In this paper, the conformable-time-fractional Klein–Fock–Gordon equation is considered and solved using the Kudryashov-expansion method to extract dual-wave solutions. Only, the quadratic and the cubic cases of the model are investigated. It has been noticed that physical changes in the construction of the obtained solutions are reported in the case of transition from the quadratic-state into the cubic-state. Additionally, the Caputo-time-fractional quadratic–cubic Klein–Fock–Gordon is also considered and studied by implementing the residual power series method. A comparison between these two types of fractional derivatives is discussed and the 2D–3D plots are provided to support the findings of this work.

在本文中，使用Kudryashov展开法提取双波解，并考虑并求解了适度时间分数的Klein-Fock-Gordon方程。仅研究模型的二次和三次情况。已经注意到，在从二次态向立方态过渡的情况下，报告了所获得溶液的构造的物理变化。此外，还通过实施残差幂级数方法来考虑和研究Caputo时间分数阶二次-三次Klein-Fock-Gordon。讨论了这两种类型的分数导数之间的比较，并提供了2D–3D图以支持这项工作的结果。