Fractional KdV and Boussenisq-Burger's equations, reduction to PDE and stability approaches

Abstract

The generalized fractional Caputo-operator is discussed. The reduction of partial differential equations retarded or advanced with delays is obtained. The applications to the space-time-fractional KdV and time-fractional Boussenisq-Burger's equation are carried. Semi-self-similar SS wave solutions are obtained. That is, there exists a soliton wave propagates along a specific characteristic curve in the xt- plane. This may be due to the effects associated with the distributed time delay. The effects of space fractional derivative on a system are attributed to the transition states. Thus, anomalous wave transport is produced mainly near the origin.