In this paper, a truly meshless method based on the partition of unity method (PUM) is developed for the numerical solution of the two-dimensional forward-backward heat equations. We propose a novel method according to the domain decomposition scheme and RBF-PUM technique. Particularly, the physical domain needs to be separated into two subdomains each defining a forward or a backward subproblem. The subproblems have been treated by a radial basis function meshfree method based on partition of unity for spatial dimension and a finite difference scheme for the time derivative. In addition, we prove that the time discrete scheme is stable and convergent. Some numerical experiments are going to be presented to show the performance of the proposed method. % our collocation scheme and some computational aspects will be discussed.
