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Stability and convergence analysis of the quadratic spline collocation method for time-dependent fractional diffusion equations

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  • Liu, Jun
  • Fu, Hongfei
  • Chai, Xiaochao
  • Sun, Yanan
  • Guo, Hui

Abstract

A quadratic spline collocation method combined with the Crank–Nicolson time discretization is proposed for time-dependent two-sided fractional diffusion equations. By carefully analyzing the mathematical properties of the coefficient matrix, the new scheme is proved to be unconditionally stable in the sense of discrete L2-norm for α ∈ [α*, 2), where α is the order of the space-fractional derivative of the fractional diffusion equation, and α* ≈ 1.2576 (see Lemma 3.1). Furthermore, the fractional-order spline interpolation error over the collocation points is studied, and subsequently we show that the spline collocation solution of the fractional diffusion equation converges to the exact one with order O(h3−α+τ2) under the discrete L2-norm, where τ and h are the temporal and spatial step sizes, respectively. Finally, numerical experiments are given to verify the theoretical results.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:346:y:2019:i:c:p:633-648
    DOI: 10.1016/j.amc.2018.10.046
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    References listed on IDEAS

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    1. Scalas, Enrico & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
    2. Sayevand, K. & Arjang, F., 2016. "Finite volume element method and its stability analysis for analyzing the behavior of sub-diffusion problems," Applied Mathematics and Computation, Elsevier, vol. 290(C), pages 224-239.
    3. 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.
    4. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
    5. Feng, L.B. & Zhuang, P. & Liu, F. & Turner, I., 2015. "Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 52-65.
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    Cited by:

    1. Liu, Jun & Fu, Hongfei & Zhang, Jiansong, 2020. "A QSC method for fractional subdiffusion equations with fractional boundary conditions and its application in parameters identification," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 174(C), pages 153-174.

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