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http://dx.doi.org/10.4134/CKMS.c170258

STABLE APPROXIMATION OF THE HEAT FLUX IN AN INVERSE HEAT CONDUCTION PROBLEM  

Alem, Leila (Laboratory of Applied Mathematics Badji Mokhtar University)
Chorfi, Lahcene (Laboratory of Applied Mathematics Badji Mokhtar University)
Publication Information
Communications of the Korean Mathematical Society / v.33, no.3, 2018 , pp. 1025-1037 More about this Journal
Abstract
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 $h_{\delta}(t)=u_x(1+{\delta},\;t)$, ${\delta}{\rightarrow}0$. When noise is present, the data is $g_{\epsilon}$ leading to a corresponding heat $h_{{\delta},{\epsilon}}$. We obtain an estimate of the error ${\parallel}h-h_{{\delta},{\epsilon}}{\parallel}$, 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;
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