On the Theory and Application of the Fractional-Order Dirac-Delta Function

In this brief, we study a generalized fractional-order Dirac delta function defined using the
$M$
-Wright function
$M_{\alpha }(t)$
. The function
$M_{\alpha }(t)$
is the inverse Laplace transform of the single-parameter Mittag-Leffler function
$E_{\alpha }({-}s)$
, which itself can be viewed as the fractional-order generalization of the exponential function for
$0 < \alpha < 1$
. At the limiting case of
$\alpha = 1$
the
$M$
-Wright function reduces to
$M_{1}(t)=\delta (t-1)$
, which is the inverse Laplace transform of
$e^{-s}$
. We investigate numerically the behavior of this fractional-order delta function as well as its integral, the fractional-order unit-step function. Both functions are critical in obtaining the impulse-response and step-response of dynamical systems. We validate our results with experimental impulse response of a supercapacitor subjected to a fractional-order delta function current signal.