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Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations: A Second-Order Scheme

  • Yonggui Yan (a1), Zhi-Zhong Sun (a2) and Jiwei Zhang (a1)
Abstract
Abstract

The fractional derivatives include nonlocal information and thus their calculation requires huge storage and computational cost for long time simulations. We present an efficient and high-order accurate numerical formula to speed up the evaluation of the Caputo fractional derivative based on the L2-1 σ formula proposed in [A. Alikhanov, J. Comput. Phys., 280 (2015), pp. 424-438], and employing the sum-of-exponentials approximation to the kernel function appeared in the Caputo fractional derivative. Both theoretically and numerically, we prove that while applied to solving time fractional diffusion equations, our scheme not only has unconditional stability and high accuracy but also reduces the storage and computational cost.

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Corresponding author
*Corresponding author. Email addresses: yan_yonggui@csrc.ac.cn (Y. Yan), zzsun@seu.edu.cn (Z.-Z. Sun), jwzhang@csrc.ac.cn (J. Zhang)
References
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[1] Oldham K. and Spanier J., The fractional calculus, Academic Press, San Diego, (1974).
[2] Podlubny I., Fractional differential equations, Academic Press, New York, (1999).
[3] Hilfer R., Applications of fractional calculus in physics, World Scientific, (2000).
[4] Metzler R. and Klafter J., The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 177.
[5] Langlands T.A.M. and Henry B.I., Fractional chemotaxis diffusion equations. Phys. Rev. E, 81 (2010), 051102.
[6] Olver F.W.. NIST Handbook of Mathematical Functions Hardback and CD-ROM. Cambridge University Press, (2010).
[7] Shih M., Momoniat E., and Mahomed F.M., Approximate conditional symmetries and approximate solutions of the perturbed fitzhugh–nagumo equation, J. Math. Phys., 46 (2005), 023503.
[8] Wang H. and Basu T.S., A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 34 (2012), A2444A2458.
[9] Barkai E., Fractional Fokker-Planck equation, solution, and application, Phys. Rev. E, 63 (2001), 046118.
[10] Eidelman S.D. and Kochubei A.N., Cauchy problem for fractional diffusion equations. J. Differ. Equations, 199 (2004), 211255.
[11] Gorenflo R., Iskenderov A., and Luchko Y., Mapping between solutions of fractional diffusion-wave equations, Fract. Calc. Appl. Anal., 3 (2000), 7586.
[12] Kilbas A., Trujillo J., and Voroshilov A., Cauchy-type problem for diffusion-wave equation with the Riemann-Liouville partial derivative, Fract. Calc. Appl. Anal., 8 (2005), 403430.
[13] Mainardi F., Pagnini G., and Luchko Y., The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., (2007), 153192.
[14] Metzler R. and Nonnenmacher T.F., Space-and time-fractional diffusion andwave equations, fractional Fokker-Planck equations, and physical motivation, Chem. Phys., 284 (2002), 6790.
[15] Wyss W., The fractional diffusion equation, J. Math. Phys., 27 (1986), 27822785.
[16] Cao J. and Xu C., A high order schema for the numerical solution of the fractional ordinary differential equations, J. Comput. Phys., 238 (2013), 154168.
[17] Li C., Chen A., and Ye J., Numerical approaches to fractional calculus and fractional ordinary differential equation, J. Comput. Phys., 230 (2011), 33523368.
[18] Gao G.H., Sun Z.Z., and Zhang H.W., A new fractional numerical differentiation formula to approximate the caputo fractional derivative and its applications, J. Comput. Phys., 259 (2014), 3350.
[19] Alikhanov A.A., A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280 (2015), 424438.
[20] Basu T.S. and Wang H., A fast second-order finite difference method for space-fractional diffusion equations, Int. J. Nume. Anal. Mod., 9 (2012), 658666.
[21] Pang H.K. and Sun H.W., Multigrid method for fractional diffusion equations, J. Comput. Phys., 231 (2012), 693703.
[22] Lei S.L. and Sun H.W., A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242 (2013), 715725.
[23] Lubich C. and Schädle A., Fast convolution for nonreflecting boundary conditions, SIAM J. Sci. Comput., 24 (2002), 161182.
[24] Jiang S., Zhang J., Zhang Q., and Zhang Z., Fast evaluation of the caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21(3) (2017), 650678.
[25] Alpert B., Greengard L., and Hagstrom T., Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation, SIAM J. Num. Anal., 37 (2000), 11381164.
[26] Greengard L. and Lin P., Spectral approximation of the free-space heat kernel, Appl. Comput. Harm. Anal., 9 (2000), 8397.
[27] Beylkin G. and Monzón L., Approximation by exponential sums revisited, Appl. Comput. Harm. Anal., 28 (2010), 131149.
[28] Jiang S. and Greengard L., Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension, Comput. Math. Appl., 47 (2004), 955966.
[29] Li J.R., A fast time stepping method for evaluating fractional integrals, SIAM J. Sci. Comput., 31 (2010), 46964714.
[30] Li D., Liao H., Sun W., Wang J., Zhang J., Analysis of 1-Galerkin FEMs for time-fractional nonlinear parabolic problems. Accepted by Commun. Comput. Phys., 2017.
[31] Liao H., Li D., Zhang J., Zhao Y., Sharp error estimate of nonuniform 1 formula for time-fractional reaction-subdiffusion equations. Submited.
[32] Zhang Q., Zhang J., Jiang S. and Zhang Z., Numerical solution to a linearized time fractional KdV equation on unbounded domains, Accepted by Math. Comput. 2017.
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
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