Abstract
This study is specially designed to solve the forward-backward heat equation in two dimension to make use of thin plate splines and finite difference method. It should be noted that the problem had more than one part categorization. The results as a whole covered a large proportion solutions by a radial basis function method for spatial and finite difference scheme along with interface boundary usage . This method was presented based on updating interface boundary. In addition, the interpolation and the collocation methods have supported through using of thin plate splines and finite difference method to update interface boundary. Furthermore, it showed that the time discretization scheme was unconditionally stable and convergent. Finally, some numerical examples were assigned to graded methods to support their efficiency.
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