Under investigations into this paper is a higher-dimensional model, namely the time fractional (3 + 1)-dimensional Korteweg–de Vries (KdV)-type equation, which can be usually used to express shallow water wave phenomena. At the beginning, the symmetry of the time fractional (3 + 1)-dimensional KdV-type equation via the group analysis scheme is obtained. The definition of the fractional derivative in the sense of the Riemann–Liouville is considered. Then, the one-parameter Lie group and invariant solutions of this considered equation are constructed. Subsequently, we applied a direct method to construct the optimal system of one-dimensional of this considered equation. Next, this considered higher-dimensional model can be reduced into the lower-dimensional fractional differential equations (FDEs) with the help of the three-parameter and two-parameter Erdélyi–Kober fractional differential operators (FDOs). Lastly, conservation laws of this discussed equation by using a new conservation theorem are also found. A series of results of the above obtained can provide strong support for us to reveal the mysterious veil of this viewed equation.

中文翻译：

---

时间分数(3 + 1)-维KDV型方程组分析

本文研究的是一个更高维的模型，即时间分数(3 + 1)维 Korteweg-de Vries (KdV) 型方程，通常可用于表达浅水波浪现象。一开始，时间分数的对称性(3 + 1)通过群分析方案得到一维KdV型方程。考虑了黎曼-刘维尔意义上的分数导数的定义。然后，构造该考虑方程的单参数李群和不变量解。随后，我们应用直接方法来构造该考虑方程的一维最优系统。接下来，在三参数和两参数 Erdélyi-Kober 分数微分算子 (FDO) 的帮助下，可以将这个考虑过的高维模型简化为低维分数微分方程 (FDE)。最后，通过使用新的守恒定理，还发现了这个讨论的方程的守恒定律。