An Implicit Difference Scheme for the Fourth-Order Nonlinear Evolution Equation with Multi-Term Riemann–Liouvile Fractional Integral Kernels
An Implicit Difference Scheme for the Fourth-Order Nonlinear Evolution Equation with Multi-Term Riemann–Liouvile Fractional Integral Kernels
复制标题
具有多项Riemann-Liouvile分数阶积分核的四阶非线性演化方程的隐式差分格式
DOI：
10.3390/fractalfract6080443
复制
发表时间：
2022-08
影响因子：
5.4
通讯作者：
Qingqing Tian
中科院分区：
文献类型：
--
作者：
Xiaoxuan Jiang;Xuehua Yang;Haixiang Zhang;Qingqing Tian
本文提出并分析了一类具有多项Riemann-Liouvile(R-L)分数次积分核的四阶非线性方程的隐式差分格式。对于非线性对流项，采用基于分段线性检验函数的Galerkin方法进行隐式处理，得到了一组非线性代数方程组。Riemann-Liouvile分数次积分项用卷积求积法处理。为了得到一种完全离散的方法，采用标准中心差分近似对空间导数进行离散。用离散能量法严格证明了算法的稳定性和收敛性。此外，严格证明了非线性系统数值解的存在唯一性。此外，我们还介绍并比较了求解非线性系统的Besse松弛算法、牛顿迭代算法和线性化迭代算法。数值结果验证了理论分析，表明了该方法的有效性。
In this paper, an implicit difference scheme is proposed and analyzed for a class of nonlinear fourth-order equations with the multi-term Riemann–Liouvile (R–L) fractional integral kernels. For the nonlinear convection term, we handle implicitly and attain a system of nonlinear algebraic equations by using the Galerkin method based on piecewise linear test functions. The Riemann–Liouvile fractional integral terms are treated by convolution quadrature. In order to obtain a fully discrete method, the standard central difference approximation is used to discretize the spatial derivative. The stability and convergence are rigorously proved by the discrete energy method. In addition, the existence and uniqueness of numerical solutions for nonlinear systems are proved strictly. Additionally, we introduce and compare the Besse relaxation algorithm, the Newton iterative method, and the linearized iterative algorithm for solving the nonlinear systems. Numerical results confirm the theoretical analysis and show the effectiveness of the method.