The fractional derivative of the Dirac delta function and new results on the inverse Laplace transform of irrational functions

TL;DR

This paper establishes that the memory function in complex materials with power-law creep is a fractional derivative of the Dirac delta, leading to new formulas for inverse Laplace transforms of irrational functions involving fractional derivatives.

Contribution

It introduces a novel connection between fractional derivatives of the Dirac delta and inverse Laplace transforms of irrational functions, expanding analytical tools for complex material modeling.

Findings

Inverse Laplace transform of s^q is a fractional derivative of delta

New formulas for inverse Laplace transforms involving Rabotnov functions

Fractional derivatives produce singularities analyzed via Mittag-Leffler functions

Abstract

Motivated from studies on anomalous diffusion, we show that the memory function $M(t)$ of complex materials, that their creep compliance follows a power law, $J(t)\sim t^q$ with $q\in \mathbb{R}^+$, is the fractional derivative of the Dirac delta function, $\frac{\mathrm{d}^q\delta(t-0)}{\mathrm{d}t^q}$ with $q\in \mathbb{R}^+$. This leads to the finding that the inverse Laplace transform of $s^q$ for any $q\in \mathbb{R}^+$ is the fractional derivative of the Dirac delta function, $\frac{\mathrm{d}^q\delta(t-0)}{\mathrm{d}t^q}$. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of $\frac{s^q}{s^{\alpha}\mp\lambda}$ where $\alpha<q\in\mathbb{R}^+$ which is the fractional derivative of order $q$ of the Rabotnov function $\varepsilon_{\alpha-1}(\pm\lambda, t)=t^{\alpha-1}E_{\alpha, \alpha}(\pm\lambda t^{\alpha})$. The fractional derivative of order $q\in \mathbb{R}^+$ of the Rabotnov function, $\varepsilon_{\alpha-1}(\pm\lambda, t)$ produces singularities which are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of $q$ in association with the recurrence formula of the two-parameter Mittag-Leffler function.