Fractional convection-diffusion systems in complex 2D and 3D geometries: A Bernoulli polynomial-based kernel method

This study presents an accurate meshless method for the efficient solution of nonlinear time-fractional convection-diffusion systems in complex two- and three-dimensional geometries. The proposed approach combines spatial discretization using a Bernoulli polynomial kernel function with temporal discretization via the backward differentiation formula. By employing positive definite kernels, the method achieves high spatial accuracy, while the use of the backward differentiation formula ensures high-order temporal accuracy. Convergence conditions and error bounds are rigorously analyzed using the Mittag-Leffler function. Error estimates are derived based on the spectral properties of the associated matrices, and inequalities describing error propagation over time are established. The method is tested on a variety of benchmark problems, including the Brusselator model and nonlinear coupled convection-diffusion systems, across both 2D and 3D domains. Extensive numerical experiments are carried out on various geometries-such as rectangular, circular, and spherical shapes-demonstrating the method’s robustness and accuracy in handling both regular and irregular computational domains.