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

In the present work, first of all, a new numerical fractional differentiation formula (called the CF۲ formula) to approximate the Caputo-Fabrizio fractional derivative of order α, (۰ < α < ۱) is developed. It is established by means of the quadratic interpolation approximation using three points (tj−۲,y(tj−۲)),(tj−۱,y(tj−۱)), and (tj, y(tj)) on each interval [tj−۱,tj] for (j ≥ ۲), while the linear interpolation approximation are applied on the first interval [t۰,t۱]. As a result, the new formula can be formally viewed as a modification of the classical CF۱ 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 CF۱ formula. The coefficients and truncation errors of this formula are discussed in detail. Two test examples show the numerical accuracy of the CF۲ formula. The CF۱ formula demonstrates that the new CF۲ is much more effective and more accurate than the CF۱ when solving fractional differential equations. Detailed stability analysis and region stability of the CF۲ are also carefully investigated.

Keywords:

fractional differential equation , Stability , Caputo-Fabrizio fractional derivative , Numerical methods , Error analysis