Optimal modified method for a fractional-diffusion inverse heat conduction problem

We consider the determination of the boundary temperature from one measured transient data temperature at some interior point of a one-dimensional semi-infinite conductor. Mathematically, it can be formulated as a fractional-diffusion inverse heat conduction problem where data are given at x = l and we want to determine a solution for 0 x l. This problem arises in several contexts and has important applications in science and engineering. The difficulty of the problem is its severe ill-posedness, i.e. the solution (if it exists) does not depend continuously on the data. In this article, we consider an optimal modified method from the frequency domain and obtain a Holder-type convergence estimate with the coefficient c = 1, which is optimal. The method can be implemented numerically using discrete Fourier transforms. Three kinds of examples illustrate the behaviour of the proposed method.