{
  "id": "1707.04733",
  "version": "v1",
  "published": "2017-07-15T13:40:52.000Z",
  "updated": "2017-07-15T13:40:52.000Z",
  "title": "The general form of the Euler--Poisson--Darboux equation and application of transmutation method",
  "authors": [
    "Elina L. Shishkina",
    "Sergei M. Sitnik"
  ],
  "comment": "Electronic Journal of Differential Equations, 2017",
  "categories": [
    "math.CA"
  ],
  "abstract": "In the paper we find solution representations in the compact integral form to the Cauchy problem for a general form of the Euler--Poisson--Darboux equation with Bessel operators via generalized translation and spherical mean operators for all values of the parameter $k$, including also not studying before exceptional odd negative values. We use a Hankel transform method to prove results in a unified way. Under additional conditions we prove that a distributional solution is a classical one too. A transmutation property for connected generalized spherical mean is proved and importance of applying transmutation methods for differential equations with Bessel operators is emphasized. The paper also contains a short historical introduction on differential equations with Bessel operators and a rather detailed reference list of monographs and papers on mathematical theory and applications of this class of differential equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-15T13:40:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A33",
      "44A15"
    ],
    "keywords": [
      "general form",
      "euler-poisson-darboux equation",
      "transmutation method",
      "bessel operators",
      "differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}