Two Regularization Methods For Solving The Time-inverse Heat Conduction Problems In The Two-dimensional Space And Error Estimations

The time-inverse heat conduction problem is concerned in the two-dimensional space,which retrieves the temperature distribution from the data of the final moment.There is one of the most important applications which is the image restoration.This is a serious ill-posed problem in inverse problems,i.e.its solution is not continuously dependent on the data under certain conditions.To overcome the defects of traditional regularization methods,the quasi-reversibility regularization method and the fractional Tikhonov regularization method are proposed to restore the dependence of the solution on the data.Meanwhile,the errors between the approximate solutions and the exact solutions for the ill-posed problem are estimated,and both the priori and the posteriori regularization parameter selection rules are given.Numerical Examples show that the regularization methods are effective for solving such a problem.