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 Mα (t). The function Mα (t) is the inverse Laplace transform of the single- parameter Mittag-Leffler function Eα (−s), which itself can be viewed as the fractional-order generalization of the exponential function for 0 < α < 1. At the limiting case of α = 1 the M-Wright function reduces to M1(t) = δ(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. © 2004-2012 IEEE.