A new numerical fractional differentiation formula to approximate the Caputo-Fabrizio fractional derivative: error analysis and stability

IF 1.1 Q2 MATHEMATICS, APPLIED

Computational Methods for Differential Equations Pub Date : 2021-01-05 DOI:10.22034/CMDE.2020.37595.1664

Leila Moghadam Dizaj Herik, M. Javidi, M. Shafiee

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Abstract

In the present work, first of all, a new numerical fractional differentiation formula (called the CF2 formula) to approximate the Caputo-Fabrizio fractional derivative of order $alpha,$ $(0

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一个新的近似Caputo-Fabrizio分数导数的数值分数微分公式：误差分析和稳定性

本文首先提出了一个新的数值分数阶微分公式(称为CF2公式)来近似阶$ α，$ $(0< α <1)$的Caputo-Fabrizio分数阶导数。对于$(j geq 2)$，在每个区间$[t_{j-1}，t_{j}]$上使用$(t_{j}) $， $(t_{j}) $， $(t_{j}) $和$(t_{j}，y(t_{j})) $三个点进行二次插值逼近，而在第一个区间$[t_{0}，t_{1}]$上应用线性插值逼近。因此，新公式可以形式上看作是对经典CF1公式的修正，经典CF1公式是通过y(t)的分段线性近似得到的。新公式的计算效率和数值精度均优于CF1公式。详细讨论了该公式的系数和截断误差。{两个测试实例显示}CF2公式的数值精度。CF1公式表明，在求解分数阶微分方程时，新的CF2比CF1更有效、更精确。对CF2进行了详细的稳定性分析和区域稳定性研究。

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