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The fundamental solution of the space-time fractional advection-dispersion equation

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Abstract

A space-time fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the first-order time derivative is replaced with Caputo derivative of order α ∈ (0, 1], and the second-order space derivative is replaced with a Riesz-Feller derivative of order β ∈ (0, 2]. We derive the solution of its Cauchy problem in terms of the Green functions and the representations of the Green function by applying its Fourier-Laplace transforms. The Green function also can be interpreted as a spatial probability density function (pdf) evolving in time. We do the same on another kind of space-time fractional advection-dispersion equation whose space and time derivatives both replacing with Caputo derivatives.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Xiamen University, 361005, Xiamen, China

    F. Huang & F. Liu

  2. School of Mathematical Sciences, Queensland University of Technology, 4001, Qld., Australia

    F. Huang

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

    F. Liu

Authors
  1. F. Huang
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  2. F. Liu
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Correspondence to F. Liu.

Additional information

Fawang Liu received his MSc from Fuzhou University in 1982 and PhD from Trinity College, Dublin, in 1991. Since graduation, he has been working in computational and applied mathematics at Fuzhou University, Trinity College Dublin and University College Dublin, University of Queensland, Queensland University of Technology and Xiamen University. Now he is a Professor at Xiamen University. His research interest is numerical analysis and techniques for solving a wide variety of problems in applicable mathematics, including semiconductor device equations, microwave heating problems, gas-solid reactions, singular perturbation problem, saltwater intrusion into aquifer systems and fractional differential equations.

Fenghui Huang received her MSc and PhD from Xiamen University, Xiamen, China, in 2001 and 2004. Now She is a lecturer at South China University of Technology. Her research interest is numerical computation for PDE, especially, solving the variety of problems in the Computational Fluid Dynamics, such as incompressible fluid flow and turbulence. She also pay respect to some applied problems in fractional differential equations.

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Huang, F., Liu, F. The fundamental solution of the space-time fractional advection-dispersion equation. JAMC 18, 339–350 (2005). https://doi.org/10.1007/BF02936577

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  • DOI: https://doi.org/10.1007/BF02936577

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