A Fast Averaged L1 Finite Difference Method for Time Fractional Mobile/Immobile Diffusion Equation with Weakly Singular Solution

doi:10.1007/s42967-025-00493-3

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (3): 1207-1225.doi: 10.1007/s42967-025-00493-3

A Fast Averaged L1 Finite Difference Method for Time Fractional Mobile/Immobile Diffusion Equation with Weakly Singular Solution

Haili Qiao¹, Aijie Cheng²

1. School of Mathematical Sciences, Liaocheng University, Liaocheng, 252059, Shandong, China;
2. School of Mathematics, Shandong University, Jinan, 250100, Shandong, China

收稿日期:2024-10-27 修回日期:2025-01-29 出版日期:2026-06-20 发布日期:2026-05-29
通讯作者: Aijie Cheng, Email: aijie@sdu.edu.cn E-mail:aijie@sdu.edu.cn
作者简介:Haili Qiao, Email: QHLmath@163.com
基金资助:
This work was supported in part by the Shandong Provincial Natural Science Foundation of China under Grant No. ZR2022QA038 and the Doctoral Research Foundation of Liaocheng University of China under Grant No.318052155.

A Fast Averaged L1 Finite Difference Method for Time Fractional Mobile/Immobile Diffusion Equation with Weakly Singular Solution

Haili Qiao¹, Aijie Cheng²

1. School of Mathematical Sciences, Liaocheng University, Liaocheng, 252059, Shandong, China;
2. School of Mathematics, Shandong University, Jinan, 250100, Shandong, China

Received:2024-10-27 Revised:2025-01-29 Online:2026-06-20 Published:2026-05-29
Contact: Aijie Cheng, Email: aijie@sdu.edu.cn E-mail:aijie@sdu.edu.cn
Supported by:
The research is supported partly by the National Natural Science Foundation of China (Grant No. 12071392).

摘要/Abstract

摘要： In this paper, the time fractional mobile/immobile diffusion equation with the weak singular solution at the initial time is studied. The averaged L1 finite difference scheme is established for the equation. The stability of the numerical scheme is analyzed by the Fourier analysis method. The convergence order of the scheme is method. The convergence order of the scheme is $O\left(\tau^2|\ln \tau|+h^2\right)$, where $\tau$ and $h$ are the sizes of the time and space steps, respectively. In addition, due to the historical dependence of the time fractional derivative, we establish a fast method based on the exponential-sum-approximation, effectively reducing computation and storage. Furthermore, we provide an error estimate of the fast algorithm. Finally, a numerical experiment verifies the effectiveness of the algorithm.

关键词: Time fractional mobile/immobile diffusion equation, Averaged L1 scheme, Exponential-sum-approximation, Weak singularity

Abstract: In this paper, the time fractional mobile/immobile diffusion equation with the weak singular solution at the initial time is studied. The averaged L1 finite difference scheme is established for the equation. The stability of the numerical scheme is analyzed by the Fourier analysis method. The convergence order of the scheme is method. The convergence order of the scheme is $O\left(\tau^2|\ln \tau|+h^2\right)$, where $\tau$ and $h$ are the sizes of the time and space steps, respectively. In addition, due to the historical dependence of the time fractional derivative, we establish a fast method based on the exponential-sum-approximation, effectively reducing computation and storage. Furthermore, we provide an error estimate of the fast algorithm. Finally, a numerical experiment verifies the effectiveness of the algorithm.

Key words: Time fractional mobile/immobile diffusion equation, Averaged L1 scheme, Exponential-sum-approximation, Weak singularity

中图分类号:

Haili Qiao, Aijie Cheng. A Fast Averaged L1 Finite Difference Method for Time Fractional Mobile/Immobile Diffusion Equation with Weakly Singular Solution[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 1207-1225.

参考文献

[1] Fardi, M., Ghasemi, M.: A numerical solution strategy based on error analysis for time-fractional mobile/immobile transport model. Soft. Comput. 25(16), 11307-11331 (2021)
[2] Huang, Y., Li, Q., Li, R., Zeng, F., Guo, L.: A unified fast memory-saving time-stepping method for fractional operators and its applications. Numer. Math.: Theory Methods Appl. 15 (3), 679-714 (2022)
[3] Huang, Y., Zeng, F., Guo, L.: Error estimate of the fast L1 method for time-fractional subdiffusion equations. Appl. Math. Lett. 133, 108288 (2022)
[4] Jiang, H., Xu, D., Qiu, W., Zhou, J.: An ADI compact difference scheme for the two-dimensional semilinear time-fractional mobile-immobile equation. Comput. Appl. Math. 39, 1-17 (2020)
[5] Liu, Z., Li, X., Zhang, X.: A fast high-order compact difference method for the fractal mobile/immobile transport equation. Int. J. Comput. Math. 97(9), 1860-1883 (2020)
[6] Maryshev, B., Joelson, M., Lyubimov, D., Lyubimova, T., Néel, M.-C.: Non Fickian flux for advection-dispersion with immobile periods. J. Phys. A: Math. Theor. 42(11), 115001 (2009)
[7] Nikan, O., Machado, J.T., Golbabai, A., Nikazad, T.: Numerical approach for modeling fractal mobile/immobile transport model in porous and fractured media. Int. Commun. Heat Mass Transfer 111, 104443 (2020)
[8] Rusagara, I., Baleanu, D.: Numerical solution of a kind of fractional parabolic equations via two difference schemes. Abstract and Applied Analysis, 2013, 324-331 (2013)
[9] Schumer, R., Benson, D.A., Meerschaert, M.M., Baeumer, B.: Fractal mobile/immobile solute transport. Water Resour. Res. 39 (10) (2003). https://doi.org/10.1029/2003WR002141
[10] Shen, J., Zeng, F., Stynes, M.: Second-order error analysis of the averaged L1 scheme $\overline{\mathrm{L} 1}$ for time-fractional initial-value and subdiffusion problems. Sci. China Math. 67(7), 1641-1664 (2024)
[11] Yang, Z., Zeng, F.: A corrected L1 method for a time-fractional subdiffusion equation. J. Sci. Comput. 95(3), 85 (2023)
[12] Yin, B., Liu, Y., Li, H.: A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations. Appl. Math. Comput. 368, 124799 (2020)
[13] Yu, F., Chen, M.: Second-order error analysis for fractal mobile/immobile Allen-Cahn equation on graded meshes. J. Sci. Comput. 96(2), 49 (2023)
[14] Zhang, H., Jiang, X., Liu, F.: Error analysis of nonlinear time fractional mobile/immobile advection-diffusion equation with weakly singular solutions. Fract. Calculus Appl. Anal. 24(1), 202-224 (2021)
[15] Zhang, J., Fang, Z., Sun, H.: Exponential-sum-approximation technique for variable-order time-fractional diffusion equations. J. Appl. Math. Comput. 68(1), 323-347 (2022)
[16] Zhao, J., Fang, Z., Li, H., Liu, Y.: Finite volume element method with the WSGD formula for nonlinear fractional mobile/immobile transport equations. Adv. Differ. Equ. 2020(1), 1-20 (2020)
[17] Zheng, Z., Wang, Y.: An averaged L1-type compact difference method for time-fractional mobile/immobile diffusion equations with weakly singular solutions. Appl. Math. Lett. 131, 108076 (2022)

A Fast Averaged L1 Finite Difference Method for Time Fractional Mobile/Immobile Diffusion Equation with Weakly Singular Solution

A Fast Averaged L1 Finite Difference Method for Time Fractional Mobile/Immobile Diffusion Equation with Weakly Singular Solution

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