivatives. Here we present a continuous variant of Grünwald-Letnikov's formula, which is useful to compute the flux of particles performing random walks, allowing for heavy-tailed jump distributions. In fact, we set a definition of fractional derivatives yielding the operators which enable us to retrieve the space fractional variant of Fick's law, for enhanced diffusion in disordered media, without passing through any partial differential equation for the space and time evolution of the concentration.
Citation: Wafa Hamrouni, Ali Abdennadher. Random walk's models for fractional diffusion equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2509-2530. doi: 10.3934/dcdsb.2016058
References:
[1]

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II. Second edition John Wiley and sons,Inc., New York-London-Sydney, 1971.  Google Scholar

[2]

R. Gorenflo and F. Mainardi, Random Walk models for space-fractional diffusion processes, Fractional Calculus and Applied Analysis, 1 (1998), 167-191.  Google Scholar

[3]

N. Heymans, Fractional calculus description of non-linear viscoelastic behaviour of polymers, in Non-linear Dynamics, 38 (2004), 221-231. doi: 10.1007/s11071-004-3757-5.  Google Scholar

[4]

N. Krepysheva, Transport anormal de traceurs passifs en milieux poreux hétérogènes: équations fractionnaires, simulation numérique et conditions aux limites, Ph.D thesis, Université D'Avignon et des Pays de Vaucluse, 2005. Google Scholar

[5]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies 204, Ed. Jan van Mill, Amsterdam, 2006.  Google Scholar

[6]

F. Mainardi, Fractional calculus: Some basic problems in countinum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien and New York, 378 (1997), 291-348, arXiv:1201.0863. doi: 10.1007/978-3-7091-2664-6_7.  Google Scholar

[7]

R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, in J. Phys. A, 37 (2004), 161-208. doi: 10.1088/0305-4470/37/31/R01.  Google Scholar

[8]

M. C. Néel, A. Abdennadher and M. joelson, Fractional Fick's law: The direct way, in Journal of Physics A: Mathematical and Theoretical, 40 (2007), 8299-8314.  Google Scholar

[9]

B. Rubin, Fractional Integrals and Potentials, Longman Green, Harlow, 1996.  Google Scholar

[10]

S. G. Samko, Hypersingular integrals and differences of fractional order, in Trudy Mat. Inst. Steklov, 192 (1990), 164-182.  Google Scholar

[11]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gorden and Breach, New York, 1993.  Google Scholar

show all references

References:
[1]

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II. Second edition John Wiley and sons,Inc., New York-London-Sydney, 1971.  Google Scholar

[2]

R. Gorenflo and F. Mainardi, Random Walk models for space-fractional diffusion processes, Fractional Calculus and Applied Analysis, 1 (1998), 167-191.  Google Scholar

[3]

N. Heymans, Fractional calculus description of non-linear viscoelastic behaviour of polymers, in Non-linear Dynamics, 38 (2004), 221-231. doi: 10.1007/s11071-004-3757-5.  Google Scholar

[4]

N. Krepysheva, Transport anormal de traceurs passifs en milieux poreux hétérogènes: équations fractionnaires, simulation numérique et conditions aux limites, Ph.D thesis, Université D'Avignon et des Pays de Vaucluse, 2005. Google Scholar

[5]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies 204, Ed. Jan van Mill, Amsterdam, 2006.  Google Scholar

[6]

F. Mainardi, Fractional calculus: Some basic problems in countinum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien and New York, 378 (1997), 291-348, arXiv:1201.0863. doi: 10.1007/978-3-7091-2664-6_7.  Google Scholar

[7]

R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, in J. Phys. A, 37 (2004), 161-208. doi: 10.1088/0305-4470/37/31/R01.  Google Scholar

[8]

M. C. Néel, A. Abdennadher and M. joelson, Fractional Fick's law: The direct way, in Journal of Physics A: Mathematical and Theoretical, 40 (2007), 8299-8314.  Google Scholar

[9]

B. Rubin, Fractional Integrals and Potentials, Longman Green, Harlow, 1996.  Google Scholar

[10]

S. G. Samko, Hypersingular integrals and differences of fractional order, in Trudy Mat. Inst. Steklov, 192 (1990), 164-182.  Google Scholar

[11]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gorden and Breach, New York, 1993.  Google Scholar

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