四阶非线性时间分数阶方程的混合虚拟元方法
The Mixed Virtual Element Method for Nonlinear Fourth-Order Time-Fractional Equations
摘要: 本文主要研究二维非线性四阶时间分数阶方程的混合虚拟元解法。通过引入辅助变量，将四阶时间分数阶方程转化为低阶的耦合方程。在时间方向上采用广义BDF2-θ得到半离散格式，进而在空间方向上应用混合虚拟元方法得到该方程的全离散格式。本文将给出具体的数值求解过程，并通过一个二维数值算例验证该算法的有效性和收敛性。
Abstract: This article primarily investigates the mixed virtual element method of two-dimensional nonlinear fourth-order time-fractional equations. By introducing an auxiliary variable, the fourth-order time-fractional equation is transformed into lower-order coupled equations. The semi-discrete scheme is obtained by employing the generalized BDF2-θ in the temporal direction. Subsequently, a fully discrete scheme for the equation is achieved by applying the mixed virtual element method in the spatial direction. This article will present the specific numerical process and validate the effectiveness and convergence of algorithm through a two-dimensional numerical example.
文章引用：肖文娟, 侯耀. 四阶非线性时间分数阶方程的混合虚拟元方法[J]. 应用数学进展, 2025, 14(4): 755-764. https://doi.org/10.12677/aam.2025.144203

参考文献

[1] Deng, W. (2007) Numerical Algorithm for the Time Fractional Fokker-Planck Equation. Journal of Computational Physics, 227, 1510-1522. [Google Scholar] [CrossRef] 
[2] Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J.J. (2012) Fractional Calculus: Models and Numerical Methods. World Scientific Publishing Co. Pte. Ltd. [Google Scholar] [CrossRef] 
[3] Gatto, P. and Hesthaven, J.S. (2014) Numerical Approximation of the Fractional Laplacian via hp-Finite Elements, with an Application to Image Denoising. Journal of Scientific Computing, 65, 249-270. [Google Scholar] [CrossRef] 
[4] Kirchner, J.W., Feng, X. and Neal, C. (2000) Fractal Stream Chemistry and Its Implications for Contaminant Transport in Catchments. Nature, 403, 524-527. [Google Scholar] [CrossRef] [PubMed]
[5] Lin, Y. and Xu, C. (2007) Finite Difference/spectral Approximations for the Time-Fractional Diffusion Equation. Journal of Computational Physics, 225, 1533-1552. [Google Scholar] [CrossRef] 
[6] Li, C. and Zhao, S. (2016) Efficient Numerical Schemes for Fractional Water Wave Models. Computers & Mathematics with Applications, 71, 238-254. [Google Scholar] [CrossRef] 
[7] Yuan, W., Chen, Y. and Huang, Y. (2020) A Local Discontinuous Galerkin Method for Time-Fractional Burgers Equations. East Asian Journal on Applied Mathematics, 10, 818-837. [Google Scholar] [CrossRef] 
[8] Liu, Y., Yu, Z., Li, H., Liu, F. and Wang, J. (2018) Time Two-Mesh Algorithm Combined with Finite Element Method for Time Fractional Water Wave Model. International Journal of Heat and Mass Transfer, 120, 1132-1145. [Google Scholar] [CrossRef] 
[9] Hu, X. and Zhang, L. (2012) On Finite Difference Methods for Fourth-Order Fractional Diffusion-Wave and Subdiffusion Systems. Applied Mathematics and Computation, 218, 5019-5034. [Google Scholar] [CrossRef] 
[10] Ran, M. and Lei, X. (2021) A Fast Difference Scheme for the Variable Coefficient Time-Fractional Diffusion Wave Equations. Applied Numerical Mathematics, 167, 31-44. [Google Scholar] [CrossRef] 
[11] Fei, M. and Huang, C. (2019) Galerkin-Legendre Spectral Method for the Distributed-Order Time Fractional Fourth-Order Partial Differential Equation. International Journal of Computer Mathematics, 97, 1183-1196. [Google Scholar] [CrossRef] 
[12] Huang, J., Qiao, Z., Zhang, J., Arshad, S. and Tang, Y. (2020) Two Linearized Schemes for Time Fractional Nonlinear Wave Equations with Fourth-Order Derivative. Journal of Applied Mathematics and Computing, 66, 561-579. [Google Scholar] [CrossRef] 
[13] Liu, Y., Du, Y., Li, H., Li, J. and He, S. (2015) A Two-Grid Mixed Finite Element Method for a Nonlinear Fourth-Order Reaction-Diffusion Problem with Time-Fractional Derivative. Computers & Mathematics with Applications, 70, 2474-2492. [Google Scholar] [CrossRef] 
[14] Li, A. and Yang, Q. (2025) The Finite Volume Element Method for Time-Fractional Nonlinear Fourth-Order Diffusion Equation with Time Delay. Engineering, 17, 53-72. [Google Scholar] [CrossRef] 
[15] Du, Y., Liu, Y., Li, H., Fang, Z. and He, S. (2017) Local Discontinuous Galerkin Method for a Nonlinear Time-Fractional Fourth-Order Partial Differential Equation. Journal of Computational Physics, 344, 108-126. [Google Scholar] [CrossRef] 
[16] Guo, L., Wang, Z. and Vong, S. (2015) Fully Discrete Local Discontinuous Galerkin Methods for Some Time-Fractional Fourth-Order Problems. International Journal of Computer Mathematics, 93, 1665-1682. [Google Scholar] [CrossRef] 
[17] Wei, L. and He, Y. (2014) Analysis of a Fully Discrete Local Discontinuous Galerkin Method for Time-Fractional Fourth-Order Problems. Applied Mathematical Modelling, 38, 1511-1522. [Google Scholar] [CrossRef] 
[18] Beirão Da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D. and Russo, A. (2012) Basic Principles of Virtual Element Methods. Mathematical Models and Methods in Applied Sciences, 23, 199-214. [Google Scholar] [CrossRef] 
[19] 孟健, 梅立泉. 特征值问题虚拟元方法发展综述[J]. 工程数学学报, 2022, 39(6): 851-861.
[20] Gatica, G.N., Munar, M. and Sequeira, F.A. (2018) A Mixed Virtual Element Method for the Navier-Stokes Equations. Mathematical Models and Methods in Applied Sciences, 28, 2719-2762. [Google Scholar] [CrossRef] 
[21] Xu, Y., Zhou, Z. and Zhao, J. (2022) Conforming Virtual Element Methods for Sobolev Equations. Journal of Scientific Computing, 93, Article No. 32. [Google Scholar] [CrossRef] 
[22] Antonietti, P.F., da Veiga, L.B., Scacchi, S. and Verani, M. (2016) A Virtual Element Method for the Cahn—Hilliard Equation with Polygonal Meshes. SIAM Journal on Numerical Analysis, 54, 34-56. [Google Scholar] [CrossRef] 
[23] Zhang, Y. and Feng, M. (2022) The Virtual Element Method for the Time Fractional Convection Diffusion Reaction Equation with Non-Smooth Data. Computers & Mathematics with Applications, 110, 1-18. [Google Scholar] [CrossRef] 
[24] Li, M., Zhao, J., Huang, C. and Chen, S. (2019) Nonconforming Virtual Element Method for the Time Fractional Reaction-Subdiffusion Equation with Non-Smooth Data. Journal of Scientific Computing, 81, 1823-1859. [Google Scholar] [CrossRef] 
[25] Zhang, B. and Zhao, J. (2024) The Virtual Element Method with Interior Penalty for the Fourth-Order Singular Perturbation Problem. Communications in Nonlinear Science and Numerical Simulation, 133, Article ID: 107964. [Google Scholar] [CrossRef] 
[26] Zhang, Y. and Feng, M. (2022) A Mixed Virtual Element Method for the Time-Fractional Fourth-Order Sub-diffusion Equation. Numerical Algorithms, 90, 1617-1637.
[27] Yin, B., Liu, Y. and Li, H. (2020) A Class of Shifted High-Order Numerical Methods for the Fractional Mobile/immobile Transport Equations. Applied Mathematics and Computation, 368, Article ID: 124799. [Google Scholar] [CrossRef] 
[28] Gao, G., Sun, H. and Sun, Z. (2015) Stability and Convergence of Finite Difference Schemes for a Class of Time-Fractional Sub-Diffusion Equations Based on Certain Superconvergence. Journal of Computational Physics, 280, 510-528. [Google Scholar] [CrossRef] 
[29] Liu, Y., Du, Y., Li, H., Liu, F. and Wang, Y. (2018) Some Second-Order Schemes Combined with Finite Element Method for Nonlinear Fractional Cable Equation. Numerical Algorithms, 80, 533-555. [Google Scholar] [CrossRef] 
[30] Dar, Z.M. and Chandru, M. (2024) A Virtual Element Scheme for the Time-Fractional Parabolic PDEs over Distorted Polygonal Meshes. Alexandria Engineering Journal, 106, 611-619. [Google Scholar] [CrossRef] 
[31] Beirão da Veiga, L., Brezzi, F., Marini, L.D. and Russo, A. (2014) The Hitchhiker’s Guide to the Virtual Element Method. Mathematical Models and Methods in Applied Sciences, 24, 1541-1573. [Google Scholar] [CrossRef] 
[32] Yang, N. and Liu, Y. (2025) Mixed Finite Element Method for a Time-Fractional Generalized Rosenau-RLW-Burgers Equation. AIMS Mathematics, 10, 1757-1778. [Google Scholar] [CrossRef] 

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