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A characteristic difference method for the variable-order fractional advection-diffusion equation

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Abstract

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term (VOFADE-NST) on a finite domain. Combining the characteristic method and the finite difference method, a characteristic finite difference method for solving the VOFADE-NST is presented. Its stability and convergence are analyzed. This new method is shown to be more efficient and superior to the standard finite difference method. Numerical experiments are carried out and the results demonstrate the effectiveness of theoretical analysis.

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Acknowledgements

The work was supported by the National Natural Science Foundation of China grant 11001090 and 11101344, the Natural Science Foundation of Huaqiao University grant 08BS507, the Australian Research Council grant DP120103770, and the Natural Science Foundation of Fujian province grant 2010J05009.

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Authors and Affiliations

  1. School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian, China

    S. Shen

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

    F. Liu, V. Anh & I. Turner

  3. School of Sciences, Jimei University, Xiamen, Fujian, China

    J. Chen

Authors
  1. S. Shen
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  2. F. Liu
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  3. V. Anh
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  4. I. Turner
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  5. J. Chen
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Corresponding author

Correspondence to F. Liu.

Appendix

Appendix

The implicit finite difference method for the VOFADE-NST developed in [36] is given by

where \(\mu_{i}^{n}=\upsilon_{i}^{n}\tau h^{-1}\), \(\gamma _{i,n}^{(1)}=\kappa_{i}^{n}c_{+,i}^{n}\tau h^{-\alpha_{i+1}^{n}}\), \(\gamma_{i,n}^{(2)}=\kappa_{i}^{n}c_{-,i}^{n}\tau h^{-\alpha_{i-1}^{n}}\) and \(g_{i,n}^{(j)}=g_{\alpha_{i}^{n}}^{(j)}\).

This scheme was proved to be unconditionally stable and convergent, and the convergence order is O(τ+h).

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Shen, S., Liu, F., Anh, V. et al. A characteristic difference method for the variable-order fractional advection-diffusion equation. J. Appl. Math. Comput. 42, 371–386 (2013). https://doi.org/10.1007/s12190-012-0642-0

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  • DOI: https://doi.org/10.1007/s12190-012-0642-0

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