{
  "id": "1410.4183",
  "version": "v1",
  "published": "2014-10-15T19:44:04.000Z",
  "updated": "2014-10-15T19:44:04.000Z",
  "title": "Explicit solutions for a non-classical heat conduction problem for a semi-infinite strip with a non-uniform heat source",
  "authors": [
    "Andrea N. Ceretani",
    "Domingo A. Tarzia",
    "Luis T. Villa"
  ],
  "comment": "25 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "A non-classical initial and boundary value problem for a non-homogeneous one-dimensional heat equation for a semi-infinite material with a zero temperature boundary condition at the face $x=0$ is studied with the aim of finding explicit solutions. It is not a standard heat conduction problem because a heat source $-\\Phi(x)F(V(t),t)$ is considered, where $V$ represents the heat flux at $x=0$. Explicit solutions independents of the space or temporal variables are given. Solutions with separated variables when the data functions are defined from the solution $X=X(x)$ of a linear initial value problem of second order and the solution $T=T(t)$ of a non-linear (in general) initial value problem of first order which involves the function $F$, are also given and explicit solutions corresponding to different definitions of $F$ are obtained. A solution by an integral representation depending on the heat flux at $x=0$ for the case in which $F=F(V,t)=\\nu V$, $\\nu>0$, is obtained and explicit expressions for the heat flux at $x=0$ and for its corresponding solution are calculated when $h=h(x)$ is a potential function and $\\Phi=\\Phi(x)$ is given by $\\Phi(x)=\\lambda x$, $\\Phi(x)=-\\mu\\sinh{(\\lambda x)}$ or $\\Phi(x)=-\\mu\\sin{(\\lambda x)}$, $\\lambda>0$ and $\\mu>0$. The limit when the temporal variable $t$ tends to $+\\infty$ of each explicit solution obtained in this paper is studied and the \"controlling\" effects of the source term $-\\Phi F$ are analysed by comparing the asymptotic behavior of each solution with the asymptotic behavior of the solution to the same problem but in absence of source term. Finally, a relationship between this problem with another non-classical initial and boundary value problem for the heat equation is established and explicit solutions for this second problem are also obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-15T19:44:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35C05",
      "35C15",
      "35C20",
      "35K55",
      "45D05",
      "80A20"
    ],
    "keywords": [
      "explicit solution",
      "non-classical heat conduction problem",
      "non-uniform heat source",
      "semi-infinite strip",
      "initial value problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.4183C"
    }
  }
}