An improved finite difference and Galerkin spectral method for the fourth-order time fractional partial differential equations

Hanmei Jian; Junying Cao; Ziqiang Wang; Hanmei Jian; Junying Cao; Ziqiang Wang

doi:10.3934/math.20251202

AIMS Mathematics

2025, Volume 10, Issue 11: 27338-27363. doi: 10.3934/math.20251202

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Research article

An improved finite difference and Galerkin spectral method for the fourth-order time fractional partial differential equations

School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China

Received: 30 September 2025 Revised: 09 November 2025 Accepted: 14 November 2025 Published: 24 November 2025
MSC : 65L12, 65M06, 65M12

We investigated a numerical scheme for the fourth-order time fractional partial differential equations by employing the temporal $ L2 $ scheme and spatial Legendre-Galerkin spectral method. The stability and error estimation of the present scheme were proved by combining partial integration, the properties of the inverse Laplace operator with respect to initial values, and the right-hand function. The error estimation indicated that the present scheme has spectral accuracy convergence in space and ($ 3-\kappa $), $\kappa\in (0, 1)$, uniform accuracy in time. Three numerical examples verified the theoretical convergence order in time and space.
- fourth-order partial differential equation,
- high-order temporal accuracy,
- Legendre-Galerkin method,
- error estimation,
- stability analysis
Citation: Hanmei Jian, Junying Cao, Ziqiang Wang. An improved finite difference and Galerkin spectral method for the fourth-order time fractional partial differential equations[J]. AIMS Mathematics, 2025, 10(11): 27338-27363. doi: 10.3934/math.20251202

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Abstract

We investigated a numerical scheme for the fourth-order time fractional partial differential equations by employing the temporal $ L2 $ scheme and spatial Legendre-Galerkin spectral method. The stability and error estimation of the present scheme were proved by combining partial integration, the properties of the inverse Laplace operator with respect to initial values, and the right-hand function. The error estimation indicated that the present scheme has spectral accuracy convergence in space and ($ 3-\kappa $), $\kappa\in (0, 1)$, uniform accuracy in time. Three numerical examples verified the theoretical convergence order in time and space.

References

[1]	J. B. Greer, A. L. Bertozzi, G. Sapiro, Fourth order partial differential equations on general geometries, J. Comput. Phys., 216 (2006), 216–246. https://doi.org/10.1016/j.jcp.2005.11.031 doi: 10.1016/j.jcp.2005.11.031
[2]	X. Yang, W. Wang, Z. Zhou, H. Zhang, An efficient compact difference method for the fourth-order nonlocal subdiffusion problem, Taiwanese J. Math., 29 (2025), 35–66. https://doi.org/10.11650/tjm/240906 doi: 10.11650/tjm/240906
[3]	H. Tariq, G. Akram, Quintic spline technique for time fractional fourth-order partial differential equation, Numer. Methods Partial Differential Equations, 33 (2017), 445–466. https://doi.org/10.1002/num.22088 doi: 10.1002/num.22088
[4]	M. Fei, C. Huang, Galerkin-Legendre spectral method for the distributed-order time fractional fourth-order partial differential equation, Int. J. Comput. Math., 97 (2020), 1183–1196. https://doi.org/10.1080/00207160.2019.1608968 doi: 10.1080/00207160.2019.1608968
[5]	Y. Liu, Z. Fang, H. Li, S. He, A mixed finite element method for a time-fractional fourth-order partial differential equation, Appl. Math. Comput., 243 (2014), 703–717. https://doi.org/10.1016/j.amc.2014.06.023 doi: 10.1016/j.amc.2014.06.023
[6]	M. Dehghan, M. Safarpoor, Application of the dual reciprocity boundary integral equation approach to solve fourth-order time-fractional partial differential equations, Int. J. Comput. Math., 95 (2018), 2066–2081. https://doi.org/10.1080/00207160.2017.1365141 doi: 10.1080/00207160.2017.1365141
[7]	Y. Chen, J. Zhou, Error estimates of spectral Legendre-Galerkin methods for the fourth-order equation in one dimension, Appl. Math. Comput., 268 (2015), 1217–1226. https://doi.org/10.1016/j.amc.2015.06.082 doi: 10.1016/j.amc.2015.06.082
[8]	D. Xu, W. Qiu, J. Guo, A compact finite difference scheme for the fourth-order time-fractional integro-differential equation with a weakly singular kernel, Numer. Methods Partial Differential Equations, 36 (2020), 439–458. https://doi.org/10.1002/num.22436 doi: 10.1002/num.22436
[9]	M. Cui, Compact difference scheme for time-fractional fourth-order equation with first Dirichlet boundary condition, East Asian J. Appl. Math., 9 (2019), 45–66. https://doi.org/10.4208/eajam.260318.220618 doi: 10.4208/eajam.260318.220618
[10]	P. Roul, V. M. K. P. Goura, A high order numerical method and its convergence for time-fractional fourth order partial differential equations, Appl. Math. Comput., 366 (2020), 124727. https://doi.org/10.1016/j.amc.2019.124727 doi: 10.1016/j.amc.2019.124727
[11]	F. Fakhar-Izadi, Fully Petrov-Galerkin spectral method for the distributed-order time-fractional fourth-order partial differential equation, Eng. Comput., 37 (2021), 2707–2716. https://doi.org/10.1007/s00366-020-00968-2 doi: 10.1007/s00366-020-00968-2
[12]	F. Fakhar-Izadi, Fully spectral-Galerkin method for the one- and two- dimensional fourth-order time-fractional partial integro-differential equations with a weakly singular kernel, Numer. Methods Partial Differential Equations, 38 (2022), 160–176. https://doi.org/10.1002/num.22634 doi: 10.1002/num.22634
[13]	M. Haghi, M. Ilati, M. Dehghan, A fourth-order compact difference method for the nonlinear time-fractional fourth-order reaction-diffusion equation, Eng. Comput., 39 (2023), 1329–1340. https://doi.org/10.1007/s00366-021-01524-2 doi: 10.1007/s00366-021-01524-2
[14]	Y. Zhang, M. Feng, A mixed virtual element method for the time-fractional fourth-order subdiffusion equation, Numer. Algor., 90 (2022), 1617–1637. https://doi.org/10.1007/s11075-021-01244-0 doi: 10.1007/s11075-021-01244-0
[15]	D. A. Koç, A numerical scheme for time-fractional fourth-order reaction-diffusion model, J. Appl. Math. Comput. Mech., 22 (2023), 15–25. https://doi.org/10.17512/jamcm.2023.2.02 doi: 10.17512/jamcm.2023.2.02
[16]	Z. Wang, Numerical analysis of local discontinuous Galerkin method for the time-fractional fourth-order equation with initial singularity, Fractal Fract., 6 (2022), 206. https://doi.org/10.3390/fractalfract6040206 doi: 10.3390/fractalfract6040206
[17]	Y. Wang, S. Yi, A compact difference-Galerkin spectral method of the fourth-order equation with a time-fractional derivative, Fractal Fract., 9 (2025), 155. https://doi.org/10.3390/fractalfract9030155 doi: 10.3390/fractalfract9030155
[18]	M. Abbaszadeh, M. Dehghan, Direct meshless local Petrov-Galerkin (DMLPG) method for time-fractional fourth-order reaction-diffusion problem on complex domains, Comput. Math. Appl., 79 (2020), 876–888. https://doi.org/10.1016/j.camwa.2019.08.001 doi: 10.1016/j.camwa.2019.08.001
[19]	M. A. Abdelkawy, M. M. Babatin, A. M. Lopes, Highly accurate technique for solving distributed-order time-fractional-sub-diffusion equations of fourth order, Comput. Appl. Math., 39 (2020), 65. https://doi.org/10.1007/s40314-020-1070-7 doi: 10.1007/s40314-020-1070-7
[20]	J. Zheng, J. An, Spectral Galerkin approximation and error analysis based on a mixed scheme for fourth-order problems in complex regions, Numer. Methods Partial Differential Equations, 41 (2025), e23154. https://doi.org/10.1002/num.23154 doi: 10.1002/num.23154
[21]	J. Zheng, J. An, A Legendre spectral-Galerkin method for fourth-order problems in cylindrical regions, Numer. Methods Partial Differential Equations, 40 (2024), e23071. https://doi.org/10.1002/num.23071 doi: 10.1002/num.23071
[22]	X. Yu, B. Guo, Spectral method for fourth-order problems on quadrilaterals, J. Sci. Comput., 66 (2016), 477–503. https://doi.org/10.1007/s10915-015-0031-6 doi: 10.1007/s10915-015-0031-6
[23]	H. Fakhari, A. Mohebbi, Galerkin spectral and finite difference methods for the solution of fourth-order time fractional partial integro-differential equation with a weakly singular kernel, J. Appl. Math. Comput., 70 (2024), 5063–5080. https://doi.org/10.1007/s12190-024-02173-6 doi: 10.1007/s12190-024-02173-6
[24]	Y. Du, Y. Liu, H. Li, Z. Fang, S. He, Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation, J. Comput. Phys., 344 (2017), 108–126. https://doi.org/10.1016/j.jcp.2017.04.078 doi: 10.1016/j.jcp.2017.04.078
[25]	S. S. Siddiqi, S. Arshed, Numerical solution of time-fractional fourth-order partial differential equations, Inter. J. Comput. Math., 92 (2015), 1496–1518. https://doi.org/10.1080/00207160.2014.948430 doi: 10.1080/00207160.2014.948430
[26]	H. Zhu, C. 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
[27]	F. Fakhar-Izadi, N. Shabgard, Time-space spectral Galerkin method for time-fractional fourth-order partial differential equations, J. Appl. Math. Comput., 68 (2022), 4253–4272. https://doi.org/10.1007/s12190-022-01707-0 doi: 10.1007/s12190-022-01707-0
[28]	J. Cao, Z. Wang, Z. Wang, A uniform accuracy high-order finite difference and FEM for optimal problem governed by time-fractional diffusion equation, Fractal Fract., 6 (2022), 475. https://doi.org/10.3390/fractalfract6090475 doi: 10.3390/fractalfract6090475
[29]	J. Shen, Efficient spectral-Galerkin method Ⅰ. Direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 1489–1505. https://doi.org/10.1137/0915089 doi: 10.1137/0915089
[30]	Y. Lin, C. 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
[31]	H. Qiao, A. Cheng, A fast high order method for time fractional diffusion equation with non-smooth data, Discrete Contin. Dyn. Syst. Ser. B, 27 (2022), 903–920. https://doi.org/10.3934/dcdsb.2021073 doi: 10.3934/dcdsb.2021073
[32]	Z. Tao, B. Sun, Galerkin spectral method for a fourth-order optimal control problem with $H^1$-norm state constraint, Comput. Math. Appl., 97 (2021), 1–17. https://doi.org/10.1016/j.camwa.2021.05.023 doi: 10.1016/j.camwa.2021.05.023
[33]	J. Zhang, X. Yang, S. Wang, A three-layer FDM for the Neumann initial-boundary value problem of 2D Kuramoto-Tsuzuki complex equation with strong nonlinear effects, Commun. Nonlinear Sci. Numer. Simul., 152 (2026), 109255. https://doi.org/10.1016/j.cnsns.2025.109255 doi: 10.1016/j.cnsns.2025.109255
[34]	Z. Zhang, X. Yang, S. Wang, The alternating direction implicit difference scheme and extrapolation method for a class of three dimensional hyperbolic equations with constant coefficients, Electron. Res. Arch., 33 (2025), 3348–3377. https://doi.org/10.3934/era.2025148 doi: 10.3934/era.2025148
[35]	J. Zhang, X. Yang, S. Wang, The ADI difference and extrapolation scheme for high-dimensional variable coefficient evolution equations, Electron. Res. Arch., 33 (2025), 3305–3327. https://doi.org/10.3934/era.2025146 doi: 10.3934/era.2025146
[36]	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 doi: 10.1007/s10915-024-02616-z
[37]	M. Abdelhakem, D. Abdelhamied, M. El-Kady, Y. H. Youssri, Two modified shifted Chebyshev-Galerkin operational matrix methods for even-order partial boundary value problems, Bound. Value Probl., 2025 (2025), 34. https://doi.org/10.1186/s13661-025-02021-x doi: 10.1186/s13661-025-02021-x
[38]	A. G. Atta, J. F. Soliman, E. W. Elsaeed, M. W. Elsaeed, Y. H. Youssri, Spectral collocation algorithm for the fractional Bratu equation via Hexic shifted Chebyshev polynomials, Comput. Methods Differ. Equ., 13 (2025), 1102–1116. https://doi.org/10.22034/cmde.2024.61045.2621 doi: 10.22034/cmde.2024.61045.2621
[39]	Y. H. Youssri, S. M. Sayed, Efficient spectral collocation algorithm for pantograph integro-differential equations involving delay and weakly singular kernels, J.Umm Al-Qura Univ. Appll. Sci., 2025. https://doi.org/10.1007/s43994-025-00274-x doi: 10.1007/s43994-025-00274-x
[40]	Y. H. Youssri, A. G. Atta, A spectral Tau method based on Lucas polynomial approximation for solving the nonlinear fractional Riccati equation, Comput. Methods Differ. Equ., 2025, In press. https://doi.org/10.22034/cmde.2025.67681.3237
[41]	S. G. Kamel, A. G. Atta, Y. H. Youssri, An explicit spectral Tau method with Delannoy polynomials for the time-fractional diffusion equation, Open. J. Math. Anal., 9 (2025), 134–144. https://doi.org/10.30538/psrp-oma2025.0169 doi: 10.30538/psrp-oma2025.0169
[42]	A. M. Abbas, Y. H. Youssri, M. El-Kady, M. Abdelhakem, Cutting-edge spectral solutions for differential and integral equations utilizing Legendre's derivatives, Iran. J. Numer. Anal. Optim., 15 (2025), 1036–1074. https://doi.org/10.22067/ijnao.2025.91804.1590 doi: 10.22067/ijnao.2025.91804.1590

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