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An α-robust fast algorithm for distributed-order time–space fractional diffusion equation with weakly singular solution

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  • Sun, Lu-Yao
  • Lei, Siu-Long
  • Sun, Hai-Wei
  • Zhang, Jia-Li

Abstract

A fast algorithm is proposed for solving two-dimensional distributed-order time–space fractional diffusion equation where the solution has a weak singularity at initial time. The distributed-order fractional problem is firstly transformed into multi-term fractional problem by the Gauss–Legendre quadrature formula. Then the exponential-sum-approximation method on graded mesh is utilized to discretize time Caputo fractional derivatives in time direction, and a standard finite difference method is employed to approximate the spatial Riesz fractional derivatives. The scheme is proved to be α-robust convergent analytically. The discrete linear system possesses symmetric positive definite block-Toeplitz–Toeplitz-block structure and is efficiently solved by conjugate gradient method with the state-of-the-art sine-transformed based preconditioner. Numerical examples confirm the error analysis and the effectiveness of the preconditioner.

Suggested Citation

  • Sun, Lu-Yao & Lei, Siu-Long & Sun, Hai-Wei & Zhang, Jia-Li, 2023. "An α-robust fast algorithm for distributed-order time–space fractional diffusion equation with weakly singular solution," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 437-452.
    • Handle: RePEc:eee:matcom:v:207:y:2023:i:c:p:437-452
    DOI: 10.1016/j.matcom.2023.01.011
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    1. Sun, Lu-Yao & Fang, Zhi-Wei & Lei, Siu-Long & Sun, Hai-Wei & Zhang, Jia-Li, 2022. "A fast algorithm for two-dimensional distributed-order time-space fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 425(C).
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