The finite differences method for solving systems on irregular shapes

A relatively simple and efficient symbolic-numerical procedure based on the finite differences method for solving partial differential equations on systems of irregular shapes is presented. The new concept is based on the spline parameterization of the irregular domain. The curvilinear domain of the real system is transformed to the rectangular domain by spline functions where the finite differences method is used to solve the transformed system of depended variables. The numerical results are then transported back to the original irregular shape of the system. In order to present the symbolic-numerical technique effectively, the Laplace's equation of heat transfer with the Dirichlet and the Neumann boundary conditions in different 2D curvilinear domains is considered. The proposed technique is applied for the non-steady-state heat transfer by conduction as well. Numerical experiments were performed to justify the proposed method.