Numerical investigation of coupled distributed-order Bessel fractional PDEs using meshless radial basis function methods

Abstract

This study introduced a novel fully coupled distributed-order Bessel fractional model for two interacting variables, capturing memory effects through temporal distributed-order Bessel derivatives and anomalous spatial diffusion via Riesz fractional operators. A fully discrete numerical scheme combined convolution-based temporal approximations with meshless radial basis function discretizations in space, providing accurate and efficient simulations. Convergence and stability analyses employed energy methods and demonstrated unconditional stability and near-second-order convergence in both time and space. Two numerical examples illustrated the performance of the proposed approach: one with an exact solution validated the error estimates and convergence rates, while the second highlighted the scheme's effectiveness for problems without closed-form solutions. The results confirmed the high accuracy and robustness of the method, offering a new tool for modeling complex spatio-temporal phenomena with memory and nonlinear interactions.