Caputo-Hadamard时间分数阶反应扩散方程的L1格式差分逼近  

L 1 Scheme Difference Approximation for Caputo-Hadamard Time Fractional Reaction-Diffusion Equation

作　　者：刘欣然 陈景华[1] 龚珊珊 LIU Xinran;CHEN Jinghua;GONG Shanshan(School of Science,Jimei University,Xiamen 361021,China)

机构地区：[1]集美大学理学院,福建厦门361021

出　　处：《集美大学学报（自然科学版）》2024年第6期481-487,共7页Journal of Jimei University：Natural Science

基　　金：福建省自然科学基金项目“非光滑数据下分数阶微分方程的高精度数值算法及其应用研究”(2020J01338);福建省教育厅基金项目“基于统计分析与显著性检测的图像处理”(JAT210231);福建省高校数学学科联盟项目“非局部优化约束问题的高阶数值方法”(2024SXLMMS03);集美大学数字福建大数据建模与智能计算研究所开放基金项目“具有吸收边界和反射边界的调和分数阶微分方程的谱延迟校正法及数值模拟”。

摘　　要：提出一种求Caputo-Hadamard时间分数阶反应扩散方程的数值解法。将一阶的时间导数用Caputo-Hadamard导数替换,再对Caputo-Hadamard时间分数阶导数采用L1插值逼近离散;利用中心差分公式离散空间二阶导数,构造方程的数值离散格式,并证明该数值格式具有稳定性和收敛性。之后利用Richardson外推法进一步提高空间精度,并给出具体算法,使方程新的差分格式达到空间方向四阶收敛。最后给出一个数值算例,证明该数值格式的有效性。A numerical method for solving Caputo-Hadamard time fractional reaction-diffusion equation was presented in this paper.The first order time derivative was replaced by the Caputo-Hadamard derivative and then the Caputo-Hadamard time fractional derivative was approximated by L1 interpolation.The second derivative of space was discretized by the central difference formula,and the numerical discretization scheme of the equation was constructed.It was proved that the numerical scheme was stability and convergence.Then Richardson extrapolation was applied to further improve the spatial accuracy,and a specific algorithm was presented to make the new difference scheme reach fourth order convergence in space direction.Finally,a numerical example was implemented to test the efficiency of the numerical scheme.

关 键 词：分数阶反应扩散方程 Caputo-Hadamard导数 L1格式 RICHARDSON外推法 稳定性 收敛性 

分 类 号：O241.82[理学—计算数学]