Spatio-temporal fractional shock waves solution for fractional Korteweg-de Vries burgers equations

Abstract

ABSTRACT We examine the nonlinear characteristics of localized spatial (temporal) fractional shock excitations in a viscous fluid. By applying the natural transform decomposition method and with the aid of standard Caputo's operators, the stationary shock solutions for the spatial (temporal) fractional Korteweg-de Vries Burgers equations are derived. The results reveal that the temporal (Spatial) fractional index significantly modifies the localized solutions for the shocks. Moreover, The Banach contraction analysis shows that the wave solutions converge at , where stands for the temporal (Spatial) variables. Numerical analysis confirms that the index α effect nonlinear waves coupling and therefore causes the steeping of the fractional shocks. Variations in γ on the other hand reduce the wave dispersion and in turn localized the shock potentials. This study is important to understand the nonlinear evolution of shock waves in collisional plasma where the spatial/temporal fractional shocks suffer modifications in the pulse amplitude as well as in the spatial extensions.