Regularization Methods And Error Estimation For 2-D Sideways Heat Equation

In this paper,we consider a 2-D sideways heat equation in the quarter plane.The exact solution of this problem is obtained by using Fourier method.It is seriously ill-posed.If the classical quasi-reversibility method is adopted,we find this method is not suitable for the current problem.So the internal stability of the solution can be restored by modifying the denominator of the solution.Then,a new regularization method is proposed.The study of such problems can be generalized to fractional orders,and we give the fractional Tikhonov method for the 2-D sideways fractional heat equation.And the H¨older-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.