Communications of the
Korean Mathematical Society CKMS

We consider an ill-posed problem for the heat equation $u_{xx}=u_{t}$ in the quarter plane $\{x>0, \ t>0\}$. We propose a new method to compute the heat flux $h(t)=u_x(1,t)$ from the boundary temperature $g(t)=u(1,t)$. The operator $g\mapsto h=Hg$ is unbounded in $L^2(\mathbb{R})$, so we approximate $h(t)$ by $h_\delta(t)=u_x(1+\delta,t)$, $\delta\rightarrow0$. When noise is present, the data is $g_\epsilon$ leading to a corresponding heat $ h_{\delta,\epsilon}$. We obtain an estimate of the error $\|h-h_{\delta,\epsilon}\|$, as well as the error when $h_{\delta,\epsilon}$ is approximated by the trapezoidal rule. With an a priori choice rule $\delta=\delta(\epsilon)$ and $\tau=\tau(\epsilon)$, the step size of the trapezoidal rule, the main theorem gives the error of the heat flux as a function of noise level $\epsilon$. Numerical examples show that the proposed method is effective and stable.

Keywords: inverse problems, ill-posed problem, stable approximation, error estimate

MSC numbers: Primary 35K05, 65M32