On the Computational Efficiency of Two-Dimensional Haar Wavelet Schemes for Solving Nonlinear Evolutionary PDEs

This paper presents collocation methods employing two-dimensional uniform and non-uniform Haar wavelet, integrated with the Newton–Raphson iterative algorithm, for the efficient solution of nonlinear partial differential equations (PDEs). The proposed approaches are specifically applied to the generalized Benjamin–Bona–Mahony–Burgers’ equation and the generalized Rosenau–KdV–RLW equation. A rigorous theoretical analysis is provided to establish the convergence behavior of both methods. To assess their accuracy and computational performance, a series of benchmark problems are examined across a range of parameter values. The numerical results reveal remarkable accuracy, even with coarse spatial discretization. Furthermore, a comparative evaluation against existing methods demonstrates the enhanced efficiency and robustness of the proposed techniques.