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Interval Shannon Wavelet Collocation Method for Fractional Fokker‐Planck Equation

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  • Shu-Li Mei
  • De-Hai Zhu

Abstract

Metzler et al. introduced a fractional Fokker‐Planck equation (FFPE) describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a Boltzmann thermal heat bath. In this paper, we present an interval Shannon wavelet numerical method for the FFPE. In this method, a new concept named “dynamic interval wavelet” is proposed to solve the problem that the numerical solution of the fractional PDE is usually sensitive to boundary conditions. Comparing with the traditional wavelet defined in the interval, the Newton interpolator is employed instead of the Lagrange interpolation operator, so, the extrapolation points in the interval wavelet can be chosen dynamically to restrict the boundary effect without increase of the calculation amount. In order to avoid unlimited increasing of the extrapolation points, both the error tolerance and the condition number are taken as indicators for the dynamic choice of the extrapolation points. Then, combining with the finite difference technology, a new numerical method for the time fractional partial differential equation is constructed. A simple Fokker‐Planck equation is taken as an example to illustrate the effectiveness by comparing with the Grunwald‐Letnikov central difference approximation (GL‐CDA).

Suggested Citation

  • Shu-Li Mei & De-Hai Zhu, 2013. "Interval Shannon Wavelet Collocation Method for Fractional Fokker‐Planck Equation," Advances in Mathematical Physics, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlamp:v:2013:y:2013:i:1:n:821820
    DOI: 10.1155/2013/821820
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    References listed on IDEAS

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    6. Shu-Li Mei, 2013. "Construction of Target Controllable Image Segmentation Model Based on Homotopy Perturbation Technology," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    7. T. Frank, 2004. "Autocorrelation functions of nonlinear Fokker-Planck equations," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 37(2), pages 139-142, January.
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