{
  "id": "1211.0935",
  "version": "v2",
  "published": "2012-10-31T13:04:24.000Z",
  "updated": "2017-11-19T14:40:06.000Z",
  "title": "Family of self-similar solutions of diffusion equation --- Structure and Properties",
  "authors": [
    "Ken Sekimoto",
    "Takahiko Fujita"
  ],
  "comment": "7 pages, 7 figures, 2nd version, unpublished",
  "categories": [
    "math.AP",
    "physics.data-an"
  ],
  "abstract": "The aim of the this rather technical note is to summarize the properties and utilities of the family of self-similar solutions of the diffusion equation in 1D and the similar parabolic partial-differential equations. In the first part we analyze the family of self-similar solutions focusing on the relationship among them. In the second part we describe the utility of the \"exotic\" self-similar solutions that decay or grow algebraically for large argument of the scaling variable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-31T13:04:24.000Z",
      "title": "Exotic similarity solutions with power-law tails",
      "abstract": "The diffusion equation is known to have the similarity solutions of the form, $\\theta^\\nu u(\\theta x,\\theta^2 t)=u(x,t)$ ($\\theta\\neq 0$). While we usually deal with those similarity solutions that tends rapidly to constant values for large $|x|$, we evoke the existence of uncountably many other exotic similarity solutions having long tails as $u(x,t)\\sim |x|^{-\\nu}$ for large $|x|$. While these solutions are often ignored by physical reasons, their existence is worth bearing in mind for the mathematical consistency on the one hand, but also for possible experimental realizations on the other hand. We present an example in the context of slow relaxation of gel accompanying the permeation of solvent.",
      "comment": "10 pages, 5 figures, submitted",
      "journal": null,
      "doi": null,
      "authors": [
        "Ken Sekimoto"
      ]
    },
    {
      "version": "v2",
      "updated": "2017-11-19T14:40:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "self-similar solutions",
      "diffusion equation",
      "properties",
      "similar parabolic partial-differential equations",
      "first part"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.0935S"
    }
  }
}