Research Article
Open Access
The TGCNMFE Method for the Generalized Nonlinear Time Fractional Fourth-Order Reaction Diffusion Equation
1
School of Mathematics and Computer Engineering, Ordos Institute of Technology, Ordos 017000, China
2
School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
3
Academician Expert Workstation, Hunan Sany Polytechnic College, Changsha 410129, China
* Corresponding Author:
Zhendong Luo,
[email protected]
Volume 1, Issue 1
2025
Received: 29 May 2025
Accepted: 18 June 2025
Published: 29 June 2025
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Article Information
Volume/Issue
Volume 1, Issue 1, 2025
Pages
18-31
Abstract
Herein, we mainly focus on developing a new two-grid Crank-Nicolson (CN) mixed finite element (MFE) (TGCNMFE) method for the generalized nonlinear time fractional fourth-order reaction diffusion equation. To do so, by introducing an auxiliary function, the nonlinear time fractional fourth-order reaction diffusion equation is first split into two second-order nonlinear equations. Thereafter, a new time semi-discrete mixed CN (TSDMCN) scheme is constructed through discretizing the time derivative and time fractional derivative by the CN difference quotient, and the existence, steadiness, and errors of the TSDMCN solutions are analysed. Next, a new TGCNMFE method is developed through using two-grid MFE technique to discretize the spacial variables, and the existence, steadiness, and error estimations for the TGCNMFE solutions are discussed. Lastly, the correctness of theory results and the superiority of the TGCNMFE method are verified by some numerical experiments.
Keywords
numerical simulations
finite element method
finite difference scheme
finite volume element method
Data Availability Statement
Data will be made available on request.
Funding
This work was supported by the National Natural Science Foundation of China under Grant 11671106.
Conflicts of Interest
The authors declare no conflicts of interest.
Ethical Approval and Consent to Participate
Not applicable.
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APA Style
Li, Y., & Luo, Z. (2025). The TGCNMFE Method for the Generalized Nonlinear Time Fractional Fourth-Order Reaction Diffusion Equation. Journal of Numerical Simulations in Physics and Mathematics, 1(1), 18-31. https://doi.org/10.62762/JNSPM.2025.256666
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TY - JOUR AU - Li, Yuejie AU - Luo, Zhendong PY - 2025 DA - 2025/06/29 TI - The TGCNMFE Method for the Generalized Nonlinear Time Fractional Fourth-Order Reaction Diffusion Equation JO - Journal of Numerical Simulations in Physics and Mathematics T2 - Journal of Numerical Simulations in Physics and Mathematics JF - Journal of Numerical Simulations in Physics and Mathematics VL - 1 IS - 1 SP - 18 EP - 31 DO - 10.62762/JNSPM.2025.256666 UR - https://www.icck.org/article/abs/JNSPM.2025.256666 KW - numerical simulations KW - finite element method KW - finite difference scheme KW - finite volume element method AB - Herein, we mainly focus on developing a new two-grid Crank-Nicolson (CN) mixed finite element (MFE) (TGCNMFE) method for the generalized nonlinear time fractional fourth-order reaction diffusion equation. To do so, by introducing an auxiliary function, the nonlinear time fractional fourth-order reaction diffusion equation is first split into two second-order nonlinear equations. Thereafter, a new time semi-discrete mixed CN (TSDMCN) scheme is constructed through discretizing the time derivative and time fractional derivative by the CN difference quotient, and the existence, steadiness, and errors of the TSDMCN solutions are analysed. Next, a new TGCNMFE method is developed through using two-grid MFE technique to discretize the spacial variables, and the existence, steadiness, and error estimations for the TGCNMFE solutions are discussed. Lastly, the correctness of theory results and the superiority of the TGCNMFE method are verified by some numerical experiments. SN - 3068-9082 PB - Institute of Central Computation and Knowledge LA - English ER -
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@article{Li2025The,
author = {Yuejie Li and Zhendong Luo},
title = {The TGCNMFE Method for the Generalized Nonlinear Time Fractional Fourth-Order Reaction Diffusion Equation},
journal = {Journal of Numerical Simulations in Physics and Mathematics},
year = {2025},
volume = {1},
number = {1},
pages = {18-31},
doi = {10.62762/JNSPM.2025.256666},
url = {https://www.icck.org/article/abs/JNSPM.2025.256666},
abstract = {Herein, we mainly focus on developing a new two-grid Crank-Nicolson (CN) mixed finite element (MFE) (TGCNMFE) method for the generalized nonlinear time fractional fourth-order reaction diffusion equation. To do so, by introducing an auxiliary function, the nonlinear time fractional fourth-order reaction diffusion equation is first split into two second-order nonlinear equations. Thereafter, a new time semi-discrete mixed CN (TSDMCN) scheme is constructed through discretizing the time derivative and time fractional derivative by the CN difference quotient, and the existence, steadiness, and errors of the TSDMCN solutions are analysed. Next, a new TGCNMFE method is developed through using two-grid MFE technique to discretize the spacial variables, and the existence, steadiness, and error estimations for the TGCNMFE solutions are discussed. Lastly, the correctness of theory results and the superiority of the TGCNMFE method are verified by some numerical experiments.},
keywords = {numerical simulations, finite element method, finite difference scheme, finite volume element method},
issn = {3068-9082},
publisher = {Institute of Central Computation and Knowledge}
}
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