{
  "id": "1706.07266",
  "version": "v1",
  "published": "2017-06-22T11:35:32.000Z",
  "updated": "2017-06-22T11:35:32.000Z",
  "title": "Fractional Partial Differential Equations with Boundary Conditions",
  "authors": [
    "Boris Baeumer",
    "Mihály Kovács",
    "Harish Sankaranarayanan"
  ],
  "categories": [
    "math.AP",
    "math.NA",
    "math.PR"
  ],
  "abstract": "We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posedness of the associated Cauchy problems in $C_0(\\Omega)$ and $L_1(\\Omega)$. In order to do so we develop a new method of embedding finite state Markov processes into Feller processes and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax-Richtmyer Equivalence Theorem to prove convergence without checking well-posedness.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-22T11:35:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boundary conditions",
      "finite state markov processes",
      "algorithm articles approximating solutions",
      "one-sided fractional partial differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}