Kansa method for solving initial-value problem of hyperbolic heat conduction in nonhomogeneous medium

The Kansa method is a well-established method for solving partial differential equation-based problems. Nevertheless, it is still a relatively young, 30-year-old method, especially compared to the finite difference method. Consequently, there are still many problems that it is not sure that the Kansa method is suitable for solving them. An example of such a situation is the unsteady-state hyperbolic heat conduction problem described by Cattaneo-Vernotte model in a medium with spatially-variable thermal parameters alongside with time-dependent Neumann boundary conditions or irregularly shaped computational domain. In this paper, such problems were considered in the one- and two-dimensional variants using the Kansa method in the so-called coefficient formulation. The study conducted in the article showed that the Kansa method is highly likely suitable for solving problems of this type. A good agreement of the Kansa method solution with the finite difference method, the finite element method and the analytical solution was obtained. Also, characteristic features of hyperbolic heat flow were observed in the solution: delay in the heat flux transmission, the introduction of strong damping into the system, the intensification of the amplitude of periodic temperature and heat flux changes, the phase shift in the temperature oscillation. The article also addresses the problem of selecting the p exponent. It has been shown that the values recommended in the literature −1/2 and 1/2 may not be appropriate for the considered class of the problem. It has also been shown that for this class of problem, much better results are obtained when using very small positive values of the p exponent.