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.

A study of double general transform for solving fractional partial differential equations

“…Examples include a smooth transition from wave propagation to diffusion or from local to nonlocal dynamics [6–8]. There are several methods that are developed in literature to solve fractional differential equations, such as homotopy analysis method [9], Laplace residual power series [10], iterative method [11], and integral transforms [12].…”

A New Perspective on the Stochastic Fractional Order Materialized by the Exact Solutions of Allen-Cahn Equation

“…In recent decades, fractional derivatives have generated substantial interest because of their possible use in several domains, including telegraph transmission (Cascaval et al, 2002), atmospheric science (Korn, 2019), chaotic oscillations (Tavazoei et al, 2008), optical fibers (Yokus and Baskonus, 2022), two-scale thermal science (He, 2021), ecological and economic systems (Saadeh et al, 2023), mechanics (Zhang and Bilige, 2019), chemistry (Yuste et al, 2004), and hydrology (Benson et al, 2000). physics (Abdoon and Hasan, 2022;Prakasha et al, 2023), biology (Amourah et al, 2023;Saadeh et al, 2022), and finance (Wyss, 2000;Raberto et al, 2002). These fractional-order equations represent the memory and heirship of different substances using fractional-order derivatives (Podlubny, 1999) and are preferred over integer-order equations.…”