Explicit exact solutions and bifurcation analysis for the mZK equation with truncated M-fractional derivatives utilizing two reliable methods

The ( 2 + 1 ) \left(2+1) -dimensional modified Zakharov-Kuznetsov (mZK) partial differential equation is of importance as a model for phenomena in various physical fields such as discrete electrical lattices, electrical waves in cold plasmas, nonlinear optical waves, deep ocean-waves, and the propagation of solitary gravity waves. In this study, the main objective is to give a detailed analysis of exact traveling wave solutions of the mZK equation with truncated M-fractional spatial-temporal partial derivatives. Using an appropriate traveling wave transformation and the homogeneous balance rule, the mZK equation is converted into a corresponding ordinary differential equation (ODE). The ( G ' / G , 1 / G ) \left(G<^>{\prime} /G,1/G) -expansion and Sardar subequation methods are then used to derive exact solutions of the ODE in the form of functions such as hyperbolic, trigonometric, and special generalized hyperbolic and trigonometric functions. The two methods give some novel solutions of the proposed model and are presented here for the first time. The fractional-order effects are studied through numerical simulations, including three-dimensional (3D), two-dimensional (2D), and contour plots. These numerical simulations clearly show physical interpretations of selected solutions. In particular, the generalized hyperbolic and trigonometric function solutions that have been derived by the Sardar subequation method are important as they provide examples of exact traveling wave solutions of various physical types. Furthermore, the results include examples of bifurcations and chaotic behaviors of the model through 2D and 3D plots when parameter values are varied. Finally, the methods of solution described in this study are reliable, powerful, and efficient and can be recommended to obtain traveling wave solutions of other nonlinear partial differential equations with truncated M-fractional derivatives.