On the Construction and Analysis of a Fractional-Order Dirac Delta Distribution with Application

We introduce the generalized fractional-order Dirac delta distribution δGFODDF, defined by applying the generalized fractional derivative (GFD) operator to the Heaviside function. This construction extends the classical Dirac delta to non-integer orders, allowing modeling of systems with memory and non-local effects. We establish fundamental properties—including shifting, scaling, evenness, derivative, and convolution—within a rigorous distributional framework and present explicit proofs. Applications are demonstrated by solving linear fractional differential equations and by modeling drug release with fractional kinetics, where the new delta captures impulse responses with long-term memory. Numerical illustrations confirm that δGFODDF reduces to the classical delta when η=1, while providing additional flexibility for 0<η<1. These results show that δGFODDF is a powerful tool for fractional-order analysis in mathematics, physics, and biomedical engineering.