Fast finite difference/Legendre spectral collocation approximations for a tempered time-fractional diffusion equation

Zunyuan Hu; Can Li; Shimin Guo; Zunyuan Hu; Can Li; Shimin Guo

doi:10.3934/math.20241650

AIMS Mathematics

2024, Volume 9, Issue 12: 34647-34673. doi: 10.3934/math.20241650

Previous Article Next Article

Research article

Fast finite difference/Legendre spectral collocation approximations for a tempered time-fractional diffusion equation

1.
Department of Applied Mathematics, Xi'an University of Technology, Xi'an, Shaanxi 710054, China
2.
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, China

Received: 20 September 2024 Revised: 16 November 2024 Accepted: 20 November 2024 Published: 11 December 2024
MSC : 65M70, 65M06, 65M12

The present work is concerned with the efficient numerical schemes for a time-fractional diffusion equation with tempered memory kernel. The numerical schemes are established by using a $ L1 $ difference scheme for generalized Caputo fractional derivative in the temporal variable, and applying the Legendre spectral collocation method for the spatial variable. The sum-of-exponential technique developed in [Jiang et al., Commun. Comput. Phys., 21 (2017), 650-678] is used to discrete generalized fractional derivative with exponential kernel. The stability and convergence of the semi-discrete and fully discrete schemes are strictly proved. Some numerical examples are shown to illustrate the theoretical results and the efficiency of the present methods for two-dimensional problems.
- spectral collocation method,
- generalized memory kernel,
- sum-of-exponential,
- error estimates
Citation: Zunyuan Hu, Can Li, Shimin Guo. Fast finite difference/Legendre spectral collocation approximations for a tempered time-fractional diffusion equation[J]. AIMS Mathematics, 2024, 9(12): 34647-34673. doi: 10.3934/math.20241650

Related Papers:

Abstract

The present work is concerned with the efficient numerical schemes for a time-fractional diffusion equation with tempered memory kernel. The numerical schemes are established by using a $ L1 $ difference scheme for generalized Caputo fractional derivative in the temporal variable, and applying the Legendre spectral collocation method for the spatial variable. The sum-of-exponential technique developed in [Jiang et al., Commun. Comput. Phys., 21 (2017), 650-678] is used to discrete generalized fractional derivative with exponential kernel. The stability and convergence of the semi-discrete and fully discrete schemes are strictly proved. Some numerical examples are shown to illustrate the theoretical results and the efficiency of the present methods for two-dimensional problems.

References

[1]	G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2022), 461–580. https://doi.org/10.1016/S0370-1573(02)00331-9 doi: 10.1016/S0370-1573(02)00331-9
[2]	R. L. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng., 32 (2004), 1–104. https://doi.org/10.1615/critrevbiomedeng.v32.i1.10 doi: 10.1615/critrevbiomedeng.v32.i1.10
[3]	W. H. Deng, R. Hou, W. L. Wang, P. B. Xu, Modeling anomalous diffusion: From statistics to mathematics, Singapore: World Scientific, 2020. https://doi.org/10.1142/11630
[4]	T. Sandev, A. Chechkin, H. Kantz, R. Metzler, Diffusion and Fokker-Planck-Smoluchowski equations with generalized memory kernel, Fract. Calc. Appl. Anal., 18 (2015), 1006–1038. https://doi.org/10.1515/fca-2015-0059 doi: 10.1515/fca-2015-0059
[5]	A. A. Alikhanov, A time-fractional diffusion equation with generalized memory kernel in differential and difference settings with smooth solutions, Comput. Meth. Appl. Mat., 17 (2017), 1–14. https://doi.org/10.1515/cmam-2017-0035 doi: 10.1515/cmam-2017-0035
[6]	Y. Luchko, M. Yamamoto, The general fractional derivative and related fractional differential equations, Mathematics, 8 (2020), 2115. https://doi.org/10.3390/math8122115 doi: 10.3390/math8122115
[7]	J. G. Liu, F. Z. Geng, An explanation on four new definitions of fractional operators, Acta Math. Sci., 44 (2024), 1271–1279. https://doi.org/10.1007/s10473-024-0405-7 doi: 10.1007/s10473-024-0405-7
[8]	B. I. Henry, T. A. M. Langlands, S. L. Wearne, Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations, Phys. Rev. E, 74 (2006), 031116. https://doi.org/10.1103/PhysRevE.74.031116 doi: 10.1103/PhysRevE.74.031116
[9]	L. Zhao, C. Li, F. Q. Zhao, Efficient diference schemes for the Caputo-tempered fractional difusion equations based on polynomial interpolation, Com. Appl. Math. Comput., 3 (2021), 1–40. https://doi.org/10.1007/s42967-020-00067-5 doi: 10.1007/s42967-020-00067-5
[10]	M. H. Chen, Z. S. Jiang, W. P. Bu, Two $L1$ schemes on graded meshes for fractional Feynman-Kac equation, J. Sci. Comput., 88 (2021), 1–24. https://doi.org/10.1007/s10915-021-01581-1 doi: 10.1007/s10915-021-01581-1
[11]	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
[12]	L. Greengard, P. Lin, Spectral approximation of the free-space heat kernel, Appl. Comput. Harmon. A., 9 (2000), 83–97. https://doi.org/10.1006/acha.2000.0310 doi: 10.1006/acha.2000.0310
[13]	S. D. Jiang, J. W. Zhang, Q. Zhang, Z. M. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21 (2017), 650–678. https://doi.org/10.4208/cicp.OA-2016-0136 doi: 10.4208/cicp.OA-2016-0136
[14]	Y. G. Yan, Z. Z. Sun, J. W. Zhang, Fast evaluation 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
[15]	H. Y. Zhu, C. J. Xu, A fast high order method for the time-fractional diffusion equation, SIAM. J. Numer. Anal., 57 (2019), 2829–2849. https://doi.org/10.1137/18M1231225 doi: 10.1137/18M1231225
[16]	X. M. Gu, T. Z. Huang, Y. L. Zhao, P. Lyu, B. Carpentieri, A fast implicit difference scheme for solving the generalized time-space fractional diffusion equations with variable coefficients, Numer. Meth. Part. D. E., 37 (2020), 1136–1162. https://doi.org/10.1002/num.22571 doi: 10.1002/num.22571
[17]	B. Y. Guo, Spectral methods and their applications, Singapore: World Scientific, 1998. https://doi.org/10.1142/3662
[18]	J. Shen, T. Tang, L. L. Wang, Spectral methods: Algorithms, analysis and applications, Berlin: Springer-Verlag, 2011. https://doi.org/10.1007/978-3-540-71041-7
[19]	Y. M. Lin, C. J. 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
[20]	X. J. Li, C. J. 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
[21]	F. H. Zeng, F. W. Liu, C. P. Li, K. Burrage, I. Turner, V. Anh, A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation, SIAM J. Numer. Anal., 52 (2014), 2599–2622. https://doi.org/10.1137/130934192 doi: 10.1137/130934192
[22]	S. M. Guo, L. Q. Mei, Y. Li, An efficient Galerkin spectral method for two-dimensional fractional nonlinear reaction-diffusion-wave equation, Comput. Math. Appl., 74 (2017), 2449–2465. https://doi.org/10.1016/j.camwa.2017.07.022 doi: 10.1016/j.camwa.2017.07.022
[23]	Y. Maday, B. Meétivet, Chebyshev spectral approximation of Navier-Stokes equations in a two dimensional domain, ESAIM-Math. Model. Num., 21 (1987), 93–123. https://doi.org/10.1051/m2an/1987210100931 doi: 10.1051/m2an/1987210100931
[24]	C. W. Lv, C. J. Xu, Improved error estimates of a finite difference/spectral method for time-fractional diffusion equations, Int. J. Numer. Anal. Mod., 12 (2015), 384–400.
[25]	C. Bernardi, Y. Maday, Approximations spectrales de problemes aux limites elliptiques, Berlin: Springer-Verlag, 1992.
[26]	A. Quarteroni, A. Valli, Numerical approximation of partial differential equations, Berlin: Springer-Verlag, 1994. https://doi.org/10.1007/978-3-540-85268-1
[27]	Y. L. Jing, C. Li, Block-centered finite difference method for a tempered subdiffusion model with time dependent coefficients, Comput. Math. Appl., 145 (2023), 202–223. https://doi.org/10.1016/j.camwa.2023.06.014 doi: 10.1016/j.camwa.2023.06.014
[28]	J. F. Zhou, X. M. Gu, Y. L. Zhao, H. Li, A fast compact difference scheme with unequal time-steps for the tempered time-fractional Black-Scholes model, Int. J. Comput. Math., 101 (2024), 989–1011. https://doi.org/10.1080/00207160.2023.2254412 doi: 10.1080/00207160.2023.2254412

Reader Comments

Your name:*

Email:*
© 2024 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)