Generalized Finite Difference Method for Anomalous Diffusion on Surfaces

In this study, a localized collocation method called generalized finite difference method (GFDM) is developed to solve the anomalous diffusion problems on surfaces. The expressions of the surface Laplace operator, surface gradient operator and surface divergence operator in tangent space are given explicitly, which is different from the definition of differential operators in the Euclidean space. Based on the moving least square theorem and Taylor series, GFDM shares similar properties with standard FDM and avoids mesh dependence, enabling numerical approximations of the surface operators on complex 3D surfaces. Simultaneously, a standard finite difference scheme is adopted to discretize the time fractional derivatives. By using GFDM, we succeed in solving both constant- and variable- order time fractional diffusion models on surfaces. Numerical examples show that the present meshless scheme has good accuracy and efficiency for various fractional diffusion models.