Abstract In this paper, group analysis of the time-fractional Harmonic Oscillator equation with Riemann–Liouville derivative is performed and its reduced fractional ordinary differential equations are determined. With the aid of the concept of nonlinear self-adjoint and the fractional generalization of the Noether operators are obtained conserved vector for this equation. Furthermore, by using Laplace transform and invariant subspace method, exact solutions of the Riemann–Liouville and Caputo fractional partial differential equations are also constructed.

中文翻译：

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时间分数阶谐波振荡方程的对称性、相似不变解、守恒定律和精确解

摘要 本文对具有Riemann-Liouville导数的时间分数阶谐波振荡器方程进行群分析，并确定了其约简分数阶常微分方程。借助非线性自伴随的概念和Noether算子的分数推广，得到了该方程的守恒向量。此外，利用拉普拉斯变换和不变子空间方法，还构造了黎曼-刘维尔和卡普托分数阶偏微分方程的精确解。