New Wave Behaviors Generated by Simple Equation Method with Riccati Equation of Some Fourth-Order Fractional Water Wave Equations

Abstract

More real situations have been rendered as nonlinear fractional partial differential equations (nfPDEs), which is why the seeking of the exact traveling wave solutions and studying the wave behavior of these equations are very important in many fields of mathematical physics. The nonlinear space-time fractional Ablowitz–Kaup–Newell–Segur (AKNS) equation and the nonlinear space-time fractional Estevez–Mansfield–Clarkson (EMC) equation presented the movement of waves in shallow water. Solving these equations required the Jumarie’s Riemann–Liouville derivative to transform nfPDEs to nonlinear ordinary differential equations (nODEs) and the collaboration of the simple equation (SE) method with Riccati equation. The new results of these equations are displayed in hyperbolic tangent forms and tangent forms. The wave behaviors, kink and periodic waves are shown in 2-D, 3-D, and contour graphs. Moreover, the solutions analysis also shows that these solutions have a structure that is not only more appropriate but also simpler.