In this study, we consider a time-fractional shallow water wave model which was introduced by Gear and Grimshaw. This model represents the dynamics of two-layered shallow water waves in oceans and rivers. Also it covers many shallow water wave models with special values taken to parameters. By performing the group invariant analysis approach for the governing system with Riemann–Liouville derivative, symmetry generators were found. Similarity reduction was constructed by Erdelyi–Kober sense. In addition, by adapting the Sine-Gordon expansion method, we find some exact traveling wave solutions. Furthermore, explicit solutions in form of power series are examined via the residual power series method. Exact and approximate solutions are compared for some $\alpha$ values and these are given as tables.

在这项研究中，我们考虑了由 Gear 和 Grimshaw 引入的时间分数浅水波模型。该模型代表了海洋和河流中两层浅水波的动力学。它还涵盖了许多具有特殊参数值的浅水波模型。通过对具有黎曼-刘维尔导数的控制系统执行群不变分析方法，找到了对称生成器。Erdelyi-Kober sense 构建了相似性降低。此外，通过采用 Sine-Gordon 展开法，我们找到了一些精确的行波解。此外，通过剩余幂级数方法检查幂级数形式的显式解决方案。比较了一些精确和近似的解决方案$\alpha$值，这些以表格形式给出。