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Original Articles

A novel ADI Galerkin spectral element method for the solution of two-dimensional time fractional subdiffusion equation

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Pages 845-867 | Received 30 Jul 2019, Accepted 19 Jun 2020, Published online: 19 Jul 2020

References

  • M. Abbaszadeh and M. Dehghan, A meshless numerical procedure for solving fractional reaction subdiffusion model via a new combination of alternating direction implicit (ADI) approach and interpolating element free Galerkin (EFG) method, Comput. Math. Appl. 70 (2015), pp. 2493–2512. doi: 10.1016/j.camwa.2015.09.011
  • M. Abbaszadeh and A. Mohebbi, A fourth-order compact solution of the two-dimensional modified anomalous fractional sub-diffusion equation with a nonlinear source term, Comput. Math. Appl. 66 (2013), pp. 1345–1359. doi: 10.1016/j.camwa.2013.08.010
  • C. Canuto, M.Y. Hussaini, A. Quarteroni, Spectral Methods Fundamentals in Single Domains, Springer, New york, 2006.
  • A. Chakraborty, M.S. Sivakumar, and S. Gopalakrishnan, Spectral element based model for wave propagation analysis in multi-wall carbon nanotubes, Int. J. Solids Struct. 43 (2006), pp. 279–294. doi: 10.1016/j.ijsolstr.2005.03.044
  • A. Chen and C.P. Li, A novel compact ADI scheme for the time-fractional subdiffusion equation in two space dimensions, Int. J. Comput. Math. 93 (2016), pp. 889–914. doi: 10.1080/00207160.2015.1009905
  • C.M. Chen, F. Liu, I. Turner, and V. Anh, Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation, Numer. Algor. 54 (2010), pp. 1–21. doi: 10.1007/s11075-009-9320-1
  • J. Crank, The Mathematics of Diffusion. 2nd ed.Oxford University Press, New York, 1989.
  • M. Dehghan, M. Abbaszadeh, and A. Mohebbi, Error estimate for the numerical solution of fractional reaction-subdifusion process based on a meshless method, J. Comput. Appl. Math. 280 (2015), pp. 14–36. doi: 10.1016/j.cam.2014.11.020
  • M. Dehghan, M. Abbaszadeh, and A. Mohebbi, Legendre spectral element method for solving time fractional modified anomalous sub-dusion equation, Appl. Math. Model. 40 (2016), pp. 3635–3654. doi: 10.1016/j.apm.2015.10.036
  • M.O. Deville, P.F. Fischer, and E.H. Mund, High-order Methods for Incompressible Fluid Flow, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, New York, 2002.
  • G.W. Ford, J.T. Lewis, and R.F. OConnell, Quantum Langevin equation, Phys. Rev. A. 37 (1988), pp. 4419–4428. doi: 10.1103/PhysRevA.37.4419
  • Y. Gu, P. Zhuang, and F. Liu, An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation, Comput. Model. Eng. Sci. 56 (2010), pp. 303–334.
  • J.S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral Methods for Time-dependent Problems, Cambridge University Press, Cambridge, 2007.
  • E. Kharazmi, M. Zayernouri, and G.E. Karniadakis, A Petrov-Galerkin spectral element method for fractional elliptic problems, Comput. Methods Appl. Mech. Eng. 324 (2017), pp. 512–536. doi: 10.1016/j.cma.2017.06.006
  • A. Klemm, H.P. Muller, and R. Kimmich, NMR-microscopy of pore-space backbones in rock, Phys. Rev. E. 55 (1997), pp. 4413–4422. doi: 10.1103/PhysRevE.55.4413
  • M. Li, Q. Ma, and X. Ding, A compact ADI Crank-Nicolson difference scheme for the two-dimensional time fractional subdiffusion equation, Int. J. Comput. Math. 12 (2018), pp. 2525–2538. doi: 10.1080/00207160.2017.1411590
  • C.P. Li and Z. Wang, The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: numerical analysis, Appl. Numer. Math. 140 (2019), pp. 1–22. doi: 10.1016/j.apnum.2019.01.007
  • C.P. Li and Z. Wang, The discontinuous Galerkin finite element method for Caputo-type nonlinear conservation law, Math. Comput. Simulat. 169 (2020), pp. 51–73. doi: 10.1016/j.matcom.2019.09.021
  • C.P. Li and Z. Wang, The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: mathematical analysis, Appl. Numer. Math. 150 (2020), pp. 587–606. doi: 10.1016/j.apnum.2019.11.007
  • C.P. Li and F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, New York, 2015.
  • M.M Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion ow equations, J. Comput. Appl. Math. 172 (2004), pp. 65–77. doi: 10.1016/j.cam.2004.01.033
  • R. Metzler and J. Klafter, The random walks guide to anomalous diffusion: a fractional dynamic approach, Phys. Rep. 339 (2000), pp. 1–77. doi: 10.1016/S0370-1573(00)00070-3
  • A. Mohebbi, M. Abbaszade, and M. Dehghan, A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term, J. Comput. Phys.240 (2013), pp. 36–48. doi: 10.1016/j.jcp.2012.11.052
  • O. Oru, A. Esen, and F. Bulut, A haar wavelet approximation for two-dimensional time fractional reaction-subdiffusion equation, Eng. Comput. 35 (2019), pp. 75–86. doi: 10.1007/s00366-018-0584-8
  • L.I. Palade, P. Attane, R.R. Huilgol, and B. Mena, Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models, Int. J. Eng. Sci. 37 (1999), pp. 315–329. doi: 10.1016/S0020-7225(98)00080-9
  • C. Pozrikidis, Introduction to Nite and Spectral Element Methods Using Matlab, Chapman and Hall/CRC, Boka Raton, 2005.
  • W.Y. Tian, H. Zhou, and W.H. Deng, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comput. 84 (2015), pp. 1703–1727. doi: 10.1090/S0025-5718-2015-02917-2
  • T. Wang and Y.M. Wang, A modified compact ADI method and its extrapolation for two-dimensional fractional subdiffusion equations, J. Appl. Math. Comput. 52 (2016), pp. 439–476. doi: 10.1007/s12190-015-0949-8
  • F. Zeng, H. Ma, and T. Zhao, Alternating direction implicit Legendre spectral element method for Schrodinger equations, J. Shanghai Univ. Nat. Sci. Ed. 6 (2011), pp. 006.
  • N. Zhang, W. Deng, and Y. Wu, Finite difference/element method for a two-dimensional modified fractional diffusion equation, Adv. Appl. Math. Mech. 4 (2012), pp. 496–518. doi: 10.4208/aamm.10-m1210
  • Y.N. Zhang and Z.Z. Sun, Error analysis of a compact ADI scheme for the 2D fractional subdiffusion equation, J. Sci. Comput. 59 (2014), pp. 104–128. doi: 10.1007/s10915-013-9756-2

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