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Backward difference formulae and spectral Galerkin methods for the Riesz space fractional diffusion equation

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  • Xu, Yang
  • Zhang, Yanming
  • Zhao, Jingjun

Abstract

Approximating Riesz space fractional diffusion equation in time by k-step backward difference formula and in space by spectral Galerkin method, we establish a fully discrete scheme with high order both in time and in space. For k≤5, we prove the stability of full discretization and obtain the error estimate with order O(τk+Nα2−m), which depends only on the regularity of initial value and right-hand function. Moreover, we extend the proposed method to two dimensional case and derive similar results. Finally, we illustrate the theoretical estimates by numerical examples.

Suggested Citation

  • Xu, Yang & Zhang, Yanming & Zhao, Jingjun, 2019. "Backward difference formulae and spectral Galerkin methods for the Riesz space fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 494-507.
    • Handle: RePEc:eee:matcom:v:166:y:2019:i:c:p:494-507
    DOI: 10.1016/j.matcom.2019.07.007
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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. 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.
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    Cited by:

    1. Yang, Hong & Lao, Cheng-Xue & She, Zi-Hang, 2023. "Fast solution methods for Riesz space fractional diffusion equations with non-separable coefficients," Applied Mathematics and Computation, Elsevier, vol. 445(C).

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