  We propose algorithms for solving convective-diffusion partial differential
equations (PDEs), which model surfactant concentration and heat transport on
evolving surfaces, based on intrinsic kernel-based meshless collocation
methods. The algorithms can be classified into two categories: one collocates
PDEs directly and analytically, and the other approximates surface differential
operators by meshless pseudospectral approaches. The former is specifically
designed to handle PDEs on evolving surfaces defined by parametric equations,
and the latter works on surface evolutions based on point clouds. After some
convergence studies and comparisons, we demonstrate that the proposed method
can solve challenging PDEs posed on surfaces with high curvatures with
discontinuous initial conditions with correct physics.
