Discretization of unknown functions: A new method to solve partial differential equations with mixed boundary conditions

In this paper, we develop a new numerical technique to obtain an approximate solution of partial differential equations subject to mixed boundary conditions (MBCs). The approach has been applied to a class of differential equations which frequently arise in a large variety of problems such as heat conduction, potential theory, and diffusion-controlled chemical reactions. In our approach, based on the discretization of unknown functions (DF), the solution is expressed as a series expansion and the determination of the series coefficients is reduced to the solution of a system of algebraic equations. The main advantages of the DF procedure are: (a) the smoothness of the function and of its first derivative in the different domains, whereas the other numerical methods generally show a highly oscillating behavior; (b) the fast convergence of the series expansion. This method has been applied to solve diffusion problems in different coordinate systems (trigonometric, cylindrical and spherical). The obtained results have been compared with the analytical solution (when available) as well as with other numerical methods commonly used to solve MBCs problems.