Abstract: In this paper,we propose a new numerical method which is a least squares approximaton based on pseudospectral method for the Forward-Backward heat equation. The existence and uniqueness of the solution of the least squares approximation are proved. Error estimates for this approximation are given,which show that tile order of convergence depends only on the regularity of tile solution and the right hand of the Forward-Backward heat equation.

Key words: Forward-Backward, heat equation, pseudospectral method, the least squares method