This research paper introduces a novel approach to deriving traveling wave solutions (TWSs) for the Caputo-fractional Klein–Gordon equations. This research presents a distinct methodological advancement by introducing TWSs of a particular time-fractional partial differential equation, utilizing a non-local fractional operator, specifically the Caputo derivative. To achieve our goal, a novel transformation is considered, that converts a time-fractional partial differential equation into fractional ordinary differential equations, enabling analytical solutions through various analytical methods. This paper employs the homotopy analysis method to achieve the target objectives. To demonstrate the efficiency and applicability of the proposed transform and method, two examples are discussed and analyzed in figures.

本研究论文介绍了一种推导 Caputo 分数阶 Klein-Gordon 方程行波解 (TWS) 的新方法。这项研究通过引入特定时间分数偏微分方程的 TWS，利用非局部分数算子，特别是 Caputo 导数，呈现出明显的方法论进步。为了实现我们的目标，我们考虑了一种新颖的变换，将时间分数阶偏微分方程转换为分数常微分方程，从而可以通过各种分析方法获得解析解。本文采用同伦分析方法来实现预期目标。为了证明所提出的变换和方法的效率和适用性，在图中讨论和分析了两个例子。