NUMERICAL METHODS FOR THE CAPUTO-TYPE FRACTIONAL DERIVATIVE WITH AN EXPONENTIAL KERNEL

Abstract

In the present article, several typical numerical discrete formulas for the Caputo-type fractional derivative with an exponential kernel (call "exponential Caputo derivative" for brevity) with order $ \alpha\in(0,1) $ and $ \alpha\in(1,2) $ are constructed, which are L1, L1-2, L2-1$ _\sigma $ formulas for $ \alpha\in(0,1) $, and H2N2 and L2$ _{1} $ formulas for $ \alpha\in(1,2) $, respectively. And the estimates of the truncation errors are determined. Meanwhile, the properties of the coefficients in these formulas are studied. Finally, some numerical examples are displayed which support the theoretical analysis.