We present a truly meshless method based on the thin plate splines for the numerical solution of the two dimensional forward-backward heat equation and give a robust formulation for the proposed method. The physical domain is divided into two subdomains each of which defines a forward or a backward subproblem. The resulting subproblems are treated by a radial basis function method for spatial dimension and a finite difference scheme for the time derivative followed by an iterative domain decomposition method to achieve a desired accuracy. In addition, we propose a combined use of an interpolation and the collocation method, in the iterations to update the interface boundary solution. Furthermore, we show that the time discretization scheme is unconditionally stable and convergent. Finally, some numerical examples will be presented to demonstrate the efficiency of the proposed method and some computational aspects will be discussed.

针对二维前后热方程的数值解，我们提出了一种基于薄板样条的真正的无网格方法，并为该方法给出了鲁棒的公式。物理域分为两个子域，每个子域定义一个正向或反向子问题。对于空间维度，使用径向基函数方法，对于时间导数，采用有限差分方案，然后使用迭代域分解方法，以实现所需的精度。另外，我们提出在迭代中结合使用插值和搭配方法来更新界面边界解决方案。此外，我们证明了时间离散方案是无条件稳定和收敛的。最后，