We utilize the fractional modified Riemann-Liouville derivative in the sense to develop careful arrangements of space–time fractional coupled Boussinesq equation which emerges in genuine applications, for instance, nonlinear framework waves iron sound waves in plasma and in vibrations in nonlinear string and space–time fractional-coupled Boussinesq Burger equation that emerges in the investigation of liquids stream in a dynamic framework and depicts engendering of shallow-water waves. A decent comprehension of its solutions is exceptionally useful for beachfront and engineers to apply the nonlinear water wave model to the harbor and seaside plans. A summed-up partial complex transformation is correctly used to change this equation to a standard differential equation thus, many precise logical arrangements are acquired with all the free parameters. At this point, the traveling wave arrangements are articulated by hyperbolic functions, trigonometric functions, and rational functions, if these free parameters are considered as specific values. We obtain kink wave solution, periodic solutions, singular kink type solution, and anti-kink type solutions which are shown in 3D and contour plots. The presentation of the method is dependable and important and gives even more new broad accurate arrangements.

我们在某种意义上利用分数修正黎曼-刘维尔导数来精心安排时空分数耦合 Boussinesq 方程，该方程出现在真正的应用中，例如，等离子体中的非线性框架波铁声波以及非线性弦和空间中的振动——时间分数耦合 Boussinesq Burger 方程出现在动态框架中的液体流研究中，并描述了浅水波的产生。对其解决方案的正确理解对于海滨和工程师将非线性水波模型应用于港口和海滨计划非常有用。正确地使用求和的偏复变换将该方程变为标准的微分方程，从而获得了许多具有所有自由参数的精确逻辑排列。此时，如果将这些自由参数视为特定值，则行波排列由双曲函数、三角函数和有理函数表达。我们获得了在 3D 和等高线图中显示的扭结波解、周期解、奇异扭结型解和反扭结型解。该方法的介绍是可靠且重要的，并提供了更多新的广泛准确的安排。