Nonlinearity And Memory Effects: The Interplay Between These Two Crucial Factors In The Harry Dym Model

This study investigates the nonlinear time-fractional Harry Dym (𠕋𠔽℠𠔻) equation, a model with significant applications in soliton theory and connections to various other nonlinear evolution equations. The Harry Dym (â„ ð ”») equation describes the propagation of nonlinear waves in various physical contexts, including shallow water waves, nonlinear optics, and plasma physics. The fractional-order derivative introduces a memory effect, allowing the model to capture nonlocal interactions and long-range dependencies in the wave dynamics. The primary objective of this research is to obtain accurate analytical solutions to the 𠕋𠔽℠𠔻 equation and explore its physical characteristics. We employ the Khater III method as the primary analytical technique and utilize the He’s variational iteration (â„ ð •€ð • ) method as a numerical scheme to validate the obtained solutions. The close agreement between analytical and numerical results enhances the applicability of the solutions in practical applications of the model. This research contributes to a deeper understanding of the â„ ð ”» equation’s behavior, particularly in the presence of fractional-order dynamics. The obtained solutions provide valuable insights into the complex interplay between nonlinearity and memory effects in the wave propagation phenomena described by the model. By shedding light on the physical characteristics of the 𠕋𠔽℠𠔻 equation, this study paves the way for further investigations into its potential applications in diverse physical settings.