Approximate Methods And Error Estimates For The Sideways Inverse Problem

In this paper,we consider an one-dimensional sideways heat equation in the quarter plane.It is seriously ill-posed.The classical quasi-reversibility method is applied to acquire an approximate solution of the problem.It is found that it is not a regularization solution.The internal stability of the solution can be restored by modifying the denominator of the solution.Then,a new regularization method is proposed.Because of its wide application value,we extend the study of this kind of problem to the fractional order in this paper,and give the fractional Tikhonov method for the sideways fractional heat equation.It overcomes the defects of over-smoothing of classical Tikhonov method.And the Holder-type error estimates under a priori and a posteriori regularization parameters choice rule are proved respectively.At the same time,numerical experiments show the effectiveness of the proposed method.