Study of Time-Fractional Dust Ion Acoustic Waves Propagation in Collisionless Unmagnetized Dusty Plasmas

The unmagnetized collisionless dusty plasmas consisting of Maxwell-distributed electrons, inertial ions, and negatively charged dust grains are taken into consideration in order to study the propagation characteristics of time fractional dust ion acoustic waves (DIAWs). The dust grains under consideration are believed to be extremely massive point charges with sizes far smaller than the plasma Debye length and collisional mean free path. The well-known reductive perturbation method is used to derive the Korteweg-de Vries (KdV) equation and then converted the KdV equation into time fractional (TF) KdV (TFKdV) equation using the procedure of El-Wakil et al. (AIP Adv 13:105012 2023). It is found that in the case of the first approximation solution, the amplitude and width of the TFKdV soliton increases with the increasing effect of the time fractional parameter (both in the limit \(\xi <0\) $\xi <0$ and \(\xi >0\) $\xi >0$ ) and travelling wave velocity (in the range \(-50000<\xi <25000\) $-50000<\xi <25000$ ). In addition, the amplitude and width of TFKdV soliton are increasing charged dust number and travelling wave velocity in the case of the second approximation solution. The electric field decreases with the time-fractional parameter, while it increases with travelling wave velocity in the case of the second approximation solution.