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

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 $alpha,$ $(0<alpha<1)$ is developed. It is established by means of the quadratic interpolation approximation using three points $ (t_{j-2},y(t_{j-2}))$, $(t_{j-1},y(t_{j-1})) $ and $ (t_{j},y(t_{j})) $ on each interval $[t_{j-1},t_{j}]$ for $ ( j geq 2 )$, while the linear interpolation approximation is applied on the first interval $[t_{0},t_{1}]$. As a result, the new formula can be formally viewed as a modification of the classical CF1 formula, which is obtained by the piecewise linear approximation for $y(t)$. Both the computational efficiency and numerical accuracy of the new formula is superior to that of the CF1 formula. The coefficients and truncation errors of this formula are discussed in detail. {Two test example show} the numerical accuracy of the CF2 formula. The CF1 formula demonstrates that the new CF2 is much more effective and more accurate than the CF1 when solving fractional differential equations. Detailed stability analysis and region stability of the CF2 are also carefully investigated.