Analytic Computational Method for Solving Fractional Nonlinear Equations in Magneto-Acoustic Waves

In this article, we employ a useful and intriguing method known as the ARA-homotopy transform approach to explore the fifth-order Korteweg-de Vries equations that are nonlinear and time-fractional. The study of capillary gravity water waves, magneto-sound propagation in plasma, and the motion of long waves under the effect of gravity in shallow water have all been influenced by Korteweg-de Vries equations. We discuss three instances of the fifth-order time-fractional Korteweg-de Vries equations to demonstrate the efficacy and applicability of the proposed method. Utilizing, also known as the auxiliary parameter or convergence control parameter, the ARA-homotopy transform technique which is a combination between ARA transform and the homotopy analysis method, allows us to modify the convergence range of the series solution. The obtained results show that the proposed method is very gratifying and examines the complex nonlinear challenges that arise in science and innovation.