Two unconditionally stable difference schemes for time distributed-order differential equation based on Caputo–Fabrizio fractional derivative

Advances in Difference Equations

article

Qiao, Haili¹  Liu, Zhengguang²  Cheng, Aijie¹ 

[1] School of Mathematics, Shandong University;School of Mathematics and Statistics, Shandong Normal University

关键词: Distributed-order;    Caputo–Fabrizio derivatives;    Compact finite difference;    Stability and convergence;    Numerical experiments;   

DOI  :  10.1186/s13662-020-2514-5

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

We consider distributed-order partial differential equations with time fractional derivative proposed by Caputo and Fabrizio in a one-dimensional space. Two finite difference schemes are established via Grünwald formula. We show that these two schemes are unconditionally stable with convergence rates $O(\tau ^{2}+h^{2}+ \Delta \alpha ^{2})$ and $O(\tau ^{2}+h^{4}+\Delta \alpha ^{4})$ in discrete $L^{2}$, respectively, where Δα, h, and τ are step sizes for distributed-order, space, and time variables, respectively. Finally, the performance of difference schemes is illustrated via numerical examples.

【 授权许可】

CC BY