On the Fractional Derivatives with an Exponential Kernel

Abstract

The present article mainly focuses on the fractional derivatives with an exponential kernel (“exponential fractional derivatives” for brevity). First, several extended integral transforms of the exponential fractional derivatives are proposed, including the Fourier transform and the Laplace transform. Then, the L2 discretisation for the exponential Caputo derivative with $\alpha \in (1,2)$ is established. The estimation of the truncation error and the properties of the coefficients are discussed. In addition, a numerical example is given to verify the correctness of the derived L2 discrete formula.