Wavelet regularization for an inverse heat conduction problem. (Q1421186)

The goal of this paper is to consider the following nonstandard inverse heat conduction problem in the quarter plane (\(t\geq 0\), \(x\geq 0\)): \[ \begin{cases} u_t+ u_x= u_{xx},\quad & x\geq 0,\;t\geq 0,\\ u(x,0)= 0,\quad & x\geq 0,\\ u(1,t)= g(t),\quad & t\geq 0,\end{cases}\tag{1} \] where \(g\in L^2(0,\infty)\) is the temperature history at the fixed point \(x= 1\). The authors look for such a solution \(u(x,\cdot)\in L^2(0,\infty)\) which is bounded as \(x\to\infty\) and they give out a wavelet regularization method for (1) with strict proof.