一类四阶非线性时间分数阶双曲方程的L2-1σ 有限元方法

王岩; WANG Yan; 杨益宁; YANG Yining

doi:10.48014/fcpm.20240410001

一类四阶非线性时间分数阶双曲方程的L2-1σ 有限元方法

L2-1σ Finite Element Method for A Nonlinear Time Fractional Fourth-order Hyperbolic Equation

中国理论数学前沿 2024年第2卷第2期页码[16-26] 下载全文[4.8MB] [HTML]

摘要

本文针对一类四阶非线性时间分数阶双曲方程构造了一种具有时空二阶精度的数值算法。首先引入一个低阶参数β=α-1将一个α ∈ (1, 2) 的Caputo分数阶导数降为β∈ (0, 1) 的Caputo分数阶导数, 其次引入两个辅助函数变量v=ut , ζ=Δu将四阶非线性时间分数阶双曲方程转化为一个包含积分项的低阶非线性耦合系统。在时间方向上利用L2-1σ公式对其分数阶导数项进行离散, 同时利用复化梯形公式离散积分项, 在空间上采用传统Galerkin有限元方法, 进一步得到该低价非线性耦合系统的全离散有限元格式。以二维区域上的均匀矩形剖分为例, 利用双线性基函数给出了刚度矩阵和荷载矩阵的具体形式, 得到了全离散数值格式的详细算法实现过程。最后分别对一维和二维算例进行数值模拟, 对比于原方程该低阶系统能够同时观察三个未知量u, v, ζ的变化, 并且详细地列出不同参数和不同时空步长下这三个未知量的误差、图像和收敛阶, 充分地验证了该算法的可行性和有效性。

Abstract

In this paper, a numerical algorithm with space-time second-order accuracy is constructed for a fourth-order nonlinear time fractional hyperbolic equation. Firstly, a low order parameter β =α-1 is introduced to reduce the Caputo fractional derivative ofα ∈ 1, 2 to the Caputo fractional derivative of β ∈ 0, 1 . Secondly, two auxiliary variables v=ut , ζ=Δu are introduced to transform. the fourth-order nonlinear time fractional hyperbolic equation into a low-order nonlinear coupled system with integral term. The fractional derivative term is discretized in the time direction by L2-1σ formula, and the integral term is discretized by the complex trapezoidal formula. Using the traditional Galerkin finite element method in space, the fully discrete finite element scheme of the low-order nonlinear coupling system is obtained. Taking the uniform. rectangular division on a two-dimensional region as an example, the specific forms of stiffness matrix and load matrix are given by the bilinear basis function, and the detailed algorithm implementation process of the fully discrete numerical scheme is obtained. Finally, numerical simulations are carried out for one-dimensional and two-dimensional examples. Compared with the original equation, the low-order system can observe the changes of three unknown variables u, v, ζ at the same time, and the errors, images and convergence orders of the three unknown variables are listed in detail under different parameters and different space-time steps, which fully verifies the feasibility and effectiveness of the algorithm.

DOI

10.48014/fcpm.20240410001

文章类型

研究性论文

收稿日期

2024-04-10

接收日期

2024-04-19

出版日期

2024-06-28

关键词

四阶非线性时间分数阶双曲方程, L2-1σ, 有限元方法

Keywords

Fourth-order nonlinear time fractional hyperbolic equation, L2-1σ, Finite element method

作者

王岩, 杨益宁^*

Author

WANG Yan, YANG Yining^*

所在单位

内蒙古大学数学科学学院, 呼和浩特 010021

Company

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

浏览量

547

下载量

339

基金项目

内蒙古自治区高校创新研究团队计划(NMGIRT2413)资助

参考文献

[1] Nandal S, Pandey DN. Numerical solution of non-linear fourth order fractional sub-diffusion wave equation with time delay[J]. Applied Mathematics and Computation, 2020, 369: 124900.
https://doi.org/10.1016/j.amc.2019.124900.
[2] Ran MH, Lei XJ. A fast difference scheme for the variable coefficient time-fractional diffusion wave equations[J]. Applied Numerical Mathematics, 2021, 167: 31-44.
https://doi.org/10.1016/j.apnum.2021.04.021.
[3] Huang JF, Qiao Z, Zhang JN, Arshad S, Tang YF. Two linearized schemes for time fractional nonlinear wave equations with fourth-order derivative[J]. Journal of Applied Mathematics and Computing, 2021, 66: 561-79.
https://doi.org/10.1007/s12190-020-01449-x.
[4] Liu N, Liu Y, Li H, Wang JF. Time second-order finite difference/finite element algorithm for nonlinear timefractional diffusion problem with fourth-order derivative term[J]. Computers & Mathematics with Applications, 2018, 75: 3521-36.
https://doi.org/10.1016/j.camwa.2018.02.014.
[5] Liu Y, Du YW, Li H, He S, Gao W. Finite difference/finite element method for a nonlinear time-fractional fourthorder reaction-diffusion problem[J]. Computers & Mathematics with Applications, 2015, 70(4): 573-91.
https://doi.org/10.1016/j.camwa.2015.05.015.
[6] Liu Y, Du YW, Li H, Li JC, He S. A two-grid mixed finite element method for a nonlinear fourth-order reaction- diffusion problem with time-fractional derivative[J]. Computers & Mathematics with Applications, 2015, 70: 2474-92.
https://doi.org/10.1016/j.camwa.2015.09.012.
[7] Yang XH, Wu. LJ, Zhang. HX. A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity[J]. Applied Mathematics and Computation, 2023, 457: 128192.
https://doi.org/10.1016/j.amc.2023.128192.
[8] Wang JF, Yin BL, Liu Y, Li H, Fang ZC. A mixed element algorithm based on the modified L1 Crank-Nicolson scheme for a nonlinear fourth-order fractional diffusion- wave model[J]. Fractal and Fractional, 2021, 5: 274.
https://doi.org/10.3390/fractalfract5040274.
[9] Habibirad A, Hesameddini E, Shekari Y. A suitable hybrid meshless method for the numerical solution of timefractional fourth-order reaction-diffusion model in the multi-dimensional case[J]. Engineering Analysis with Boundary Elements, 2022, 145: 149-160.
https://doi.org/10.1016/j.enganabound.2022.09.007.
[10] Alikhanov AA. A new difference scheme for the time fractional diffusion equation[J]. Journal of Computational Physics, 2015, 280: 424-438.
https://doi.org/10.1016/j.jcp.2014.09.031.
[11] Wang JR, Liu Y, Wen C, Li H. Efficient numerical algorithm with the second-order time accuracy for a two-dimensional nonlinear fourth-order fractional wave equation[J]. Results in Applied Mathematics, 2022, 14: 100264.
https://doi.org/10.1016/j.rinam.2022.100264.
[12] Wang Y, Yang YN, Wang JF, Li H, Liu Y. Unconditional analysis of the linearized second-order time-stepping scheme combined with a mixed element method for a nonlinear time fractional fourth-order wave equation[J]. Computers & Mathematics with Applications, 2024, 157: 74-91.
https://doi.org/10.1016/j.camwa.2023.12.023.
[13] Fakhar-Izadi F. Fully petrov-galerkin spectral method for the distributed-order time-fractional fourth-order partial differential equation[J]. Engineering with computers[ J], 2021, 37: 2707-2716.
https://doi.org/10.1007/s00366-020-00968-2.
[14] Santra S, Behera R. A novel higher-order numerical method for parabolic integro-fractional differential equations based on wavelets and L2-1σscheme[J]. 2023, arXiv, arXiv: 2304. 08009.
https://doi.org/10.48550/arXiv.2304.08009.

引用本文

王岩, 杨益宁. 一类四阶非线性时间分数阶双曲方程的L2-1σ 有限元方法[J]. 中国理论数学前沿, 2024, 2(2): 16-26.

Citation

WANG Yan, YANG Yining. L2-1σ finite element method for a nonlinear time fractional fourthorder hyperbolic equation[J]. Frontiers of Chinese Pure Mathematics, 2024, 2(2): 16-26.