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An efficient numerical method for the distributed order time-fractional diffusion equation with error analysis and stability

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  • Derakhshan, Mohammad Hossein
  • Rezaei, Hamid
  • Marasi, Hamid Reza

Abstract

This article proposes a numerical method to find the numerical solutions of the time-fractional diffusion equations involving fractional distributed order operator of Caputo type. Using the finite difference approach, we solve these equations by applying the semi-discrete method regarding the time variable and the fully-discrete method regarding the spatial variable. For the distributed integral part with respect to time, the Gauss–Legendre quadrature formula is applied and to estimate the multi-term time-fractional operator, including the Caputo fractional derivative, the L2 -1 approach is utilized. In addition, the error analysis and stability of the proposed numerical method are studied in this work. Finally, some numerical examples are provided to demonstrate the accuracy and efficiency of the suggested method. These examples are compared to several numerical previous methods stated in the articles, and the results show that the accuracy of our method is superior to these methods.

Suggested Citation

  • Derakhshan, Mohammad Hossein & Rezaei, Hamid & Marasi, Hamid Reza, 2023. "An efficient numerical method for the distributed order time-fractional diffusion equation with error analysis and stability," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 214(C), pages 315-333.
    • Handle: RePEc:eee:matcom:v:214:y:2023:i:c:p:315-333
    DOI: 10.1016/j.matcom.2023.07.017
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    References listed on IDEAS

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    1. González-Calderón, Alfredo & Vivas-Cruz, Luis X. & Taneco-Hernández, M.A. & Gómez-Aguilar, J.F., 2023. "Assessment of the performance of the hyperbolic-NILT method to solve fractional differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 375-390.
    2. Li, Yuanlu & Liu, Fawang & Turner, Ian W. & Li, Tao, 2018. "Time-fractional diffusion equation for signal smoothing," Applied Mathematics and Computation, Elsevier, vol. 326(C), pages 108-116.
    3. Pourbabaee, Marzieh & Saadatmandi, Abbas, 2019. "A novel Legendre operational matrix for distributed order fractional differential equations," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 215-231.
    4. Ansari, Alireza & Derakhshan, Mohammad Hossein, 2023. "On spectral polar fractional Laplacian," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 636-663.
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