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Networks and Heterogeneous Media

2023, Volume 18, Issue 1: 291-309. doi: 10.3934/nhm.2023011
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Effective difference methods for solving the variable coefficient fourth-order fractional sub-diffusion equations

  • 1.
    School of Mathematical Sciences and V.C. and V.R. Key Lab, Sichuan Normal University, Chengdu 610068, China
  • 2.
    School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China
  • 3.
    School of Mathematics, Aba Teachers University, Aba 623002, China
  • Received: 21 September 2022 Revised: 13 November 2022 Accepted: 05 December 2022 Published: 21 December 2022

  • This paper is concerned with the numerical approximations for the variable coefficient fourth-order fractional sub-diffusion equations subject to the second Dirichlet boundary conditions. We construct two effective difference schemes with second order accuracy in time by applying the second order approximation to the time Caputo derivative and the sum-of-exponentials approximation. By combining the discrete energy method and the mathematical induction method, the proposed methods proved to be unconditional stable and convergent. In order to overcome the possible singularity of the solution near the initial stage, a difference scheme based on non-uniform mesh is also given. Some numerical experiments are carried out to support our theoretical results. The results indicate that the our two main schemes has the almost same accuracy and the fast scheme can reduce the storage and computational cost significantly.

    Citation: Zhe Pu, Maohua Ran, Hong Luo. Effective difference methods for solving the variable coefficient fourth-order fractional sub-diffusion equations[J]. Networks and Heterogeneous Media, 2023, 18(1): 291-309. doi: 10.3934/nhm.2023011

    Related Papers:

  • This paper is concerned with the numerical approximations for the variable coefficient fourth-order fractional sub-diffusion equations subject to the second Dirichlet boundary conditions. We construct two effective difference schemes with second order accuracy in time by applying the second order approximation to the time Caputo derivative and the sum-of-exponentials approximation. By combining the discrete energy method and the mathematical induction method, the proposed methods proved to be unconditional stable and convergent. In order to overcome the possible singularity of the solution near the initial stage, a difference scheme based on non-uniform mesh is also given. Some numerical experiments are carried out to support our theoretical results. The results indicate that the our two main schemes has the almost same accuracy and the fast scheme can reduce the storage and computational cost significantly.


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    [1] V. Srivastava, K. N. Rai, A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues, Math. Comput. Model., 51 (2010), 616–624. https://doi.org/10.1016/j.mcm.2009.11.002 doi: 10.1016/j.mcm.2009.11.002
    [2] Y. Z. Povstenko, Fractional Cattaneo-type equations and generalized thermoelasticity, J. Therm. Stresses, 34 (2011), 97–114. https://doi.org/10.1080/01495739.2010.511931 doi: 10.1080/01495739.2010.511931
    [3] F. Mainardi, M. Reberto, R. Gorenflo, E. Scalas, Fractional caculus and continuous-time finance II: the waiting-time distribution, Physica A, 287 (2000), 468–481. https://doi.org/10.1016/S0378-4371(00)00386-1 doi: 10.1016/S0378-4371(00)00386-1
    [4] D. Benson, R. Schumer, M. Meerschaert, S. Wheatcraft, Fractional dispersion, Lévy motion, and the MADE tracer tests, Transport Porous Med., 42 (2001), 211–240. https://doi.org/10.1023/A:1006733002131
    [5] I. Podlubny, Fractional Differential Equations, New York: Academic press, 1999.
    [6] J. B. Greer, A. L. Bertozzi, G. Sapiro, Fourth order partial differential equations on general geometries, J. Comput. Phys., 216 (2006), 216–246. https://doi.org/10.1016/j.jcp.2005.11.031 doi: 10.1016/j.jcp.2005.11.031
    [7] F. Memoli, G. Sapiro, P. Thompson, Implicit brain imaging, NeuroImage, 23 (2004), 179–188. https://doi.org/10.1016/j.neuroimage.2004.07.072
    [8] M. Hofer, H. Pottmann, Energy-minimizing splines in manifolds, ACM Trans. Graph, 23 (2004), 284–293. https://doi.org/10.1145/1015706.1015716 doi: 10.1145/1015706.1015716
    [9] D. Halpern, O. E. Jensen, J. B. Grotberg, A theoretical study of surfactant and liquid delivery into the lung, J. Appl. Physiol., 85 (1998), 333–352. https://doi.org/10.1152/jappl.1998.85.1.333 doi: 10.1152/jappl.1998.85.1.333
    [10] T. G. Myers, J. P. F. Charpin, S. J. Chapman, The flow and solidification of a thin fluid film on an arbitrary three-dimensional surface, Phys. Fluids, 14 (8) (2002), 2788–2803. https://doi.org/10.1063/1.1488599
    [11] T. G. Myers, J. P. F. Charpin, A mathematical model for atmospheric ice accretion and water flow on a cold surface, Int. J. Heat Mass Transf., 47 (2004), 5483–5500. https://doi.org/10.1016/j.ijheatmasstransfer.2004.06.037 doi: 10.1016/j.ijheatmasstransfer.2004.06.037
    [12] X. Hu, L. Zhang, On finite difference methods for fourth-order fractional diffusion-wave and subdiffusion systems, Appl. Math. Comput., 218 (2012), 5019–5034. https://doi.org/10.1016/j.amc.2011.10.069 doi: 10.1016/j.amc.2011.10.069
    [13] X. Hu, L. Zhang, A compact finite difference scheme for the fourth-order fractional diffusion-wave system, Comput. Phys. Commun., 182 (2011), 1645–1650. https://doi.org/10.1016/j.cpc.2011.04.013 doi: 10.1016/j.cpc.2011.04.013
    [14] C. Ji, Z. Sun, Z. Hao, Numerical algorithms with high spatial accuracy for the fourth-order fractional sub-diffusion equations with the first Dirichlet boundary conditions, J. Sci. Comput., 66 (2015), 1148–1174. https://doi.org/10.1007/s10915-015-0059-7 doi: 10.1007/s10915-015-0059-7
    [15] P. Zhang, H. Pu, A second-order compact difference scheme for the fourth-order fractional sub-diffusion equation, Numer Algor, 76 (2017), 573–598. https://doi.org/10.1007/s11075-017-0271-7 doi: 10.1007/s11075-017-0271-7
    [16] A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280 (2015), 424–438. https://doi.org/10.1016/j.jcp.2014.09.031 doi: 10.1016/j.jcp.2014.09.031
    [17] M. Ran, C. Zhang, New compact difference scheme for solving the fourth-order time fractional sub-diffusion equation of the distributed order, Appl. Numer. Math, 129 (2018), 58–70. https://doi.org/10.1016/j.apnum.2018.03.005 doi: 10.1016/j.apnum.2018.03.005
    [18] X. Zhao, Q. Xu, Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient, Appl. Math. Model., 38 (2014), 3848–3859. https://doi.org/10.1016/j.apm.2013.10.037 doi: 10.1016/j.apm.2013.10.037
    [19] S. Vong, P. Lyu, Z. Wang, A compact difference scheme for fractional sub-diffusion equations with the spatially variable coefficient under Neumann boundary conditions, J. Sci. Comput., 66 (2016), 725–739. https://doi.org/10.1007/s10915-015-0040-5 doi: 10.1007/s10915-015-0040-5
    [20] A. A. Samarskii, V. V. Andreev, Difference methods for elliptic equations (in Russian), Moscow: Izdatel'stvo Nauka, 1976.
    [21] Z. Sun, Numerical Methods of Partial Differential Equations (in Chinese), Beijing: Science Press, 2012.
    [22] T. Yan, Z. Sun, J. Zhang, Fast evalution of the Caputo fractional derivative and its applications to fractional diffusion equations: A second-order scheme, Commun. Comput. Phys., 22 (2017), 1028–1048. https://doi.org/10.4208/cicp.OA-2016-0136 doi: 10.4208/cicp.OA-2016-0136
    [23] H. Liao, D. Li, J. Zhang, Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiffusion equations, SIAM J. Numer. Anal., 56 (2018), 1112–1133. https://doi.org/10.1137/17M1131829 doi: 10.1137/17M1131829
    [24] B. Jin, R. Lazarov, Z. Zhou, Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview, Comput. Methods Appl. Mech. Engrg., 346 (2019), 332–358. https://doi.org/10.1016/j.cma.2018.12.011 doi: 10.1016/j.cma.2018.12.011
    [25] M. Stynes, E. O'Riordan, J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057–1079. https://doi.org/10.1137/16M1082329 doi: 10.1137/16M1082329
    [26] M. Cui, A compact difference scheme for time-fractional Dirichlet biharmonic equation on temporal graded meshe, East Asian J. Appl. Math., 11 (2021), 164–180. https://doi.org/10.4208/eajam.270520.210920 doi: 10.4208/eajam.270520.210920
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