Dynamical structure of truncated M−fractional Klein–Gordon model via two integral schemes

Abstract

In this work, the Truncated M−fractional Klein–Gordon (tM-fKG) equation is integrated by implementing the exp(-ϕ(ξ))-expansion and new form of modified Kudryashov’s schemes. The novel exponential, hyperbolic, trigonometric, rational solutions and multi-soliton solutions with free parameters are achieved with the help of the schemes. Taking special values of the parameters, the multi-bell wave, periodic rogue, linked rogue, soliton solution through the exp(-ϕ(ξ))-expansion and bright and dark bell shape, periodic wave and linked rogue wave solutions through the new form of modified Kudryashov’s schemes are successfully constract, which are significantly stimulating phenomena in mathematical physics. Furthermore, the effects of Truncated M−fractional time on the obtained solutions of the tM-fK-G equation are shown with proper graphical representations.