Stable algorithm for identifying a source in the heat equation

• Main Authors: Lahcene Chorfi, Leila Alem
• Format: Article
• Language: English
• Published: Texas State University 2015-10-01
• Series: Electronic Journal of Differential Equations
• Subjects: Inverse problem; heat equation; fourier regularization; finite difference
• Online Access: http://ejde.math.txstate.edu/Volumes/2015/267/abstr.html

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.