The fractional derivative of the Dirac delta function and new results on the inverse Laplace transform of irrational 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, $\frac{\mathrm{d}^q\delta(t0)}{\mathrm{d}t^q}$ 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, $\frac{\mathrm{d}^q\delta(t0)}{\mathrm{d}t^q}$. 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 $\varepsilon_{\alpha1}(\pm\lambda, t)=t^{\alpha1}E_{\alpha, \alpha}(\pm\lambda t^{\alpha})$. The fractional derivative of order $q\in \mathbb{R}^+$ of the Rabotnov function, $\varepsilon_{\alpha1}(\pm\lambda, t)$ 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 twoparameter MittagLeffler function.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.04966
 Bibcode:
 2020arXiv200604966M
 Keywords:

 Mathematical Physics;
 Mathematics  Classical Analysis and ODEs
 EPrint:
 arXiv admin note: substantial text overlap with arXiv:2002.04581