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 is an element of (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 is an element of (0, L) , 0 < x(1) < x(2) <= L . The problem is challenging since it is severely ill-posed for x is not an element of[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.