  In this paper, we present a method based on Radial Basis Function
(RBF)-generated Finite Differences (FD) for numerically solving diffusion and
reaction-diffusion equations (PDEs) on closed surfaces embedded in
$\mathbb{R}^d$. Our method uses a method-of-lines formulation, in which surface
derivatives that appear in the PDEs are approximated locally using RBF
interpolation. The method requires only scattered nodes representing the
surface and normal vectors at those scattered nodes. All computations use only
extrinsic coordinates, thereby avoiding coordinate distortions and
singularities. We also present an optimization procedure that allows for the
stabilization of the discrete differential operators generated by our RBF-FD
method by selecting shape parameters for each stencil that correspond to a
global target condition number. We show the convergence of our method on two
surfaces for different stencil sizes, and present applications to nonlinear
PDEs simulated both on implicit/parametric surfaces and more general surfaces
represented by point clouds.
