Abro Kashif Ali, Siyal Ambreen, Atangana Abdon, Al-Mdallal Qasem M
Institute of Ground Water Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein, South Africa.
Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Pakistan.
Opt Quantum Electron. 2023;55(8):704. doi: 10.1007/s11082-023-04919-1. Epub 2023 Jun 8.
Klein-Gordon equation characterizes spin-particles through neutral charge field within quantum particle. In this context, fractionalized Klein-Gordon equation is investigated for the comparative analysis of the newly presented fractional differential techniques with non-singularity among kernels. The non-singular and non-local kernels of fractional differentiations have been employed on Klein-Gordon equation for the development of governing equation. The analytical solutions of Klein-Gordon equation have been traced out by fractional techniques by means of Laplace transforms and expressed in terms of series form and gamma function. The data analysis of fractionalized Klein-Gordon equation is observed for Pearson's correlation coefficient, probable error and regression analysis. For the sake of comparative analysis of fractional techniques, 2D sketch, 3D pie chart contour surface with projection and 3D bar sketch have been depicted on the basis of embedded parameters. Our results suggest that varying frequency has reversal trends for quantum wave and de Broglie wave.
克莱因-戈尔登方程通过量子粒子内的中性电荷场来描述自旋粒子。在此背景下，研究了分数阶克莱因-戈尔登方程，以对新提出的核内无奇异性的分数微分技术进行比较分析。分数微分的非奇异和非局部核已应用于克莱因-戈尔登方程以建立控制方程。通过拉普拉斯变换利用分数技术求出了克莱因-戈尔登方程的解析解，并以级数形式和伽马函数表示。对分数阶克莱因-戈尔登方程进行了数据分析，包括皮尔逊相关系数、可能误差和回归分析。为了对分数技术进行比较分析，并基于嵌入参数绘制了二维草图、带投影的三维饼图等高面和三维柱状草图。我们的结果表明，不同频率对量子波和德布罗意波具有相反的趋势。
