Identical approximation operator and regularization method for the Cauchy problem of 2-D heat conduction equation

The purpose of this work is to create an identical approximate regularization method for solving a Cauchy problem of two-dimensional heat conduction equation. The 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 text. The numerical results show that our method is effective, accurate, and stable to solve the ill-posed Cauchy problems.