In this paper, the semi-inverse method is applied to derive the Lagrangian of the [Formula: see text]th Korteweg de Vries equation (KdV). Then the time and space differential operators of the Lagrangian are replaced by corresponding fractional derivatives. The variation of the functional of this Lagrangian is devoted to lead the fractional Euler Lagrangian via Agrawal’s method, which gives the space-time fractional KdV equation. Jumarie derivative is used to obtain the space-time fractional KdV equations. The homotopy analysis method (HAM) is applied to solve the derived space-time fractional KdV equation. Then numerical solutions are compared with the known analytical solutions by tables and figures.

中文翻译：

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齐次和非齐次 5α 阶时空分数 KdV 方程的近似解

在本文中，应用半逆法推导[公式：见正文]th Korteweg de Vries 方程（KdV）的拉格朗日。然后将拉格朗日的时空微分算子替换为相应的分数导数。该拉格朗日函数的变分致力于通过Agrawal的方法引导分数欧拉拉格朗日，得到时空分数KdV方程。Jumarie导数用于获得时空分数KdV方程。应用同伦分析法（HAM）求解导出的时空分数KdV方程。然后通过表格和图形将数值解与已知的解析解进行比较。