Numerical analysis of the fractional nonlinear waves of fifth-order KdV and Kawahara equations under Caputo operator

This study delved into the analytical investigation of two significant nonlinear partial differential equations, namely the fractional Kawahara equation and fifth-order Korteweg-De Vries (KdV) equations, utilizing advanced analytical techniques: the Aboodh residual power series method and the Aboodh transform iterative method. Both equations were paramount in various fields of applied mathematics and physics due to their ability to describe diverse nonlinear wave phenomena. Here, we explored using the Aboodh methods to efficiently solve these equations under the framework of the Caputo operator. Through rigorous analysis and computational simulations, we demonstrated the efficacy of the proposed methods in providing accurate and insightful solutions to the time fractional Kawahara equation and fifth-order KdV equations. Our study advanced the understanding of nonlinear wave dynamics governed by fractional calculus, offering valuable insights and analytical tools for tackling complex mathematical models in diverse scientific and engineering applications.