Theory and Application of the Fractional-order Delta Function Associated with the Inverse Laplace Transform of the Mittag-Leffler Function

This paper is devoted to the study of the $M$-Wright function ($M_{\alpha }(t)$) which is the inverse Laplace transform of the single-parameter Mittag-Leffler (ML) function ($E_{\alpha }(-s)$). Because $E_{\alpha }(-s)$ can be viewed as the fractional-order generalization of the exponential function for $0<\alpha <1$, to which it reduces for $\alpha =1$, i.e. $E_1(-s)=\exp (-s)$, its inverse Laplace transform, being $M_{\alpha }(t)$, can be viewed as a generalized fractional-order Dirac delta function. At the limiting case of $\alpha =1$ the $M$-Wright function reduces to $M_1(t)=\delta (t-1)$. We investigate numerically the behavior of this fractional-order delta function as well as it integral, the fractional-order unit-step function. Subsequently, we validate our results with experimental data for the charging of a supercapacitive device.