Electronic Journal of Differential Equations,
Vol. 2015 (2015), No. 267, pp. 1-14.
Title: Stable algorithm for identifying a source in the heat equation
Authors: Lahcene Chorfi (Univ. B. M. d'Annaba, Annaba, Alg\erie)
Leila Alem (Univ. B. M. d'Annaba, Annaba, Alg\erie)
Abstract:
We consider an inverse problem for the heat equation
$u_{xx}=u_t$ in the quarter plane $\{x>0, t>0\}$ where one wants
to determine the temperature $f(t)=u(0,t)$ from the measured data
$g(t)=u(1,t)$. This problem is severely ill-posed and has been
studied before. It is well known that the central difference
approximation in time has a regularization effect, but
the backward difference scheme is not well studied in
theory and in practice. In this paper, we revisit this method
to provide a stable algorithm. Assuming an a priori bound on
$\|f\|_{H^s}$ we derive a Holder type stability result.
We give some numerical examples to show the efficiency of the
proposed method. Finally, we compare our method to one based on
the central or forward differences.
Submitted June 27, 2015. Published October 16, 2015.
Math Subject Classifications: 35K05, 65M32, 65T50.
Key Words: Inverse problem; heat equation; fourier regularization;
finite difference.