Caputo-Hadamard时间分数阶反应扩散方程的L1格式差分逼近-《集美大学学报（自然科学版）》

文章信息/Info

Title:: L1 Scheme Difference Approximation for Caputo-Hadamard Time Fractional ReactionDiffusion Equation

Author(s):: LIU Xinran; CHEN Jinghua; GONG Shanshan; School of Science，Jimei University，Xiamen 361021，China

Keywords:: fractional reactiondiffusion equation; Caputo-Hadamard derivative; L1 scheme; Richardson extrapolation; stability; convergence

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。

Abstract:: 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.