{
  "id": "2309.04988",
  "version": "v1",
  "published": "2023-09-10T10:38:38.000Z",
  "updated": "2023-09-10T10:38:38.000Z",
  "title": "Analysis of fractional Cauchy problems with some probabilistic applications",
  "authors": [
    "Fabrizio Cinque",
    "Enzo Orsingher"
  ],
  "comment": "21 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we give an explicit solution of Dzherbashyan-Caputo-fractional Cauchy problems related to equations with derivatives of order $\\nu k$, for $k$ non-negative integer and $\\nu>0$. The solution is obtained by connecting the differential equation with the roots of the characteristic polynomial and it is expressed in terms of Mittag-Leffler-type functions. Under the some stricter hypothesis the solution can be expressed as a linear combination of Mittag-Leffler functions with common fractional order $\\nu$. We establish a probabilistic relationship between the solutions of differential problems with order $\\nu/m$ and $\\nu$, for natural $m$. Finally, we use the described method to solve fractional differential equations arising in the fractionalization of partial differential equations related to the probability law of planar random motions with finite velocities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-10T10:38:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34A08",
      "35R11",
      "60K99"
    ],
    "keywords": [
      "probabilistic applications",
      "planar random motions",
      "dzherbashyan-caputo-fractional cauchy problems",
      "common fractional order",
      "fractional differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}