In this work, the regularized and the modified regularized long wave equations are solved using an improvised cubic B-spline collocation technique. Posteriori corrections are made in the cubic B-spline interpolant by forcing it to satisfy some specific interpolatory and end conditions, which results in the formation of improvised B-spline collocation technique. For temporal domain discretization, strong stability preserving Runge–Kutta scheme of third-order and four stages (SSP-RK43) is used. Seven important problems of the regularized and the modified regularized long wave equations are solved to demonstrate the applicability of the proposed technique. The motion of a single solitary wave, the interaction of two or more solitary waves, and the undulations of waves are shown graphically. These equations possess three invariants of motion namely mass, momentum, and energy, which need to remain constant with run time. For each problem, these invariants are calculated and it is observed that they coincide with exact values and remain preserved with time. The $L_2$ and $L_{\infty }$ error norms are calculated at different time levels and CPU time is noted. Also, the stability analysis of the technique is carried out and is found to be conditionally stable.

在这项工作中，正则化和修正的正则化长波方程使用改进的三次 B 样条搭配技术求解。通过强迫三次B样条插值满足某些特定的插值和结束条件，对三次B样条插值进行后验校正，从而形成了即兴的B样条搭配技术。对于时域离散化，使用了三阶四阶段的强稳定性 Runge-Kutta 方案（SSP-RK43）。解决了正则化和修正正则化长波方程的七个重要问题，以证明所提出技术的适用性。以图形方式显示单个孤立波的运动、两个或多个孤立波的相互作用以及波的起伏。这些方程具有三个运动不变量，即质量、动量和能量，它们需要随着运行时间保持恒定。对于每个问题，都会计算这些不变量，并观察到它们与精确值一致并随时间保持不变。这${升}_2$ 和 ${升}_{\infty }$错误范数是在不同的时间级别计算的，并记录了 CPU 时间。此外，对该技术进行了稳定性分析，发现其条件稳定。