The regularized and the modified regularized long wave (RLW and MRLW) equations are solved numerically by the Bernstein polynomials in both the space and time directions based on Kronecker product. In this paper, we applied a fully different Bernstein collocation method than the other methods which used Bernstein polynomials to solve the problems. The approximate solution is defined by the Bernstein polynomials in all directions. A general form for any $m$ derivative of any Bernstein polynomials is constructed. A general matrix form for the vector of any $m$ derivative of any Bernstein polynomials is also constructed. Convergence study for the proposed numerical scheme is investigated. To determine the conservation properties of the RLW and MRLW equations, three invariants of motion ($I_1$, $I_2$ and $I_3$) are computed. To test the accuracy, two error norms (${\Vert E\Vert }_2$ and ${\Vert E\Vert }_{\infty }$) are evaluated. Numerical outcomes and comparisons with other techniques for the single and the interaction of two solitary waves for RLW and MRLW equations are presented.

正则化和修正的正则化长波（RLW 和 MRLW）方程通过伯恩斯坦多项式在空间和时间方向上基于 Kronecker 乘积进行数值求解。在本文中，我们应用了与使用伯恩斯坦多项式解决问题的其他方法完全不同的伯恩斯坦搭配方法。近似解由所有方向的伯恩斯坦多项式定义。任何的一般形式$米$构造任何伯恩斯坦多项式的导数。任意向量的一般矩阵形式$米$还构造了任何伯恩斯坦多项式的导数。研究了所提出的数值方案的收敛性研究。为了确定 RLW 和 MRLW 方程的守恒性质，三个运动不变量 (${\mathrm{一世}}_1$, ${\mathrm{一世}}_2$ 和 ${\mathrm{一世}}_3$) 计算。为了测试准确性，两个误差范数（${\Vert 乙\Vert }_2$ 和 ${\Vert 乙\Vert }_{\infty }$) 进行评估。给出了 RLW 和 MRLW 方程的单个和两个孤立波相互作用的数值结果和与其他技术的比较。