Abstract: The paper is intended to study the nonlinear behavior of planar ion-acoustic waves  in an unmagnetized plasma consisting of positive ions, non-extensive electrons, positrons, and stationary negatively charged dust grains. In the basic fluid eq-uations, Jumarie's modified Riemann-Liouville fractional derivative is adopted as the time derivative for the first time. Combined with the reductive perturbation method, a Korteweg-de Vries (KdV) equation is derived for the motions of ion-acoustic waves. The Chebyshev-Legendre-Galerkin (CLG) pseudospectral method is applied to solve this KdV equation numerically for analyzing how the plasma parameters influence the soliton structures of ion-acoustic waves. It concludes that increasing the order of fractional derivatives can increase the amplitude of solitary waves, which enables a better understanding of nonlinear wave phenomena in both astrophysical environments and laboratory plasma experiments.

Key words: fractional calculus, KdV equation, spectral method, ion-acoustic waves