A two-parameter Tikhonov regularization for a fractional sideways problem with two interior temperature measurements

This paper deals with a fractional sideways problem of determining the surface temperature of a heat body from two interior temperature measurements. Mathematically, it is formulated as a problem for the one-dimensional heat equation with Caputo fractional time derivative of order $\alpha \in (0,1]$, where the data are given at two interior points, namely $x=x_1$ and $x=x_2$, and the solution is determined for $x\in (0,L),0<x_1<x_2\le L$. The problem is challenging since it is severely ill-posed for $x\notin [x_1,x_2]$. For the ill-posed problem, we apply the Tikhonov regularization method in Hilbert scales to construct stable approximation problems. Using the two-parameter Tikhonov regularization, we obtain the order optimal convergence estimates in Hilbert scales by using both a priori and a posteriori parameter choice strategies. Numerical experiments are presented to show the validity of the proposed method.