Abstract
In this paper, we consider the finite element method for time fractional partial differential equations. The existence and uniqueness of the solutions are proved by using the Lax-Milgram Lemma. A time stepping method is introduced based on a quadrature formula approach. The fully discrete scheme is considered by using a finite element method and optimal convergence error estimates are obtained. The numerical examples at the end of the paper show that the experimental results are consistent with our theoretical results.
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References
K. Adolfsson, M. Enelund, and S. Larsson, Adaptive discretization of integro-differential equation with a weakly singular convolution kernel. Comput. Methods Appl. Mech. Engrg. 192 (2003), 5285–5304.
K. Adolfsson, M. Enelund, and S. Larsson, Adaptive discretization of fractional order viscoelasticity using sparse time history. Comput. Methods Appl. Mech. Engrg. 193 (2004), 4567–4590.
F. Amblard, A. C. Maggs, B. Yurke, A. N. Pargellis, and S. Leibler, Subdiffusion and anomalous local viscoelasticity in actin networks. Phys. Rev. Lett. 77 (1996), 4470.
E. Barkai, R. Metzler, and J. Klafter, Form continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 61 (2000), 132–138.
M. Caputo, The Green function of the diffusion of fluids in porous media with memory. Rend. Fis. Acc. Lincei (Ser. 9) 7 (1996), 243–250.
K. Diethelm, Generalized compound quadrature formulae for finite-part integrals. IMA J. Numer. Anal. 17 (1997), 479–493.
K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order. Electronic Trans. on Numerical Analysis 5 (1997), 1–6.
K. Diethelm, N.J. Ford, A.D. Freed, Yu. Luchko, Algorithms for the fractional calculus: A selection of numerical methods. Computer Methods in Appl. Mechanics and Engineering 194 (2005), 743–773.
N.J. Ford and A.C. Simpson, The numerical solution of fractional differential equations: speed versus accuracy. Numer. Algorithms 26 (2001), 333–346.
V.J. Ervin and J.P. Roop, Variational formulation for the stationary fractional advection dispersion equation. Numerical Meth. P.D.E. 22 (2006), 558–576.
V.J. Ervin and J.P. Roop, Variational solution of fractional advection dispersion equations on bounded domain in R d. Numerical. Meth. P.D.E. 23 (2007), 256–281.
N.J. Ford and J.A. Connolly, Comparison of numerical methods for fractional differential equations. Comm. Pure Appl. Anal. 5 (2006), 289–307.
N.J. Ford and J.A. Connolly, Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations. J. Comput. Appl. Math. 229 (2009), 382–391.
B. Henry and S. Wearne, Fractional reaction-diffusion. Physica A 276 (2000), 448–455.
B. Henry and S. Wearne, Existence of turring instabilities in a twospecies fractional reaction-diffusion system. SIAM J. Appl. Math. 62 (2002), 870–887.
C. Li, Z. Zhao, and Y. Chen, Numerical approximation and error estimates of a time fractional order diffusion equation. In: Proc. of the ASME 2009 Intern. Design Engineering Technical Conf. and Computers and Information in Engineering Conf. IDETC/CIE 2009, San Diego, California, USA.
Y. Jiang and J. Ma, High-order finite element methods for timefractional partial differential equations. J. Comput. Appl. Math. 235 (2011), 3285–3290.
T.A.M. Langlands and B.I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205 (2005), 719–736.
X.J. Li and C.J. Xu, Existence and uniqeness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8 (2010), 1016–1051.
Y.M. Lin and C.J. Xu, Finite difference/spectral approximation for the time fractional diffusion equations. J. Comput. Phys. 2 (2007), 1533–1552.
J.L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Volume 1. Springer-Verlag, 1972.
F. Liu, S. Shen, V. Anh and I. Turner, Analysis of a discrete nonmarkovian random walk approximation for the time fractional diffusion equation. ANZIAMJ 46 (2005), 488–504.
F. Liu, V. Anh, I. Turner, and P. Zhuang, Time fractional advection dispersion equation. J. Comput. Appl. Math. 13 (2003), 233–245.
H.P. Müller, R. Kimmich, and J. Weis, NMR flow velocity mapping in random percolation model objects: Evidence for a power-law dependence of the volume-averaged velocity on the probe-volume radius. Phys. Rev. E 54 (1996), 5278–5285.
R.R. Nigmatullin, Realization of the generalized transfer equation in a medium with fractal geometry. Physica B 133 (1986), 425–430.
I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, 1999.
J.P. Roop, Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domain in R 2. J. Comp. Appl. Math. 193 (2006), 243–268.
H. Scher and M. Lax, Stochastic transport in a disordered solid. Phys. Rev. B 7 (1973), 4491–4502.
H. Scher and E. Montroll, Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12 (1975), 2455–2477.
W.R. Schneider and W. Wyss, Fractional diffusion and wave equations. J. Math. Phys. 30 (1989), 134–144.
Z.Z. Sun and X.N. Wu, A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56 (2006), 193–209.
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin, 2007.
W. Wyss, The fractional diffusion equation. J. Math. Phys. 27 (1989), 2782–2785.
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Ford, N.J., Xiao, J. & Yan, Y. A finite element method for time fractional partial differential equations. fcaa 14, 454–474 (2011). https://doi.org/10.2478/s13540-011-0028-2
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DOI: https://doi.org/10.2478/s13540-011-0028-2