This paper presents travelling wave solutions for the nonlinear time-fractional Gardner and Benjamin–Ono equations via the exp(\(- \Phi ( \varepsilon ))\)-expansion approach. Specifically, both the models are studied in the sense of conformable fractional derivative. The obtained travelling wave solutions are structured in rational, trigonometric (periodic solutions) and hyperbolic functions. Further, the investigation of symmetry analysis and nonlinear self-adjointness for the governing equations are discussed. The exact derived solutions could be very significant in elaborating physical aspects of real-world phenomena. We have 2D and 3D illustrations for free choices of the physical parameter to understand the physical explanation of the problems. Moreover, the underlying equations with conformable derivative have been investigated using the new conservation theorem.

本文通过exp（\（-\ Phi（\ varepsilon））\）给出了非线性时间分数阶Gardner和Benjamin–Ono方程的行波解。-扩展方法。具体来说，这两个模型都是在适度的分数导数意义上进行研究的。所获得的行波解由有理，三角（周期解）和双曲函数构成。此外，讨论了控制方程的对称性分析和非线性自伴性的研究。精确推导的解决方案在阐述现实世界现象的物理方面可能非常重要。我们提供2D和3D插图供您自由选择物理参数，以了解问题的物理解释。此外，已经使用新的守恒定理研究了具有相称导数的基础方程。