A mesh-free method using Pascal polynomials for analyzing space-fractional PDEs in irregular biological geometries

In recent years, various numerical methods, including finite difference method (FDM), finite volume method (FVM), and finite element method (FEM), have been devised to solve time-fractional PDEs, on different computational geometries. However, owing to the presence of integrals in the definition of space-fractional PDEs (SFPDEs), only a few numerical procedures have been developed for solving SFPDEs on irregular regions. This limitation arises because a Gauss–Legendre quadrature must be employed to approximate certain integrals in the calculation of fractional derivatives. The present paper introduces a novel numerical solution based upon Pascal polynomials and a multiple-scale idea. The strength of this technique lies in the construction of Pascal polynomials from monomials. The Pascal polynomials will be employed for estimating the spatial derivatives of SFPDEs on different non-rectangular physical areas.