High order compact difference scheme for solving the time multi-term fractional sub-diffusion equations

Lei Ren; Lei Ren

doi:10.3934/math.2022508

AIMS Mathematics

2022, Volume 7, Issue 5: 9172-9188. doi: 10.3934/math.2022508

Previous Article Next Article

Research article Special Issues

High order compact difference scheme for solving the time multi-term fractional sub-diffusion equations

Lei Ren ^,

School of Mathematics and Statistics, Shangqiu Normal University, Shangqiu 476000, China

Received: 28 December 2021 Revised: 18 February 2022 Accepted: 24 February 2022 Published: 09 March 2022
MSC : 65M06, 65M12, 65M15, 35R11

In this paper, a high order compact finite difference is established for the time multi-term fractional sub-diffusion equation. The derived numerical differential formula can achieve second order accuracy in time and four order accuracy in space. A unconditionally stable and convergent difference scheme is presented, and a rigorous proof for the stability and convergence is given. Numerical results demonstrate the efficiency of the proposed difference schemes.
- fractional sub-diffusion equation,
- multi-term fractional derivatives,
- variable coefficient,
- compact difference method,
- convergence,
- stability
Citation: Lei Ren. High order compact difference scheme for solving the time multi-term fractional sub-diffusion equations[J]. AIMS Mathematics, 2022, 7(5): 9172-9188. doi: 10.3934/math.2022508

Related Papers:

Abstract

In this paper, a high order compact finite difference is established for the time multi-term fractional sub-diffusion equation. The derived numerical differential formula can achieve second order accuracy in time and four order accuracy in space. A unconditionally stable and convergent difference scheme is presented, and a rigorous proof for the stability and convergence is given. Numerical results demonstrate the efficiency of the proposed difference schemes.

References

[1]	K. B. Oldham, S Jerome, The fractional calculus, New York: Academic Press, 1974.
[2]	I. Podlubny, Fractional differential equations, New York: Academic Press, 1999.
[3]	A. Bueno-Orovio, D. Kay, V. Grau, B. Rodriguez, K. Burrage, Fractional diffusion models of cardiac electrical propagation: Role of structural heterogeneity in dispersion of repolarization, J. R. Soc. Interface, 11 (2014), 1–12. http://dx.doi.org/10.1098/rsif.2014.0352 doi: 10.1098/rsif.2014.0352
[4]	K. Burrage, N. Hale, D. Kay, An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations, SIAM J. Sci. Comput., 34 (2012), A2145–A2172. http://dx.doi.org/10.1137/110847007 doi: 10.1137/110847007
[5]	K. D. Dwivedi, S. Das, Rajeev, D. Baleanu, Numerical solution of highly non-linear fractional order reaction advection diffusion equation using the cubic B-spline collocation method, Int. J. Nonlin. Sci. Numer. Simul., 2021. https://doi.org/10.1515/ijnsns-2020-0112
[6]	K. D. Dwivedi, Rajeev, S. Das, Fibonacci collocation method to solve two-dimensional nonlinear fractional order advection-reaction diffusion equation, Spec. Top. Rev. Porous, 10 (2019), 569–584. https://doi.org/10.1615/SpecialTopicsRevPorousMedia.2020028160 doi: 10.1615/SpecialTopicsRevPorousMedia.2020028160
[7]	K. D. Dwivedi, Rajeev, Numerical solution of fractional order advection reaction diffusion equation with fibonacci neural network, Neural Process Lett., 53 (2021), 2687–2699. https://doi.org/10.1007/s11063-021-10513-x doi: 10.1007/s11063-021-10513-x
[8]	S. Chen, F. Liu, P. Zhuang, V. Anh, Finite difference approximations for the fractional Fokker-Planck equation, Appl. Math. Model., 33 (2009), 256–273. https://doi.org/10.1016/j.apm.2007.11.005 doi: 10.1016/j.apm.2007.11.005
[9]	M. Cui, Compact exponential scheme for the time fractional convection-diffusion reaction equation with variable coefficients, J. Comput. Phys., 280 (2015), 143–163. https://doi.org/10.1016/j.jcp.2014.09.012 doi: 10.1016/j.jcp.2014.09.012
[10]	Y. Dimitrov, Numerical approximations for fractional differential equations, J. Frac. Calc. Appl., 5 (2014), 1–45. https://doi.org/10.13140/2.1.1802.8323 doi: 10.13140/2.1.1802.8323
[11]	R. Herrmann, Fractional calculus: An introduction for physicists, Singapore: World Scientific, 2011.
[12]	R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific, 2000.
[13]	T. A. M. Langlands, B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205 (2005), 719–736. https://doi.org/10.1016/j.jcp.2004.11.025 doi: 10.1016/j.jcp.2004.11.025
[14]	X. Li, C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), 2108–2131. https://doi.org/10.1137/080718942 doi: 10.1137/080718942
[15]	C. P. Li, Z. G. Zhao, Y. Q. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl., 62 (2011), 855–875. https://doi.org/10.1016/j.camwa.2011.02.045 doi: 10.1016/j.camwa.2011.02.045
[16]	Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533–1552. https://doi.org/10.1016/j.jcp.2007.02.001 doi: 10.1016/j.jcp.2007.02.001
[17]	A. Mohebbi, M. Abbaszadeh, Compact finite difference scheme for the solution of time fractional advection-dispersion equation, Numer. Algor., 63 (2013), 431–452. https://doi.org/10.1007/s11075-012-9631-5 doi: 10.1007/s11075-012-9631-5
[18]	A. A. Samarskii, The theory of difference schemes, New York: Marcel Dekker Inc., 2001.
[19]	W. Y. Tian, H. Zhou, W. H. Deng, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comp., 84 (2015), 1703–1727. https://doi.org/10.1090/S0025-5718-2015-02917-2 doi: 10.1090/S0025-5718-2015-02917-2
[20]	M. Uddin, S. Hag, RBFs approximation method for time fractional partial differential equations, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 4208–4214. https://doi.org/10.1016/j.cnsns.2011.03.021 doi: 10.1016/j.cnsns.2011.03.021
[21]	Z. Wang, S. Vong, Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J. Comput. Phys., 277 (2014), 1–15. https://doi.org/10.1016/j.jcp.2014.08.012 doi: 10.1016/j.jcp.2014.08.012
[22]	S. B. Yuste, L. Acedo, An explicit finite difference method and a new Von-Neumann type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42 (2005), 1862–1874. https://doi.org/10.1137/030602666 doi: 10.1137/030602666
[23]	Y. Zhang, D. A. Benson, D. M. Reeves, Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications, Adv. Water Resour., 32 (2009), 561–581. https://doi.org/10.1016/j.advwatres.2009.01.008 doi: 10.1016/j.advwatres.2009.01.008
[24]	L. Zhao, W. Deng, A series of high order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives, Numer. Methods Partial Differ. Equ., 31 (2015), 1345–1381. https://doi.org/10.1002/num.21947 doi: 10.1002/num.21947
[25]	Z. Z. Sun, X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193–209. https://doi.org/10.1016/j.apnum.2005.03.003 doi: 10.1016/j.apnum.2005.03.003
[26]	G. H. Gao, Z. Z. Sun, H. W. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys., 259 (2014), 33–50. https://doi.org/10.1016/j.jcp.2013.11.017 doi: 10.1016/j.jcp.2013.11.017
[27]	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
[28]	M. Dehghan, M. Safarpoor, M. Abbaszadeh, Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations, J. Comput. Appl. Math., 290 (2015), 174–195. https://doi.org/10.1016/j.cam.2015.04.037 doi: 10.1016/j.cam.2015.04.037
[29]	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
[30]	S. C. Shiralashetti, A. B. Deshi, An efficient Haar wavelet collocation method for the numerical solution of multi-term fractional differential equations, Nolinear Dyn., 83 (2016), 293–303. https://doi.org/10.1007/s11071-015-2326-4 doi: 10.1007/s11071-015-2326-4
[31]	Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374 (2011), 538–548. https://doi.org/10.1016/j.jmaa.2010.08.048 doi: 10.1016/j.jmaa.2010.08.048
[32]	B. Jin, R. Lazarov, Y. Lin, Z. Zhou, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys., 281 (2015), 825–843. https://doi.org/10.1016/j.jcp.2014.10.051 doi: 10.1016/j.jcp.2014.10.051
[33]	F. Liu, M. M. Meerschaert, R. McGough, P. Zhang, Q. Liu, Numerical methods for solving the multi-term time-fractional wave-diffusion equations, Fract. Calc. Appl. Anal., 281 (2013), 9–25. https://doi.org/10.2478/s13540-013-0002-2 doi: 10.2478/s13540-013-0002-2
[34]	A. A. Alikhanov, Numerical methods of solutions of boundary value problems for the multi-term variable distributed order diffusion equation, Appl. Math. Comput., 268 (2015), 12–22. https://doi.org/10.1016/j.amc.2015.06.045 doi: 10.1016/j.amc.2015.06.045
[35]	G. H. Gao, A. A. Alikhanov, Z. Z. Sun, The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations, J. Sci. Comput., 73 (2017), 93–121. https://doi.org/10.1007/s10915-017-0407-x doi: 10.1007/s10915-017-0407-x
[36]	X. Zhao, Q. Xu, Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient, Appl. Math. Model., 73 (2014), 3848–3859. https://doi.org/10.1016/j.apm.2013.10.037 doi: 10.1016/j.apm.2013.10.037
[37]	Z. Z. Sun, A unconditionally stable and $O(\tau^2+h^4)$ order $L_{\infty}$ convergence difference scheme for linear parabolic equation with variable coefficients, Numer. Methods Partial Differ. Equ., 17 (2001), 619–631. https://doi.org/10.1002/num.1030 doi: 10.1002/num.1030
[38]	Y. M. Wang, L. Ren, Efficient compact finite difference methods for a class of time-fractional convection-reaction-diffusion equations with variable coefficients, Int. J. Comput. Math., 96 (2019), 264–297. https://doi.org/10.1080/00207160.2018.1437262 doi: 10.1080/00207160.2018.1437262
[39]	V. F. Morales-Delgado, J. F. Gomez-Aguilar, K. M. Saad, M. A. Khan, P. Agarwal, Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach, Phys. A: Stat. Mech. Appl., 523 (2019), 48–65. https://doi.org/10.1016/j.physa.2019.02.018 doi: 10.1016/j.physa.2019.02.018

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)