Linearized $ L1 $-Galerkin method for variable order time-fractional Schrödinger equation with unconditional convergence

Boya Zhou; Shaohong Pan; Zhiwei Fang; Min Li; Boya Zhou; Shaohong Pan; Zhiwei Fang; Min Li

doi:10.3934/math.20251166

AIMS Mathematics

2025, Volume 10, Issue 11: 26527-26544. doi: 10.3934/math.20251166

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Linearized $ L1 $-Galerkin method for variable order time-fractional Schrödinger equation with unconditional convergence

1.
School of Mathematics, Foshan University, Foshan, 52800, Guangdong, China
2.
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China

Received: 01 July 2025 Revised: 18 September 2025 Accepted: 25 September 2025 Published: 17 November 2025

The nonlinear Schrödinger equation with a nonlocal operator plays an important role in quantum mechanics, and the time-fractional Schrödinger problems have been widely studied for the case of constant exponents. In this paper, we propose a linearized unconditionally convergent $ L1 $-Galerkin method to solve the variable-exponent fractional Schrödinger equations. The optimal error convergence of the fully discrete scheme is proved without any time-space step restriction condition, even when incorporating the influence of the nonlocal operator in the temporal direction. The proof relies critically on the Sobolev embedding theorem combined with the inverse inequality. The discrete fractional Grönwall inequality is also used to obtain the error estimates. Numerical experiments are given to verify our theoretical results.
Citation: Boya Zhou, Shaohong Pan, Zhiwei Fang, Min Li. Linearized $ L1 $-Galerkin method for variable order time-fractional Schrödinger equation with unconditional convergence[J]. AIMS Mathematics, 2025, 10(11): 26527-26544. doi: 10.3934/math.20251166

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Abstract

The nonlinear Schrödinger equation with a nonlocal operator plays an important role in quantum mechanics, and the time-fractional Schrödinger problems have been widely studied for the case of constant exponents. In this paper, we propose a linearized unconditionally convergent $ L1 $-Galerkin method to solve the variable-exponent fractional Schrödinger equations. The optimal error convergence of the fully discrete scheme is proved without any time-space step restriction condition, even when incorporating the influence of the nonlocal operator in the temporal direction. The proof relies critically on the Sobolev embedding theorem combined with the inverse inequality. The discrete fractional Grönwall inequality is also used to obtain the error estimates. Numerical experiments are given to verify our theoretical results.

References

[1]	H. G. Sun, A. Chang, Y. Zhang, W. Chen, A review on variable-order fractional differential equations: Mathematical foundations, physical models, numerical methods and applications, FCAA, 22 (2019), 27–59. https://doi.org/10.1515/fca-2019-0003 doi: 10.1515/fca-2019-0003
[2]	X. Guo, M. Xu, Some physical applications of fractional Schrödinger equation, J. Math. Phys., 47 (2006), 082104. https://doi.org/10.1063/1.2235026 doi: 10.1063/1.2235026
[3]	M. Naber, Time fractional Schrödinger equation, J. Math. Phys., 45 (2004), 3339–3352. https://doi.org/10.1063/1.1769611 doi: 10.1063/1.1769611
[4]	A. Tofighi, Probability structure of time fractional Schrödinger equation, Acta Phys. Pol. A, 116 (2009), 2114–118. https://doi.org/10.12693/APhysPolA.116.114 doi: 10.12693/APhysPolA.116.114
[5]	T. M. Atanackovic, B. Stankovic, D. Zorica, Fractional Calculus with Application in Mechanics: Vibrations and Diffusion Processes, London: John Wiley and Sons, 2014. https://doi.org/10.1002/9781118577530
[6]	N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298–305. https://doi.org/10.1016/S0375-9601(00)00201-2 doi: 10.1016/S0375-9601(00)00201-2
[7]	N. Laskin, Fractional quantum mechanics, Phys. Rev. E, 62 (2000), 3135. https://doi.org/10.1103/PhysRevE.62.3135 doi: 10.1103/PhysRevE.62.3135
[8]	B. Ross, S. Samko, Fractional integration operator of variable order in the Holder space, Int. J. Math. Math. Sci., 18 (1995), 777–788. https://doi.org/10.1155/S0161171295001001 doi: 10.1155/S0161171295001001
[9]	C. Coimbra, Mechanics with variable-order differential operators, Annalen der Physik, 12 (2003), 692–703. https://doi.org/10.1002/andp.200351511-1203 doi: 10.1002/andp.200351511-1203
[10]	F. Santamaria, S. Wils, E. D. Schutter, G. J. Augustine, Anomalous diffusion in purkinjecell dendrites caused by spines, Neuron, 52 (2006), 635–648. https://doi.org/10.1016/j.neuron.2006.10.025 doi: 10.1016/j.neuron.2006.10.025
[11]	S. G. Samko, B. Ross, Integration and differentiation to a variable fractional order, Integr. Transf. Spec. F, 4 (1993), 277–300. https://doi.org/10.1080/10652469308819027
[12]	C. F. Lorenzo, T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dyn., 29 (2002), 57–98. https://doi.org/10.1023/a:1016586905654 doi: 10.1023/a:1016586905654
[13]	H. Sun, W. Chen, Y. Chen, Variable-order fractional differential operators in anomalous diffusion modeling, Phys. A, 388 (2009), 4586–4592. https://doi.org/10.1016/j.physa.2009.07.024 doi: 10.1016/j.physa.2009.07.024
[14]	H. Wang, X. Zheng, Wellposedness and regularity of the variable-order time-fractional diffusion equations, J. Math. Anal. Appl., 475 (2019), 1778–1802. https://doi.org/10.1016/j.jmaa.2019.03.052 doi: 10.1016/j.jmaa.2019.03.052
[15]	R. Lin, F. Liu, V. Anh, I. Turner, Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation, Appl. Math. Comput., 212 (2009), 435–445. https://doi.org/10.1016/j.amc.2009.02.047 doi: 10.1016/j.amc.2009.02.047
[16]	X. Zhao, Z. Sun, G. E. Karniadakis, Second-order approximations for variable order fractional derivatives: algorithms and applications, J. Comput. Phys., 293 (2015), 184–200. https://doi.org/10.1016/j.jcp.2014.08.015 doi: 10.1016/j.jcp.2014.08.015
[17]	X. Zheng, H. Wang, Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions, IMA J. Numer. Anal., 41 (2021), 1522–1545. https://doi.org/10.1093/imanum/draa013. doi: 10.1093/imanum/draa013
[18]	M. A. Zaky, K. V. Bockstal, T. R. Taha, D. Suragan, A. S. Hendy, An $L1$ type difference/Galerkin spectral scheme for variable-order time-fractional nonlinear diffusion–reaction equations with fixed delay, J. Comput. Appl. Math., 420 (2023), 114832. https://doi.org/10.1016/j.cam.2022.114832 doi: 10.1016/j.cam.2022.114832
[19]	X. Zheng, H. Wang, An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes, SIAM J. Numer. Anal., 58 (2020), 330–352. https://doi.org/10.1137/19M1245621 doi: 10.1137/19M1245621
[20]	L. Wei, Y. Yang, Optimal order finite difference/local discontinuous Galerkin method for variable-order time-fractional diffusion equation, J. Comput. Appl. Math., 383 (2021), 113129. https://doi.org/10.1016/j.cam.2020.113129 doi: 10.1016/j.cam.2020.113129
[21]	R. Du, Z. Sun, H. Wang, Temporal second-order finite difference schemes for variable-order time-fractional wave equations, SIAM J. Numer. Analy., 60 (2022), 104–132. https://doi.org/10.1137/19m1301230 doi: 10.1137/19m1301230
[22]	J. Jia, H. Wang, X. Zheng, A fast algorithm for time-fractional diffusion equation with space-time-dependent variable order, Numer. Algor., 94 (2023), 1705–1730. https://doi.org/10.1007/s11075-023-01552-7 doi: 10.1007/s11075-023-01552-7
[23]	C. Huang, H. Chen, Superconvergence analysis of finite element methods for the variable-order subdiffusion equation with weakly singular solutions, Appl. Math. Lett., 139 (2023), 108559. https://doi.org/10.1016/j.aml.2022.108559 doi: 10.1016/j.aml.2022.108559
[24]	A. H. Bhrawy, M. A. Zaky, An improved collocation method for multi-dimensional space–time variable-order fractional Schrödinger equations, Appl. Numer. Math., 111 (2017), 197–218. https://doi.org/10.1016/j.apnum.2016.09.009 doi: 10.1016/j.apnum.2016.09.009
[25]	A. Atangana, A. H. Cloot, Stability and convergence of the space fractional variable-order Schrödinger equation, Adv. Diff. Equat., 2013 (2013), 80. https://doi.org/10.1186/1687-1847-2013-80 doi: 10.1186/1687-1847-2013-80
[26]	W. Sun, J. Wang, Optimal error analysis of Crank–Nicolson schemes for a coupled nonlinear Schrödinger system in 3D, J. Comput. Appl. Math., 317 (2017), 685–699. https://doi.org/10.1016/j.cam.2016.12.004 doi: 10.1016/j.cam.2016.12.004
[27]	D. Li, W. Cao, C. Zhang, Z. Zhang, Optimal error estimates of a linearized Crank–Nicolson Galerkin FEM for the Kuramoto–Tsuzuki equations, Commun. Comput. Phys., 26 (2019), 838–854. https://doi.org/10.4208/cicp.OA-2018-0208 doi: 10.4208/cicp.OA-2018-0208
[28]	B. Li, W. Sun, Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations, 2013. https://doi.org/10.48550/arXiv.1208.4698
[29]	B. Li, W. Sun, Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media, SIAM J. Numer. Anal., 51 (2013), 1959–1977. https://doi.org/10.1137/120871821 doi: 10.1137/120871821
[30]	W. Yuan, D. Li, C. Zhang. Linearized transformed L1 Galerkin FEMs with unconditional convergence for nonlinear time fractional Schrödinger equations, Numer. Math.: Theor. Meth. Appl., 16 (2023), 348–369. https://doi.org/10.4208/nmtma.OA-2022-0087 https://doi.org/10.4208/nmtma.OA-2022-0087 doi: 10.4208/nmtma.OA-2022-0087
[31]	D. Li, J. Wang, J. Zhang, Unconditionally convergent $L1$-Galerkin FEMs for nonlinear time-fractional Schrödinger equations, SIAM J. Sci. Comput., 39 (2017), A3067–A3088. https://doi.org/10.1137/16M1105700 doi: 10.1137/16M1105700
[32]	X. L. Li, H. Y. Dong, Unconditional error analysis of an element-free Galerkin method for the nonlinear Schrödinger equation, Commu. Nonlinear. Sci., 151 (2025), 109103. https://doi.org/10.1016/j.cnsns.2025.109103 doi: 10.1016/j.cnsns.2025.109103
[33]	S. Shen, F. Liu, J. Chen, I. Turner, V. Anh, Numerical techniques for the variable order time fractional diffusion equation, Appl. Math. Comput., 218 (2012), 10861–10870. https://doi.org/10.1016/j.amc.2012.04.047 doi: 10.1016/j.amc.2012.04.047
[34]	C. M. Chen, F. Liu, V. Anh, I. Turner, Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation, Math. Comput., 81 (2012), 345–366. https://doi.org/10.1090/s0025-5718-2011-02447-6 doi: 10.1090/s0025-5718-2011-02447-6
[35]	C. Huang, N. An, H. Chen, X. Yu, $\alpha$-Robust error analysis of two nonuniform schemes for subdiffusion equations with variable-order derivatives, J. Sci. Comput., 97 (2023), 43. https://doi.org/10.1007/s10915-023-02357-5 doi: 10.1007/s10915-023-02357-5
[36]	V. Thomée, Galerkin finite element methods for parabolic problems, Berlin, Heidelberg: Springer, 2007. https://doi.org/10.1007/978-3-662-03359-3

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