An implicit difference scheme for the fourth-order nonlinear non-local PIDEs with a weakly singular kernel
An implicit difference scheme for the fourth-order nonlinear non-local PIDEs with a weakly singular kernel
复制标题
弱奇异核四阶非线性非局部PIDE的隐式差分格式
DOI：
10.1007/s40314-022-02040-9
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发表时间：
2022-09
影响因子：
2.6
通讯作者：
Xiaoxuan Jiang
中科院分区：
文献类型：
--
作者：
Qingqing Tian;Haixiang Zhang;Xuehua Yang;Xiaoxuan Jiang
本文构造了具有弱奇异核的四阶非线性非局部偏积分-微分方程的隐式差分格式。时间方向上的Caputo导数项用l1离散化公式逼近。用一阶卷积正交来处理Riemann-Liouville （R-L）分数阶积分项。结合标准中心差分近似，建立了一种完全离散差分格式。对于非线性对流项，采用基于分段线性测试函数的伽辽金方法隐式处理，得到一个非线性代数方程组。对于四阶项，我们用泰勒积分余数展开来处理。然后，证明了数值解的存在性。分别在范数和范数上严格证明了数值解的稳定性和收敛性。同时证明了该非线性系统的唯一性。最后，我们介绍并比较了求解隐式差分格式的两种迭代算法。数值计算结果与理论分析相吻合。
In this paper, an implicit difference scheme is constructed for the fourth-order nonlinear non-local partial integro-differential equations (PIDEs) with a weakly singular kernel. The Caputo derivative term in temporal direction is approximated by L1-discretization formula. And the first-order convolution quadrature is used to deal with the Riemann–Liouville (R–L) fractional integral terms. By combining the standard central difference approximation, a fully discrete difference scheme is established. For the nonlinear convection term, the Galerkin method based on piecewise linear test functions is used to handle it implicitly and attain a system of nonlinear algebraic equations. For the fourth-order term, we use the Taylor expansion with integral remainder to deal with it. Then, the existence of the numerical solutions is proved. The stability and the convergence of the numerical solutions are strictly proved in the-norm and-norm, respectively. The uniqueness is also proved for the nonlinear system. Finally, we introduce and compare two iterative algorithms of solving the implicit difference scheme. And the numerical results are consistent with the theoretical analysis.
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DOI：
10.22436/jnsa.010.07.06
发表时间：
2017-07
期刊：
The Journal of Nonlinear Sciences and Applications
影响因子：
--
作者：
A. Yokuş;D. Kaya
通讯作者：
A. Yokuş;D. Kaya
影响因子：
2.9
作者：
J. Sanz-Serna
通讯作者：
J. Sanz-Serna
DOI：
--
发表时间：
2008
期刊：
Journal of Systems Science and Mathematical Sciences
影响因子：
--
作者：
Chen Hong-bin
通讯作者：
Chen Hong-bin
DOI：
10.1016/j.cpc.2011.04.013
发表时间：
2011-08
期刊：
Comput. Phys. Commun.
影响因子：
--
作者：
Xiuling Hu;Lu-ming Zhang
通讯作者：
Xiuling Hu;Lu-ming Zhang
影响因子：
3
作者：
Zhi-yuan Li;Yuri Luchko;Masahiro Yamamoto
通讯作者：
Zhi-yuan Li;Yuri Luchko;Masahiro Yamamoto