In this paper, we solve the conformal time-fractional Klein-Gordon equation with high-order nonlinearities via the complete discrimination system for polynomial method, namely with the quadratic and cubic nonlinearities. Our results contain all the existing solutions, and some new exact solutions such as Jacobian elliptic function solutions, are also presented. Moreover, to ensure the existence of each solution, the concrete examples are given and the discussions of the influence of the fractional order are also presented.

中文翻译：

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具有高阶非线性的相容时间分数阶Klein-Gordon方程的精确解

本文通过多项式方法的完全判别系统，即二次和三次非线性方程，求解了具有高阶非线性的保形时间分数阶Klein-Gordon方程。我们的结果包含了所有现有的解决方案，并且还提出了一些新的精确解，例如Jacobian椭圆函数解。此外，为确保每个解的存在，给出了具体示例，并对分数阶的影响进行了讨论。