The ill-posedness of the forward-backward heat problem arises in boundary layer problems, fluid dynamics, plasma physics, astrophysics and the study of propagation of an electron beam through the solar corona. In this study, a new truly meshless method is developed for the numerical solution of the two dimensional forward-backward heat equation. We propose a novel method based on the domain decomposition scheme and RBF method with an adaptive nodes technique. Specifically, the physical domain is divided into two subdomains each defining a forward or a backward subproblem. The resulting subproblems are dealt with by a radial basis function meshfree method for spatial dimension and a finite difference scheme for the time derivative followed by an adaptive algorithm to achieve a desired accuracy. In addition, we prove that the time discrete scheme is stable and convergent. Some numerical experiments will be presented to show the performance of our collocation scheme.
