  In this paper, a novel Hermite radial basis function-based differential
quadrature method (H-RBF-DQ) is presented. This new method is designed to treat
derivative boundary conditions accurately. The developed method is very
different from the original Hermite RBF method. In order to illustrate the
specific process of this method, although the method can be used to study most
of partial differential equations, the numerical simulation of two-dimensional
variable-order time fractional advection-diffusion equation is chosen as an
example. For the general case of irregular geometry, the meshless local form of
RBF-DQ was used and the multiquadric type of radial basis functions are
selected for the computations. The method is validated by the documented test
examples involving variable-order fractional modeling of air pollution. The
numerical results demonstrate the robustness and the versatility of the
proposed approach.
