{
  "id": "1803.07732",
  "version": "v1",
  "published": "2018-03-21T03:36:44.000Z",
  "updated": "2018-03-21T03:36:44.000Z",
  "title": "Superconvergence Points of Integer and Fractional Derivatives of Special Hermite Interpolations and Its Applications in Solving FDEs",
  "authors": [
    "Beichuan Deng",
    "Jiwei Zhang",
    "Zhimin Zhang"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "In this paper, we study convergence and superconvergence theory of integer and fractional derivatives of the one-point and the two-point Hermite interpolations. When considering the integer-order derivative, exponential decay of the error is proved, and superconvergence points are located, at which the convergence rates are $O(N^{-2})$ and $O(N^{-1.5})$, respectively, better than the global rate for the one-point and two-point interpolations. Here $N$ represents the degree of interpolation polynomial. It is proved that the $\\alpha$-th fractional derivative of $(u-u_N)$ with $k<\\alpha<k+1$, is bounded by its $(k+1)$-th derivative. Furthermore, the corresponding superconvergence points are predicted for fractional derivatives, and an eigenvalue method is proposed to calculate the superconvergence points for the Riemann-Liouville fractional derivative. In the application of the knowledge of superconvergence points to solve FDEs, we discover that a modified collocation method makes numerical solutions much more accurate than the traditional collocation method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-21T03:36:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N35",
      "65M15",
      "26A33",
      "41A05",
      "41A10"
    ],
    "keywords": [
      "superconvergence points",
      "fractional derivative",
      "special hermite interpolations",
      "solving fdes",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}