82
Views
2
CrossRef citations to date
0
Altmetric
Original Articles

Two efficient Galerkin finite element methods for the modified anomalous subdiffusion equation

Pages 1834-1851 | Received 04 Mar 2020, Accepted 16 Oct 2020, Published online: 25 Nov 2020

References

  • M. Al-Maskari and S. Karaa, Galerkin FEM for a time-fractional Oldroyd-B fluid problem, Adv. Comput. Math. 45 (2019), pp. 1005–1029.
  • E. Awad and R. Metzler, Crossover dynamics from superdiffusion to subdiffusion: Models and solutions, Frac. Calculus and Appl. Anal. 23 (2020), pp. 55–102.
  • E. Bazhlekova, B. Jin, R. Lazarov, and Z. Zhou, An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid, Numer. Math. 131 (2015), pp. 1–31.
  • E.B. Brown, E.S. Wu, W. Zipfel, and W.W. Webb, Measurement of molecular diffusion in solution by multiphoton fluorescence photobleaching recovery, Biophys. J. 77 (1999), pp. 2837–2849.
  • Y. Chen and C.M. Chen, Numerical simulation with the second order compact approximation of first order derivative for the modified fractional diffusion equation, Appl. Math. Comput. 320 (2018), pp. 319–330.
  • A. Chen and C. Li, A novel compact ADI scheme for the time-fractional subdiffusion equation in two space dimensions, Int. J. Comput. Math. 93 (2015), pp. 889–914.
  • A. Chen, Q. Du, C. Li, and Z. Zhou, Asymptotically compatible schemes for space-time nonlocal diffusion equations, Chaos, Solitons Fractals 102 (2017), pp. 361–371.
  • E. Cuesta, C. Lubich, and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations, Math. Comput. 75 (2006), pp. 673–697.
  • H. Ding and C. Li, High-order compact difference schemes for the modified anomalous subdiffusion equation, Numer. Methods Partial Differ. Equ. 32 (2015), pp. 213–242.
  • Q. Du, An invitation to nonlocal modeling, analysis and computation, Proc. Int. Cong. Math. – Rio De Janeiro 3 (2018), pp. 3523–3552.
  • Q. Du, J. Yang, and Z. Zhou, Analysis of a nonlocal-in-time parabolic equation, Discrete Contin. Dyn. Syst. B 22 (2017), pp. 339–368.
  • W. He, H. Song, Y. Su, L. Geng, B.J. Ackerson, H.B. Peng, and P. Tong, Dynamic heterogeneity and non-Gaussian statistics for acetylcholine receptors on live cell membrane, Nat. Commun. 7 (2016), pp. 11701.
  • W. Jiang and Z. Chen, A collocation method based on reproducing kernel for a modified anomalous subdiffusion equation, Numer. Methods Partial Differ. Equ. 30 (2014), pp. 289–300.
  • B. Jin and Z. Zhou, Multigrid methods for time-Fractional evolution equations: A numerical study, Commun. Appl. Math. Comput. 2 (2020), pp. 163–177.
  • T. Langlands, Solution of a modified fractional diffusion equation, Physica A 367 (2006), pp. 136–144.
  • C. Li and A. Chen, Numerical methods for fractional partial differential equations, Int. J. Comput. Methods Eng. Sci. Mech. 95 (2018), pp. 1048–1099.
  • C. Li, F. Zeng, Numerical Methods for Fractional Calculus, Chapman and Hall/CRC, Boca Raton, 2015.
  • C. Lubich, Convolution quadrature and discretized operational calculus. I, BIT Numer. Math. 52 (1988), pp. 129–145.
  • C. Lubich, Convolution quadrature revisited, BIT Numer. Math. 44 (2004), pp. 503–514.
  • F. Mainardi, A. Mura, G. Pagnini, and R. Gorenflo, Time-fractional diffusion of distributed order, J. Vib. Control 14 (2008), pp. 1267–1290.
  • J. Masoliver, M. Montero, and G.H. Weiss, Continuous-time random-walk model for financial distributions, Phys. Rev. E 67 (2003), pp. 021112.
  • R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339 (2000), pp. 1–77.
  • R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A: Math. General 37 (2004), pp. R161–R208.
  • R. Metzler, J.H. Jeon, A.G. Cherstvy, and E. Barkai, Anomalous diffusion models and their properties: Non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Phys. Chem. Chem. Phys. 16 (2014), pp. 24128–24164.
  • P.R. Smith, I.E.G. Morrison, K.M. Wilson, N. Fernández, and R.J. Cherry, Anomalous diffusion of major histocompatibility complex class I molecules on heLa cells determined by single particle tracking, Biophys. J. 76 (1999), pp. 3331–3344.
  • I.M. Sokolov and J. Klafter, From diffusion to anomalous diffusion: A century after Einstein's Brownian motion, Chaos 15 (2005), pp. 026103.
  • V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd ed., Springer, Berlin, 2006.
  • Z. Wang and S. Vong, Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J. Comput. Phys. 277 (2014), pp. 1–15.
  • B. Yin, Y. Liu, and H. Li, A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations, Appl. Math. Comput. 368 (2020), pp. 124799.
  • B. Yin, Y. Liu, and H. Li, Necessity of introducing non-integer shifted parameters by constructing high accuracy finite difference algorithms for a two-sided space-fractional advection–diffusion model, Appl. Math. Lett. 105 (2020), pp. 106347.
  • F. Zeng and C. Li, A new Crank–Nicolson finite element method for the time-fractional subdiffusion equation, Appl. Numer. Math. 121 (2017), pp. 82–95.
  • F. Zeng, I. Turner, K. Burrage, and G.E. Karniadakis, A new class of semi-Implicit methods with linear complexity for nonlinear fractional differential equations, SIAM J. Sci. Comput. 40 (2018), pp. A2986–A3011.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.