This article is devoted to solving numerically the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation that has several applications in physics and applied sciences. First, the time derivative is approximated by using a finite difference formula. Afterward, the stability and convergence analyses of the obtained time semi‐discrete are proven by applying the energy method. Also, it has been demonstrated that the convergence order in the temporal direction is O(dt). Second, a fully discrete formula is acquired by approximating the spatial derivatives via Legendre spectral element method. This method uses Lagrange polynomial based on Gauss–Legendre–Lobatto points. An error estimation is also given in detail for full discretization scheme. Ultimately, the GBBMB equation in the one‐ and two‐dimension is solved by using the proposed method. Also, the calculated solutions are compared with theoretical solutions and results obtained from other techniques in the literature. The accuracy and efficiency of the mentioned procedure are revealed by numerical samples.

中文翻译：

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基于勒让德谱元法求解非线性广义本杰明-波纳-马洪尼-伯格斯方程的数值和理论讨论

本文致力于用数值方法求解非线性广义本杰明-波纳-马洪尼-伯格斯（GBBMB）方程，该方程在物理学和应用科学中有多种应用。首先，通过使用有限差分公式来近似时间导数。然后，通过应用能量法证明了所获得的半离散时间的稳定性和收敛性分析。此外，已经证明，时间方向上的收敛阶为O（dt）。其次，通过勒让德谱元法近似空间导数，得到一个完全离散的公式。该方法使用基于高斯-勒让德-洛巴托点的拉格朗日多项式。对于完全离散化方案，还将详细给出误差估计。最终，通过使用所提出的方法来解决一维和二维中的GBBMB方程。此外，将计算出的解与理论解以及从文献中其他技术获得的结果进行比较。数字样本显示了所提到的过程的准确性和效率。