{
  "id": "1004.1577",
  "version": "v1",
  "published": "2010-04-09T15:30:07.000Z",
  "updated": "2010-04-09T15:30:07.000Z",
  "title": "Fractional Cauchy problems on bounded domains: survey of recent results",
  "authors": [
    "Erkan Nane"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.AP",
    "math.MP"
  ],
  "abstract": "In a fractional Cauchy problem, the usual first order time derivative is replaced by a fractional derivative. This problem was first considered by \\citet{nigmatullin}, and \\citet{zaslavsky} in $\\mathbb R^d$ for modeling some physical phenomena. The fractional derivative models time delays in a diffusion process. We will give a survey of the recent results on the fractional Cauchy problem and its generalizations on bounded domains $D\\subset \\rd$ obtained in \\citet{m-n-v-aop, mnv-2}. We also study the solutions of fractional Cauchy problem where the first time derivative is replaced with an infinite sum of fractional derivatives. We point out a connection to eigenvalue problems for the fractional time operators considered. The solutions to the eigenvalue problems are expressed by Mittag-Leffler functions and its generalized versions. The stochastic solution of the eigenvalue problems for the fractional derivatives are given by inverse subordinators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-04-09T15:30:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G18"
    ],
    "keywords": [
      "fractional cauchy problem",
      "bounded domains",
      "derivative models time delays",
      "fractional derivative",
      "eigenvalue problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.1577N"
    }
  }
}