In this paper, we employ the time fractional higher order nonlinear Boussinesq–Burgers system to investigate shallow water waves based on the Riemann–Liouville and Caputo fractional partial derivatives. The Lie algebra of infinitesimal generators associated with the symmetry groups that leave the studied system invariant is systematically constructed, and the similarity reductions are established. Furthermore, conserved quantities of the system under consideration are formulated. Next, the power series solution, including the convergence analysis, is obtained. Based on the fractional Sumudu–Adomian decomposition method, some approximate solutions are derived. Finally, in order to show the efficiency of the considered methods, the effect of the fractional order, as well as the dynamical behavior of solutions, numerical validation and graphical representations are designed.

中文翻译：

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时间分数高阶 Boussinesq-Burgers 系统的不变分析、近似解和守恒定律

在本文中，我们采用时间分数高阶非线性 Boussinesq-Burgers 系统来研究基于 Riemann-Liouville 和 Caputo 分数偏导数的浅水波。系统地构造了与使研究系统保持不变的对称群相关的无穷小生成元的李代数，并建立了相似性约简。此外，还制定了所考虑的系统的守恒量。接下来，获得幂级数解，包括收敛分析。基于分数式Sumudu-Adomian分解方法，推导了一些近似解。最后，为了显示所考虑方法的效率，设计了分数阶的效果以及解的动态行为、数值验证和图形表示。