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Fundamental solution and discrete random walk model for a time-space fractional diffusion equation of distributed order

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Abstract

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β 1∈(0,1) and β 2∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361006, China

    S. Shen

  2. School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, QLD 4001, Australia

    F. Liu & V. Anh

  3. School of Mathematical Sciences, South China University of Technology, Guangzhou, 510640, China

    F. Liu

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  1. S. Shen
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  2. F. Liu
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  3. V. Anh
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Correspondence to F. Liu.

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Shen, S., Liu, F. & Anh, V. Fundamental solution and discrete random walk model for a time-space fractional diffusion equation of distributed order. J. Appl. Math. Comput. 28, 147–164 (2008). https://doi.org/10.1007/s12190-008-0084-x

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  • DOI: https://doi.org/10.1007/s12190-008-0084-x

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