Optimal regularization for a general multi-dimensional time-fractional sideways problem

Abstract This paper addresses a general multi-dimensional time-fractional sideways problem, with a focus on determining spatial derivatives of the temperature distribution from internally measured data. The problem is known to be severely ill-posed, as its solution does not depend continuously on the data. By selecting a suitable spectral source set, we establish lower bounds on the worst-case error associated with the problem. Furthermore, we demonstrate that a general Tikhonov regularization method can achieve the optimal convergence rates derived from our analysis. To the best of our knowledge, this represents a new optimal result for the multi-dimensional sideways problem. Numerical experiments are included to validate the theoretical results.