{
  "id": "1704.04987",
  "version": "v1",
  "published": "2017-04-13T03:33:28.000Z",
  "updated": "2017-04-13T03:33:28.000Z",
  "title": "Reconstruction of the Temporal Component in the Source Term of a (Time-Fractional) Diffusion Equation",
  "authors": [
    "Yikan Liu",
    "Zhidong Zhang"
  ],
  "comment": "26 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "math.NA"
  ],
  "abstract": "In this article, we consider the reconstruction of $\\rho(t)$ in the (time-fractional) diffusion equation $(\\partial_t^\\alpha-\\triangle)u(x,t)=\\rho(t)g(x)$ ($0<\\alpha \\le 1$) by the observation at a single point $x_0$. We are mainly concerned with the situation of $x_0 \\notin$ supp g, which is practically important but has not been well investigated in literature. Assuming the finite sign changes of $\\rho$ and an extra observation interval, we establish the multiple logarithmic stability for the problem based on a reverse convolution inequality and a lower estimate for positive solutions. Meanwhile, we develop a fixed point iteration for the numerical reconstruction and prove its convergence. The performance of the proposed method is illustrated by several numerical examples.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-13T03:33:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R11",
      "35R30",
      "26A33",
      "26D10",
      "65M32"
    ],
    "keywords": [
      "diffusion equation",
      "source term",
      "temporal component",
      "reconstruction",
      "time-fractional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}