Abstract: We considered the inverse problem of Cauchy problem of two dimensional parabolic equations, which was seriously ill-posed. Firstly, a regular approximate solution of the problem was obtained by using Landweber iterative regularization method, and  Fourier transform was used to obtain  the exact solution of the problem. Secondly, the Holder type error estimation between the exact solution and the regular solution was given under the selection rules of the posterior regularization parameters, and stronger prior conditions were used to give  the error estimation at the end point x=1. Finally, numerical examples were given to demonstrate the effectiveness of the proposed method. The results show that the proposed method has a faster  convergence rate than existing methods.

Key words: Cauchy problem, ill-posed problem, Landweber iterative regularization, error estimation, posterior estimation