A New Computational Technique for Analytic Treatment of Time‐Fractional Nonlinear Equations Arising in Magneto‐Acoustic Waves

Abstract

This paper presents the study of time‐fractional nonlinear fifth‐order Korteweg–de Vries equations by utilizing an adequate novel technique, namely, the q‐homotopy analysis transform method. The fifth‐order Korteweg–de Vries equation has got its importance in the study of magneto‐sound propagation in plasma, capillary gravity water waves, and the motion of long waves under the influence of gravity in shallow water. To justify the effectiveness and pertinence of the contemplated technique, we take a look at three examples of the time‐fractional fifth‐order Korteweg–de Vries equations. The q‐homotopy analysis transform method offers us to modulate the range of convergence of the series solution using ℏ, called the auxiliary parameter or convergence control parameter. The study of the fractional behaviour of the considered equations expresses the originality of the presented work. There is a visible variation in the obtained solutions for different fractional orders and which can lead to different consequences for future work. As a future research direction, readers can use the hybrid methodologies merging with our projected scheme to achieve better consequences. Additionally, to validate the precision and reliability of the proposed method, we organized suitable numerical simulations. The obtained findings show that the proposed method is very gratifying and examines the complex nonlinear challenges that arise in science and innovation.