Some Regularization Methods For The One-dimensional Time Fractional Derivative Inverse Heat Conduction Problem

This paper investigates an one-dimensional inverse heat conduction problem in a quarter-plane with time fractional derivative, in other words, we want to deter-mine the temperature or heat flux for 0< x< 1 from the temperature measured at x= 1. This problem is ill-posed: its solution (if it exists) does not depend continu-ously on the data and the ill-posedness becomes sharper as the unknown solutions are closer to the boundary x= 0. The problem is not only contributive for practical applications but also very important for theoretical research. Hence it is valuable to give some effective regularization methods. In this paper, the modified Tikhonov method, filtering method, the finite difference method and quasi-reversibility method are proposed to solve this problem. We have obtained the convergence estimates for all these regularization methods, and restored stability of the solutions in the inner. Meanwhile, numerical examples show that our methods are effective.