An improvement to the traditional Finite Volume Method (FVM) for the solution of boundary value problems is presented. The new method applies the local Hermitian interpolation with Radial Basis Functions (RBF) as an interpolation scheme to the FVM discretization. This approach, called the Control Volume-Radial Basis Function (CV-RBF) method, uses an interpolation scheme based on the meshless Symmetric method, in which the numerical solution is approximated by employing the governing equation and the boundary condition operators. The RBF implemented is the Multiquadric (MQ) with a shape parameter found experimentally. The two-dimensional solutions to the Dirichlet problem for linear heat conduction, heat transfer in the Poiseuille flow and the non-linear conduction situations are obtained by the CV-RBF method. The numerical results are in agreement with the corresponding analytical and numerical solutions found in the literature.