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The TGCNMFE Method for the Generalized Nonlinear Time Fractional Fourth-Order Reaction Diffusion Equation
Research Article | Published: 29 June 2025
Journal of Numerical Simulations in Physics and Mathematics Volume 1, Issue 1, 2025: 18-31
Volume 1, Issue 1, Journal of Numerical Simulations in Physics and Mathematics
Volume 1, Issue 1, 2025
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Journal of Numerical Simulations in Physics and Mathematics, Volume 1, Issue 1, 2025: 18-31

Open Access | Research Article | 29 June 2025
The TGCNMFE Method for the Generalized Nonlinear Time Fractional Fourth-Order Reaction Diffusion Equation
by
Yuejie Li Yuejie Li 1
Zhendong Luo Zhendong Luo 2,3 *
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]
DOI: 10.62762/JNSPM.2025.256666
Received: 29 May 2025, Accepted: 18 June 2025, Published: 29 June 2025  
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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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Cite This Article
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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