A novel shifted Vieta–Lucas spectral collocation approach for multidimensional generalized Benjamin–Bona–Mahony–Burgers equations

Here, we put forward a numerical method for solving the (1+1)- and (2+1)-dimensional nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation using a collocation approach based on shifted Vieta–Lucas polynomials (SVLPs). The proposed technique involves expressing the approximate solution as a finite series expansion in terms of SVLPs and enforcing the governing equation at specific collocation points. This process leads to a nonlinear algebraic system amenable to efficient solution. The method offers high accuracy and fast convergence. An error analysis and convergence proof of the proposed algorithm are established, thereby ensuring its theoretical soundness. Several numerical experiments were conducted to validate the effectiveness of the approach, and the results demonstrate its superiority over existing methods in terms of accuracy and simplicity. This makes the proposed SVLP-based collocation method a versatile and stable technique for tackling nonlinear partial differential equations in applied sciences and engineering.