α-robust error analysis of two nonuniform schemes for Caputo-Hadamard fractional reaction sub-diffusion problems

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

In this paper, we focused on the Caputo-Hadamard fractional reaction sub-diffusion equations. By using the nonuniform L1 scheme and nonuniform Alikhanov scheme in the temporal domain, we formulated two efficient numerical schemes, where the second order difference method was used in the spatial dimension. Furthermore, we derived the stability and convergence of these proposed schemes. Remarkably, both derived numerical methods exhibited α-robustness, that is, it remained valid when α → 1⁻. Numerical experiments were given to demonstrate the theoretical statements.

Keywords

Caputo-Hadamard derivative / reaction subdiffusion equations / the nonuniform L1 scheme / the nonuniform Alikhanov scheme / α-robust error estimates

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] K. Diethelm, The Analysis of Fractional Differential Equations , Springer-Verlag, 2010. https://doi.org/10.1007/978-3-642-14574-2

[2] A. Kilbas, H Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations , Elsevier, 2006. https://doi.org/10.3182/20060719-3-PT-4902.00008

[3] K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations , Wiley, 1993.

[4] I. Podlubny, Fractional Differential Equations , Acad. Press, 1999.

[5] R. Garra, F. Mainardi, G. Spada, A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus, Chaos Solitons Fractals, 102 (2017), 333-338. https://doi.org/10.1016/j.chaos.2017.03.032

[6] C. Lomnitz, Application of the logarithmic creep law to stress wave attenuation in the solid earth, J. Geophys. Res., 67 (1962), 365-368. https://doi.org/10.1029/JZ067i001p00365

[7] S. Denisov, H. Kantz, Continuous-time random walk theory of superslow diffusion, Europhys. Lett., 92 (2010), 30001. https://doi.org/10.1209/0295-5075/92/30001

[8] J. Dräger, J. Klafter, Strong anomaly in diffusion generated by iterated maps, Phys. Rev. Lett., 84 (2000), 5998-6001. https://doi.org/10.1103/PhysRevLett.84.5998

[9] F. Iglói, L. Turban, H. Rieger, Anomalous diffusion in aperiodic environments, Phys. Rev. E, 59 (1999), 1465. https://doi.org/10.1103/PhysRevE.59.1465

[10] L. Sanders, M. Lomholt, L. Lizana, K. Fogelmark, R. Metzler, T. Ambjornsson, Severe slowing-down and universality of the dynamics in disordered interacting many-body systems: Ageing and ultraslow diffusion, New J. Phys, 16 (2014), 113050. https://doi.org/10.1088/1367-2630/16/11/113050

[11] T. Sandev, A. Iomin, H. Kantz, R. Metzler, A. Chechkin, Comb model with slow and ultraslow diffusion, Math. Model. Nat. Phenom., 11 (2016), 18-33. https://doi.org/10.1051/mmnp/201611302

[12] C. Li, D. Li, J. Wang, L1/LDG method for the generalized time-fractional Burgers equation, Commun. Appl. Math. Comput., 5 (2023), 1299-1322. https://doi.org/10.1016/j.matcom.2021.03.005

[13] E. Fan, C. Li, Z. Li, Numerical approaches to Caputo-Hadamard fractional derivatives with applications to long-term integration of fractional differential systems, Commun. Nonlinear Sci. Numer. Simul., 106 (2022), 106096. https://doi.org/10.1016/j.cnsns.2021.106096

[14] T. Zhao, C. Li, D Li, Efficient spectral collocation method for fractional differential equation with Caputo-Hadamard derivative, Fract. Calc. Appl. Anal., 26 (2023), 2903-2927. https://doi.org/10.1007/s13540-023-00216-6

[15] X. Ye, J. Cao, C. Xu, A high order scheme for fractional differential equations with the Caputo-Hadamard derivative, J. Comput. Math., 434 (2025), 615-640. https://doi.org/10.4208/jcm.2312-m2023-0098

[16] 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

[17] M. Stynesy, E. O’riordanz, Z. Graciax, 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

[18] H. Liao, W Mclean, J. Zhang, A second-order scheme with nonuniform time steps for a linear reaction-sudiffusion problem, Commun. Comput. Phys., 30 (2021), 567-601. https://doi.org/10.4208/cicp.OA-2020-0124

[19] H. Liao, W Mclean, J. Zhang, A discrete Gronwall inequality with applications to numerical schemes for subdiffusion problems, SIAM J. Numer. Anal., 57 (2019), 218-237. https://doi.org/10.1137/16M1175742

[20] B. Zhou, X. Chen, D. Li, Nonuniform Alikhanov linearized Galerkin finite element methods for nonlinear time-fractional parabolic equations, J. Sci. Comput., 85 (2020), 39. https://doi.org/10.1007/s10915-020-01350-6

[21] X. Yang, H. Zhang, The uniform l¹ long-time behavior of time discretization for time-fractional partial differential equations with nonsmooth data , Appl. Math. Lett., 124 (2022), 107644. https://doi.org/10.1016/j.aml.2021.107644

[22] D. Hou, M. Azaiez, C. Xu, Müntz spectral method for two-dimensional space-fractional convection-diffusion equation, Commun. Comput. Phys., 26 (2019), 1415-1443. https://doi.org/10.4208/cicp.2019.js60.04

[23] X. Yang, W. Qiu, H. Chen, H. Zhang, Second-order BDF ADI Galerkin finite element method for the evolutionary equation with a nonlocal term in three-dimensional space, Appl. Numer. Math., 172 (2022), 497-513. https://doi.org/10.1016/j.apnum.2021.11.004

[24] Z. Sun, L. Ling, A kernel-based meshless conservative Galerkin method for solving Hamiltonian wave equations, SIAM J. Sci. Comput., 44 (2022), A2789-A2807. https://doi.org/10.1137/21M1436981

[25] H. Zhang, Y. Liu, X. Yang, An efficient ADI difference scheme for the nonlocal evolution problem in three-dimensional space, J. Appl. Math. Comput., 69 (2023), 651-674. https://doi.org/10.1007/s12190-022-01760-9

[26] X. Yang, Z. Zhang, Superconvergence analysis of a robust orthogonal Gauss collocation method for 2D fourth-order subdiffusion equations, J. Sci. Comput., 100 (2024), 62. https://doi.org/10.1007/s10915-024-02616-z

[27] Z. Sun, L. Ling, A high-order meshless linearly implicit energy-preserving method for nonlinear wave equations on Riemannian manifolds, SIAM J. Sci. Comput., 46 (2024), A3779-A3802. https://doi.org/10.1137/24M1654245

[28] C. Li, Z. Li, Z. Wang, Mathematical analysis and the local discontinuous Galerkin method for Caputo-Hadamard fractional partial differential equation, J. Sci. Comput., 85 (2020), 41. https://doi.org/10.1007/s10915-020-01353-3

[29] Z. Wang, C. Ou, S. Vong, A second-order scheme with nonuniform time grids for Caputo-Hadamard fractional sub-diffusion equations, J. Comput. Appl. Math., 414 (2022), 114448. https://doi.org/10.1016/j.cam.2022.114448

[30] J. Zhang, Z. Zhang, C. Zhao, α-robust error estimates of general non-uniform time-step numerical schemes for reaction-subdiffusion problems, preprint, arXiv:2305.07383. https://doi.org/10.48550/arXiv.2305.07383

[31] Z. Yang, X. Zheng, H. Wang, Well-posedness and regularity of Caputo-Hadamard fractional stochastic differential equations, Z. Angew. Math. Phys., 72 (2021), 141. https://doi.org/10.1007/s00033-021-01566-y