二维非线性四阶分数阶波动方程的BDF2-WSGI有限元算法
BDF2-WSGI Finite Element Algorithm for a Two-Dimensional Nonlinear Fourth-Order Fractional Wave Equation
摘要: 本文主要研究了二维非线性四阶分数阶波动方程的有效数值算法。通过结合二阶BDF2-WSGI时间离散格式与有限元方法对二维非线性四阶分数阶方程进行求解。首先，引入辅助变量，将分数阶四阶波动问题转化为低阶耦合方程，然后利用Riemann-Liouville分数阶积分对所得方程进行积分，最后使用WSGI逼近公式逼近分数阶积分，形成二阶BDF2有限元格式。本文给出了详细的数值算法，并通过一个二维算例进行了数值试验，验证了算法的有效性和收敛性。
Abstract: This article mainly studies effective numerical algorithms for two-dimensional nonlinear fourth-order fractional wave equations. We combine the second-order BDF2-WSGI time discretization scheme with the finite element method to solve two-dimensional nonlinear fourth-order fractional equations. Firstly, introducing auxiliary variables transforms the fractional fourth-order wave problem into a low-order coupled equation. Then, the Riemann-Liouville fractional integration is used to integrate the resulting equation. Finally, the WSGI approximation formula approximates the fractional integration, forming a second-order BDF2 finite element scheme. This article provides a detailed numerical algorithm and conducts numerical experiments on a two-dimensional example to verify the effectiveness and convergence of the algorithm.
文章引用：刘心愿. 二维非线性四阶分数阶波动方程的BDF2-WSGI有限元算法[J]. 应用数学进展, 2024, 13(4): 1217-1225. https://doi.org/10.12677/aam.2024.134112

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