**Solving the forward-backward heat equation in two-dimension
by Radial basis functions mesh free method**

Siamak Banei^{1*},
Kamal Shanazari^{1}, Yaqub Azari^{2}, and Yones Esmaeelzade
Aghdam^{2}

^{1}Department of Applied Mathematics, University of
Kurdistan, Sanandaj, Iran

^{2}Faculty of Science, Shahid Rajaee
Teacher Training University, Lavizan, Tehran, 16785-163

Original: 21 February 2020 Revised: 18 August 2020 Accepted: 26 September 2020 Published online: 20 December 2020

**Doi Link:**

**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 boundry usage . This method was presented based on updating interface boundry. In addition , the interpolation and the collcation methods have supported through using of thin plate splines and finite difference method to update interface boundary. Furthermore, it showed that the time dicretization scheme was unconditionally stable and convergent. . Finally, some numerical examples were assigned to graded methods to support their efficiency.

**Key Words: ***Radial
basis functions, Meshless method, Thin plate spline, Domain decomposition,
Finite difference*