On the Tantawy technique for modeling fractional dissipative dust-acoustic solitons in a polarized complex plasma, part (I): Fractional damped KdV equation

This study examines the properties of linear and nonlinear low-frequency fractional dust-acoustic waves (DAWs) in a complex plasma with polarization force. This model is composed of inertialess Maxwellian electrons and ions as well as inertial negatively charged dust grains. In this model, the dust-neutral collisions incorporate the dissipative effect of wave damping. The impact of the polarization force term, the electron concentration, and dust-neutral collisions on the linear properties of DAWs is examined. To investigate the impact of various physical parameters on the profile of nonlinear dust-acoustic solitons (DASs), the fluid model equations are reduced to the damped Korteweg-de Vries (KdV) equation via the reductive perturbation technique (RPT) in the limit of vanishingly small dust-neutral collisions. It is known that the damped KdV equation is completely nonintegrable, that is, it does not support exact solutions. Thus, in the second part of this study, the fractional damped KdV equation is derived to examine the influence of fractionality on the dynamics of dissipative DASs. For this purpose, we utilize an appropriate transformation to convert the integer-order evolutionary wave equation (damped KdV) into a damped fractional KdV (FKdV) equation. Subsequently, we employ one of the most precise methods for studying fractional differential equations, called the Tantawy technique. The time-varying solutions of both integer and damped FKdV equations yield weakly negative-polarity dissipative dust-acoustic solitons (DDASs). A numerical investigation of the evolution of (non)fractional DDASs, focusing on the polarization force term, electron concentration, and dust-neutral collisional parameter, is presented.