A hybrid radial basis functions collocation technique to numerically solve fractional advection-diffusion models

In this work, we propose a hybrid radial basis functions (RBFs) collocation technique for the numerical solution of fractional advection-diffusion models. In the formulation of hybrid RBFs (HRBFs), there exist shape parameter (c*) and weight parameter (epsilon) that control numerical accuracy and stability. For these parameters, an adaptive algorithm is developed and validated. The proposed HRBFs method is tested for numerical solutions of some fractional Black-Sholes and diffusion models. Numerical simulations performed for several benchmark problems verified the proposed method accuracy and efficiency. The quantitative analysis is made in terms of L-infinity, L-2, L-rms, and L-rel error norms as well as number of nodes N over space domain and time-step delta t. Numerical convergence in space and time is also studied for the proposed method. The unconditional stability of the proposed HRBFs scheme is obtained using the von Neumann methodology. It is observed that the HRBFs method circumvented the ill-conditioning problem greatly, a major issue in the Kansa method.