A meshless radial basis function method for 2D steady-state heat conduction problems in anisotropic and inhomogeneous media

The paper presents a new meshless numerical method for solving 2D steady-state heat conduction problems in anisotropic and inhomogeneous media. The coefficients of the governing PDEs are spatially dependent functions including the main operator part. The boundary conditions of a most general form for the temperature and the heat flux are considered. The key idea of the method is the use of the basis functions which satisfy the homogeneous boundary conditions of the problem. Each basis function used in the algorithm is a sum of a RBF and a special correcting function which is chosen to satisfy the homogeneous BC of the problem. The conical radial basis functions, the Duchon splines and the multiquadric RBFs are used in approximation of the PDE. This allows us to seek an approximate solution in the form which satisfies the boundary conditions of the initial problem with any choice of the free parameters. As a result we separate the approximation of the boundary conditions and the approximation of the PDE inside the solution domain. The numerical experiments are carried out for accuracy and convergence investigations. The comparison of the numerical results obtained in the paper with the exact solutions and with the data obtained with the use of other numerical techniques is performed. The numerical examples demonstrate that the present method is accurate, convergent, stable, and computationally efficient in solving this kind of problems.