Abstarct: In this thesis, a high-order accurate numerical method baxsed on a space–time collocation pseudo-spectral scheme is developed for the fractional regularized long-wave Burgers equation. This equation models the propagation of nonlinear waves in media exhibiting memory effects and energy dissipation. The solution method is first established for the case involving a constant-order Caputo fractional derivative, with a convergence analysis carried out in the space of continuous functions. The methodisthenextendedtoaccommodatetheequationwithavariable-order(space time-dependent) Caputo fractional derivative. To this end, a novel operational matrix is derived for the direct approximation of the variable-order Caputo derivative by means of Jacobi–Gauss quadrature. The principal novelty lies in the simultaneous application of the pseudo-spectral col location approach in both temporal and spatial dimensions, as well as the incorporation of the mixed-derivative term, which has been disregarded in most prior studies. Numerous numerical examples are presented for various configurations. The results demonstrate exponential conver gence, high accuracy, and robustness of the method with respect to variations in the order function, confirming its superiority over conventional finite-difference methods.