Numerical Simulation of Enhanced Fractional Rayleigh–Stokes Model for Heated Generalized Second-Grade Fluids Using RBF-PUM

The primary objective of this study is to demonstrate the effectiveness of the radial basis function partition of unity collocation method (RBF-PUM) for solving the fractional Rayleigh-Stokes problem in two dimensions. The time-fractional derivative is formulated in the Riemann–Liouville sense. Initially, the time-fractional derivatives in the governing equation are discretized, followed by the application of RBF-PUM to approximate the spatial derivatives. This method offers notable advantages, including local adaptivity and the ability to adjust node density within each partition. These features help mitigate the ill-conditioning commonly encountered in traditional RBF-based methods. The unconditional stability and convergence of the time-discrete scheme are established using the energy method. To validate the theoretical findings and illustrate the efficiency of RBF-PUM, two numerical experiments are conducted on regular and irregular 2D domains, alongside comparative numerical tests.