In this paper, Kudryashov and modified Kudryashov methods are implemented for the first time to compute new exact traveling wave solutions of the space-time fractional $(3+1)$-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation and Calogero–Bogoyavlenskii–Schiff and Bogoyavlensky Konopelchenko (CBS-BK) equation. With the help of wave transformation, the aforementioned fractional differential equations are converted into nonlinear ordinary differential equations. The purpose of this paper is to devise novel exact solutions for the space-time-fractional $(3+1)$-dimensional CBS and the space-time-fractional CBS-BK equations by utilizing the Kudryashov and modified Kudryashov techniques. The solutions, thus, acquired are demonstrated in figures by choosing appropriate values for the parameters. The solutions derived take the form of various wave patterns, including the kink type, the anti-kink type and the singular kink wave solutions. The obtained solutions are indeed beneficial to analyze the dynamic behavior of fractional CBS and CBS-BK equations in describing the interesting physical phenomena and mechanisms. The obtained solutions are entirely new and can be considered as a generalization of the existing results in the ordinary derivative case. The techniques presented here are very simple, efficacious and plausible and hence can be employed to attain new exact solutions for fractional PDEs.

在本文中，首次实现了 Kudryashov 和改进的 Kudryashov 方法来计算时空分数的新精确行波解$（3+1）$维Calogero–Bogoyavlenskii–Schiff (CBS) 方程和Calogero–Bogoyavlenskii–Schiff 和Bogoyavlensky Konopelchenko (CBS-BK) 方程。借助于波动变换，上述分数阶微分方程被转换为非线性常微分方程。本文的目的是为时空分数设计新颖的精确解$（3+1）$利用 Kudryashov 和改进的 Kudryashov 技术计算一维 CBS 和时空分数 CBS-BK 方程。因此，通过为参数选择适当的值，获得的解决方案在图中得到了展示。导出的解采用各种波形的形式，包括扭结型、反扭结型和奇异扭结波解。所获得的解确实有利于分析分数式 CBS 和 CBS-BK 方程的动态行为，以描述有趣的物理现象和机制。所获得的解决方案是全新的，可以被视为普通导数情况下现有结果的推广。这里介绍的技术非常简单、有效且可信，因此可用于获得分数偏微分方程的新精确解。