Comparison of fractional effects for Phi-4 equation using beta and M-truncated derivatives

摘要

Fractional Phi-4 equation plays a crucial role in many studies of nuclear and particle physics. In this paper, the fractional Phi-4 equation is studied for two recently developed definitions of fractional derivatives and the comparison of the results with both derivatives is presented. The wave solutions of fractional Phi-4 equation are obtained using the extended sinhdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$sinh$$end{document}-Gordon equation expansion method. The beta and M-truncated fractional effects are considered. New complex trigonometric and complex hyperbolic function solutions are retrieved. Dark solitons and periodic solitary waves are observed for obtained solutions. The wave profiles obtained for the beta and M-truncated derivatives are compared for different values of fractional order using graphical simulations. The presented results are novel and the extended sinh-Gordon equation expansion method has been utilized to investigate fractional Phi-4 equation with beta and M-truncated derivative for the first time in this work.