Comparison of two-dimensional wavelets techniques for variable-order time fractional Regularized Long Wave Burgers equation representing propagation of surface water wave in a channel

This paper presents novel computational approaches for the numerical solution of the variable-order time fractional Regularized Long Wave Burgers (RLW-Burgers) equation based on two-dimensional wavelets. By employing the shifted Legendre, Euler, and Gegenbauer wavelets basis for the function approximation, alongside the operational matrices together with collocation points, the variable-order time fractional RLW-Burgers equation is transformed into a system of algebraic equations. In order to verify and validate our results obtained by the two-dimensional wavelets method, we also present the spectral collocation method (SCM) for solving this equation. The SCM is based on the shifted Legendre Gauss–Lobatto collocation (SL-GL-C) and shifted Chebyshev Gauss–Radau collocation (SC-GR-C) methods. The results obtained by the two-dimensional wavelets method using various wavelets have been compared with the results obtained by SCM. To comprehensively evaluate the accuracy and computational efficiency of the proposed two-dimensional wavelet-based approach, we employ the well-known spectral collocation method (SCM) as a benchmark for comparison. This benchmarking allows for a systematic assessment of the relative performance of both methodologies. A thorough comparison of the proposed wavelet approach and SCM is carried out over a wide range of test cases, notably those requiring complicated variable-order fractional dynamics. The findings of this research clearly demonstrate that the proposed wavelet-based strategy consistently outperforms other methods in terms of numerical accuracy, effectiveness, and moderate computational cost. These results showcase the robustness and practical effectiveness of our strategy, especially when dealing with challenging problems that may pose difficulties for existing spectral methods. Furthermore, several useful theorems are explored rigorously to assess the convergence and error estimation of the two-dimensional wavelets method. Finally, several numerical examples are provided to validate the computational efficiency and demonstrate the practical applicability of this proposed numerical approach.