Analysis of Kudryashov’s equation with conformable derivative via the modified Sardar sub-equation algorithm

In the present work, we utilize a new Sardar sub-equation approach, leading to the successful derivation of several exact solutions for the time-fractional Kudryashov's equation, which describes the propagation pulses in optical fibers. These solutions encompass a range of categories, including singular, wave, bright, mixed dark-bright, and bell-shaped optical solutions. To effectively showcase these novel optical soliton solutions, we utilized contour plots, three-dimensional graphs, and three-dimensional surface plots. Through multiple graphical simulations, we provide a comprehensive demonstration of the dynamic behavior and physical significance of these optical solutions within the proposed model. Moreover, we investigate the magnitude of the time-fractional Kudryashov's equation by analyzing the influence of the fractional order derivative and the impact of the time parameter on the newly constructed optical solutions. Our findings highlight the versatility of the presented method, as it can readily be applied to other differential equations in various fields, such as non-linear optics and plasma physics. The proposed technique is a generalized form that incorporates various methods, including the improved Sardar sub-equation method, the modified Kudryashov method, the tanh-function extension method, and others. To the best of our knowledge, these solutions are novel and have not been reported in the literature and have potential application in nonlinear optics.