Time fractional modified anomalous sub-diffusion equation with a nonlinear source term through locally applied meshless radial point interpolation

In this paper, an alternative approach of spectral meshless radial point interpolation (SMRPI) is applied to the modified anomalous fractional sub-diffusion equation with a nonlinear source term in one and two dimensions. The time fractional derivative is described in the Riemann-Liouville sense. The applied approach is based on a combination of meshless methods and the spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct the shape functions which act as basis functions in the frame of the SMRPI. It is proved that the scheme is unconditionally stable with respect to the time variable in H-1 and convergent with the order of convergence O(delta t(1+gamma)), 0 < gamma < 1. In this work, the thin plate splines (TPS) are used as the radial basis functions. In order to eliminate the nonlinearity, a simple predictor-corrector (P-C) scheme is used. The results of numerical experiments are compared to the analytical solutions in order to confirm the accuracy and the efficiency of the presented scheme.