In this paper, an identical approximate regularization method is
extended to the Cauchy problem of two-dimensional heat conduction
equation, this kind of problem is severely ill-posed. The convergence
rates are obtained under a priori regularization parameter choice rule.
Numerical results are presented for two examples with smooth and
continuous but not smooth boundaries, and compared the identical
approximate regularization solutions which are displayed in paper. The
numerical results show that our method is effective, accurate and stable
to solve the ill-posed Cauchy problems.