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Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives (2307.08078v2)

Published 16 Jul 2023 in math.NA and cs.NA

Abstract: In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders $\alpha_i\in(0,1)$, $i=1,2,\cdots,n$). The proposed method employs a fast finite difference scheme to approximate multi-term fractional derivatives in time, requiring only $O(1)$ storage and $O(N_T)$ computational complexity, where $N_T$ denotes the total number of time steps. Then we use a Legendre spectral collocation method for spatial discretization. The stability and convergence of the scheme have been thoroughly discussed and rigorously established. We demonstrate that the proposed scheme is unconditionally stable and convergent with an order of $O(\left(\Delta t\right){2}+N{-m})$, where $\Delta t$, $N$, and $m$ represent the timestep size, polynomial degree, and regularity in the spatial variable of the exact solution, respectively. Numerical results are presented to validate the theoretical predictions.

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References (42)
  1. Phys. A., 484 (2018), 233–252.
  2. Comput. Appl. Math., 37 (2018), 3307–3333.
  3. Adv. Differential Equations., 2017 (2017), 315.
  4. New Trends in Mathematical Sciences, 4 (2016), 79–89.
  5. A. Atangana, On the new fractional derivative and application to nonlinear Fisher’s reaction–diffusion equation, Appl. Math. Comput., 273 (2016), 948–956.
  6. Arab. J. Geosciences., 6 (2016), 1–6.
  7. C. Bernardi, Y. Maday, Approximations spectrales de problemes aux limites elliptiques, volume 142, Berlin: Springer Press, 1992.
  8. Progr. Fract. Differ. Appl., 1(2) (2015), 73–85.
  9. Progr. Fract. Differ. Appl., 2(1) (2016), 1–11.
  10. Appl. Numer. Math., 151 (2020), 246–262.
  11. Signal Process., 92(2) (2012), 553–563.
  12. Numer. Methods Partial Differential Equations, 33(5) (2017), 1616–1627.
  13. Math. Sci., 16(4) (2022), 417–430.
  14. Phys. A., 447 (2016), 467–481.
  15. J. Comput. Phys., 417 (2020), 109576.
  16. Appl. Math. Lett., 136 (2023), 108447.
  17. Commun. Comput. Phys., 21(3) (2017), 650–678.
  18. B. Jin, Fractional Differential Equations: An Approach via Fractional Derivatives, Springer Cham Press, 2021.
  19. J. Comput. Phys., 281 (2015), 825–843.
  20. Fract. Calc. Appl. Anal., 23(3) (2020), 610–634.
  21. J. Comput. Phys., 358 (2018), 256–282.
  22. SIAM J. Numer. Anal., 47(3) (2009), 2108–2131.
  23. J. Comput. Phys., 225(2) (2007), 1533–1552.
  24. ANZIAM J., 46 (2004), C488–C504.
  25. Int. J. Comput. Math., 96(7) (2019), 1444–1460.
  26. AIMS Math., 5(3) (2020), 1729–1744.
  27. Critical Reviews in Biomedical Engineering, 32(1) (2004), 104 pages.
  28. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific: Imperial College Press, 2010.
  29. Phys. Rep., 339(1) (2000), 1–77.
  30. Comput. Math. Appl., 73(1) (2017), 1–10.
  31. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier Science: Academic Press, 1998.
  32. A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations, Springer Berlin: Heidelberg Press, 2009.
  33. East Asian J. Appl. Math., 4(3) (2014), 242–266.
  34. S. Jocelyn, Fractional-order derivatives defined by continuous kernels: are they really too restrictive?, Fractal and Fractional, 4(3) (2020), 40.
  35. Filomat, 34(11) (2020), 3609–3626.
  36. J. Comput. Appl. Math., 375 (2020), 112811.
  37. arXiv, (2019), 906.00328v1.
  38. J. Sci. Comput., 77 (2018), 283–307.
  39. Comput. Math. Appl., 73(6) (2017), 1087–1099.
  40. Appl. Math. Model., 40(7-8) (2016), 4970–4985.
  41. SIAM J. Numer. Anal., 57(6) (2019), 2829–2849.
  42. Int. J. Comput. Math., (2023), in press.

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Authors (1)

  1. Bin Fan
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