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Error estimate of a transformed L1 scheme for a multi-term time-fractional diffusion equation by using discrete comparison principle

Author

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  • Zhou, Yongtao
  • Li, Mingzhu

Abstract

This work is concerned with the multi-term time-fractional diffusion equation ∑j=0JbjDtαju−pΔu+c(x,t)u=f, where Dtαj is the Caputo derivative with 1>α0>α1>⋯>αJ>0 and bj, p are positive constants. The solution of this problem usually has a weak singularity near the initial time. To handle such difficulty, a smoothing transformation t=s1/α0 is applied so that an equivalent re-scaled fractional differential equation is obtained. Then the equivalent equation is solved by the transformed L1 scheme of the Caputo derivative and the standard 3-point discretization of the spatial derivative on uniform meshes both in time and space direction. The α-robust error estimate with the temporal convergence order O(Nα0−2) is given by using the discrete comparison principle, which does not blow up as α→1−. Finally, numerical results are given to confirm our error analysis.

Suggested Citation

  • Zhou, Yongtao & Li, Mingzhu, 2024. "Error estimate of a transformed L1 scheme for a multi-term time-fractional diffusion equation by using discrete comparison principle," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 217(C), pages 395-404.
    • Handle: RePEc:eee:matcom:v:217:y:2024:i:c:p:395-404
    DOI: 10.1016/j.matcom.2023.11.010
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    Keywords

    Error estimate; Transformed L1 scheme; Multi-term time-fractional diffusion equation; Discrete comparison principle; α-robust;
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    JEL classification:

    • L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance

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