Numerical Analysis of a High-Order Scheme with Nonuniform Time Grids for Caputo-Hadamard Fractional Reaction Sub-diffusion Equations

doi:10.1007/s42967-024-00441-7

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (1): 338-365.doi: 10.1007/s42967-024-00441-7

• ORIGINAL PAPERS • Previous Articles Next Articles

Numerical Analysis of a High-Order Scheme with Nonuniform Time Grids for Caputo-Hadamard Fractional Reaction Sub-diffusion Equations

Chunxiu Liu¹, Junying Cao², Tong Lyu¹, Xingyang Ye¹

1. School of Science, Jimei University, Xiamen, 361021, Fujian, China;
2. School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, 550025, Guizhou, China

Received:2024-02-19 Revised:2024-05-25 Online:2026-02-20 Published:2026-02-11
Contact: Xingyang Ye,E-mail:yexingyang@jmu.edu.cn E-mail:yexingyang@jmu.edu.cn
Supported by:
This work is supported by the Natural Science Foundation of Fujian Province of China (2022J01338), Fujian Alliance of Mathematics (2024SXLMMS03), the NSFC (12361083 and 11961009), and the Natural Science Research Project of Department of Education of Guizhou Province, China (QJJ2023012).

Abstract

Abstract: In this paper, we propose a numerical approach for the fractional reaction sub-diffusion equation with a Caputo-Hadamard derivative of fractional order α∈(0,1), whose solutions typically exhibit singular behavior at the initial time. The approach involves the use of an L2-type discrete operator of order 3-α to approximate the fractional derivative on nonuniform temporal meshes, while a standard second-order difference method is employed for uniform spatial meshes. Detailed discussions are presented on the truncation errors and coefficients of the proposed discrete fractional derivative operator. The stability as well as the accuracy of the resulting numerical scheme is rigorously analyzed on a unique nonuniform mesh. Several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method.

Key words: Caputo-Hadamard derivative, Fractional reaction sub-diffusion equations, Nonuniform meshes, Stability and convergence

CLC Number:

Chunxiu Liu, Junying Cao, Tong Lyu, Xingyang Ye. Numerical Analysis of a High-Order Scheme with Nonuniform Time Grids for Caputo-Hadamard Fractional Reaction Sub-diffusion Equations[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 338-365.

TrendMD

References

[1] Ahmad, B., Alsaedi, A., Ntouyas, S., Tariboon, J.: Hadamard-type Fractional Differential Equations. Inclusions and Inequalities. Springer, Cham (2017)
[2] Denisov, S., Kantz, H.: Continuous-time random walk theory of superslow diffusion. Europhys. Lett. 92(3), 30001 (2010)
[3] Diethelm, K.: The Analysis of Fractional Differential Equations. Springer-Verlag, Berlin (2010)
[4] DraGer, J., Klafter, J.: Strong anomaly in diffusion generated by iterated maps. Phys. Rev. Lett. 84(26), 5998-6001 (2000)
[5] Fan, E., Li, C., Li, Z.: Numerical approaches to Caputo-Hadamard fractional derivatives with applications to long-term integration of fractional differential systems. Commun. Nonlinear Sci. Numer. Simu. 106, 106096 (2022)
[6] Garra, R., Mainardi, F., Spada, G.: A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus. Chaos, Solitons Fractals 102, 333-338 (2017)
[7] Gohar, M., Li, C., Li, Z.: Finite difference methods for Caputo-Hadamard fractional differential equations. Mediterr. J. Math. 17(6), 194 (2020)
[8] Gohar, M., Li, C., Yin, C.: On Caputo-Hadamard fractional differential equations. Int. J. Comput. Math. 97(7), 1459-1483 (2020)
[9] Green, C., Liu, Y., Yan, Y.: Numerical methods for Caputo-Hadamard fractional differential equations with graded and non-uniform meshes. Mathematics 9, 2728 (2021)
[10] Hadamard, J.: Essai sur i’étude des fonctions données par leur development de taylor. Journal de Mathématiques Pures et Appliquées 8, 101-186 (1892)
[11] Iglói, F., Turban, L., Rieger, H.: Anomalous diffusion in aperiodic environments. Phys. Rev. E 59, 1465 (1999)
[12] Jarad, F., Abdeljawad, T., Baleanu, D.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Difference Equ. 2012(1), 142 (2012)
[13] Kilbas, A.: Hadamard-type fractional calculus. J. Korean Math. Soc. 38(6), 1191-1204 (2001)
[14] Li, C., Li, Z.: Stability and logarithmic decay of the solution to Hadamard-Type fractional differential equation. Chaos 31(2), 31 (2021)
[15] Li, C., Li, Z., Wang, Z.: Mathematical analysis and the local discontinuous Galerkin method for Caputo-Hadamard fractional partial differential equation. J. Sci. Comput. 85(2), 41 (2020)
[16] Liao, H., Mclean, W., Zhang, J.: A second-order scheme with nonuniform time steps for a linear reaction-sudiffusion problem. Commun. Comput. Phys. 30(2), 567-601 (2021)
[17] Lomnitz, C.: Application of the logarithmic creep law to stress wave attenuation in the solid earth. J. Geophys. Res. 67(1), 365-368 (1962)
[18] Lyu, C., Xu, C.: Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput. 38(5), A2699-A2724 (2016)
[19] Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
[20] Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
[21] Sanders, L., Lomholt, M., Lizana, L., Fogelmark, K., Metzler, R., Ambjornsson, T.: Severe slowing-down and universality of the dynamics in disordered interacting many-body systems: ageing and ultraslow diffusion. New J. Phys. 16, 113050 (2014)
[22] Sandev, T., Iomin, A., Kantz, H., Metzler, R., Chechkin, A.: Comb model with slow and ultraslow diffusion. Math. Model. Nat. Phenom. 11(3), 18-33 (2016)
[23] Srivastava, H., Kilbas, A., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier Science, Amsterdam (2006)
[24] Stynesy, M., O’riordanz, E., Graciax, J.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55(2), 1057-1079 (2017)
[25] Wang, Z., Ou, C., Vong, S.: A second-order scheme with nonuniform time grids for Caputo-Hadamard fractional sub-diffusion equations. J. Comput. Appl. Math. 414, 114448 (2022)

Numerical Analysis of a High-Order Scheme with Nonuniform Time Grids for Caputo-Hadamard Fractional Reaction Sub-diffusion Equations

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 7

Recommended Articles

Metrics

Comments

[1]	Enyu Fan, Yaxuan Li, Qianlan Zhao. H3N3 Approximate Formulae for Typical Fractional Derivatives [J]. Communications on Applied Mathematics and Computation, 2025, 7(6): 2485-2501.
[2]	Min Cai, Changpin Li, Yu Wang. Numerical Algorithms for Ultra-slow Diffusion Equations [J]. Communications on Applied Mathematics and Computation, 2025, 7(6): 2339-2384.
[3]	Qiaoqiao Dai, Dongxia Li. The L2-1_σ/LDG Method for the Caputo Diffusion Equation with a Variable Coefficient [J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1378-1397.
[4]	Jieying Zhang, Caixia Ou, Zhibo Wang, Seakweng Vong. An Order Reduction Method for the Nonlinear Caputo-Hadamard Fractional Diffusion-Wave Model [J]. Communications on Applied Mathematics and Computation, 2025, 7(1): 392-408.
[5]	Zhen Wang. L1/LDG Method for Caputo-Hadamard Time Fractional Diffusion Equation [J]. Communications on Applied Mathematics and Computation, 2025, 7(1): 203-227.
[6]	Yu Wang, Min Cai. Finite Difference Schemes for Time-Space Fractional Diffusion Equations in One- and Two-Dimensions [J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1674-1696.
[7]	Xue-lei Lin, Pin Lyu, Michael K. Ng, Hai-Wei Sun, Seakweng Vong. An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients [J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 215-239.