{
  "id": "1509.02603",
  "version": "v1",
  "published": "2015-09-09T02:07:07.000Z",
  "updated": "2015-09-09T02:07:07.000Z",
  "title": "Multiplicity of Solutions for Linear Partial Di?fferential Equations Using (Generalized) Energy Operators",
  "authors": [
    "J. P. Montillet"
  ],
  "comment": "21 pages, 2 figures. Submitted to EJAM",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Families of energy operators and generalized energy operators have recently been introduced in the definition of the solutions of linear Partial Differential Equations (PDEs) with a particular application to the wave equation [Montillet, 2014, doi: 10.1007/s10440-014-9978-9]. To do so, the author has introduced the notion of energy spaces included in the Schwartz space $\\mathbf{S}^-(\\mathbb{R})$. In this model, the key is to look at which ones of these subspaces are reduced to {0} with the help of energy operators (and generalized energy operators). It leads to define additional solutions for a nominated PDE. Beyond that, this work intends to develop the concept of multiplicity of solutions for a linear PDE through the study of these energy spaces (i.e. emptiness). The main concept is that the PDE is viewed as a generator of solutions rather than the classical way of solving the given equation with a known form of the solutions together with boundary conditions. The theory is applied to the wave equation with the special case of the evanescent waves. The work ends with a discussion on another concept, the duplication of solutions. The discussion takes place for the special case of waves trapped in an electromagnetic chamber (i.e. a closed tapered waveguide)",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-09T02:07:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A99",
      "34L30",
      "46A11"
    ],
    "keywords": [
      "multiplicity",
      "generalized energy operators",
      "special case",
      "wave equation",
      "energy spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}