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A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients

Author

Listed:
  • Chen, S.
  • Liu, F.
  • Jiang, X.
  • Turner, I.
  • Anh, V.

Abstract

In this paper, we derive a new nonlinear two-sided space-fractional diffusion equation with variable coefficients from the fractional Fick’s law. A semi-implicit difference method (SIDM) for this equation is proposed. The stability and convergence of the SIDM are discussed. For the implementation, we develop a fast accurate iterative method for the SIDM by decomposing the dense coefficient matrix into a combination of Toeplitz-like matrices. This fast iterative method significantly reduces the storage requirement of O(n2) and computational cost of O(n3) down to n and O(nlogn), where n is the number of grid points. The method retains the same accuracy as the underlying SIDM solved with Gaussian elimination. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

Suggested Citation

  • Chen, S. & Liu, F. & Jiang, X. & Turner, I. & Anh, V., 2015. "A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 591-601.
    • Handle: RePEc:eee:apmaco:v:257:y:2015:i:c:p:591-601
    DOI: 10.1016/j.amc.2014.08.031
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    Citations

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    Cited by:

    1. Xing, Zhiyong & Wen, Liping, 2019. "Numerical analysis and fast implementation of a fourth-order difference scheme for two-dimensional space-fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 155-166.
    2. Luo, Wei-Hua & Huang, Ting-Zhu & Wu, Guo-Cheng & Gu, Xian-Ming, 2016. "Quadratic spline collocation method for the time fractional subdiffusion equation," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 252-265.
    3. Liu, Jun & Fu, Hongfei & Chai, Xiaochao & Sun, Yanan & Guo, Hui, 2019. "Stability and convergence analysis of the quadratic spline collocation method for time-dependent fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 633-648.
    4. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.
    5. Almushaira, Mustafa, 2023. "An efficient fourth-order accurate conservative scheme for Riesz space fractional Schrödinger equation with wave operator," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 424-447.

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