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 α ∈ (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 ∈ (0 , L) , 0 < x 1 < x 2 ≤ L. The problem is challenging since it is severely ill-posed for x ∉ [ 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. [ABSTRACT FROM AUTHOR]