{
  "id": "2006.04966",
  "version": "v1",
  "published": "2020-06-08T22:21:17.000Z",
  "updated": "2020-06-08T22:21:17.000Z",
  "title": "The fractional derivative of the Dirac delta function and new results on the inverse Laplace transform of irrational functions",
  "authors": [
    "Nicos Makris"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:2002.04581",
  "categories": [
    "math-ph",
    "math.CA",
    "math.MP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-08T22:21:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dirac delta function",
      "inverse laplace transform",
      "fractional derivative",
      "irrational functions",
      "rabotnov function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}