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This study focuses on solving the Caudrey-Dodd-Gibbon ( CDG ) equation using the Khater II ( KII ) method and the Variational Iteration ( V I ) method. The CDG equation is a pivotal mathematical model in nonlinear wave dynamics, essential for understanding the evolution, interaction, and preservation of wave forms in dispersive media. Its applications span various fields, including fluid dynamics, nonlinear optics, and plasma physics, where it plays a crucial role in analyzing solitons and complex wave interactions. In this research, we meticulously implement the KII and V I methods to derive solutions for this nonlinear partial differential equation. Our findings reveal new aspects of the equation's behavior, offering deeper insights into nonlinear wave phenomena. The significance of this study lies in its contribution to advancing the understanding of these phenomena and their practical applications in the academic realm. The results underscore the effectiveness of the employed methods, their innovative contributions, and their relevance to applied mathematics., Competing Interests: Declarations Competing interests Te authors declare no competing interests., (© 2024. The Author(s).)

This study investigates the nonlinear Pochhammer-Chree equation, a model crucial for understanding wave propagation in elastic rods, through the application of the Khater III method. The research aims to derive precise analytical solutions and validate them using He's variational iteration method (VIM). The Pochhammer-Chree equation's relationship to other nonlinear evolution equations, such as the Korteweg-de Vries and nonlinear Schrödinger equations, underscores its significance in the field of nonlinear wave dynamics. The methodology employs the Khater III method for deriving analytical solutions, while He's VIM serves as a numerical validation tool, ensuring the accuracy and stability of the obtained results. This dual approach not only yields novel solutions but also provides a robust framework for analyzing complex wave phenomena in elastic media. The findings of this study have significant implications for material science and engineering applications, offering new insights into the behavior of waves in elastic rods. By bridging the gap between theoretical models and practical applications, this research contributes to the advancement of both mathematical theory and physical understanding of nonlinear wave dynamics. Situated within the domain of applied mathematics, with a focus on nonlinear wave equations, this work exemplifies the interdisciplinary nature of contemporary research in mathematical physics. The results presented herein open new avenues for future investigations in related fields and highlight the potential for innovative applications in material science and engineering., (© 2024. The Author(s).)

The Caudrey-Dodd-Gibbon ( CDG ) model, a variation of the fifth-order KdV equation (fKdV) with significant practical consequences, is solved in this study using a precise and numerical technique. This model shows how gravity-capillary waves, shallow-water waves driven by surface tension, and magneto-acoustic waves move through a plasma medium. With a focus on accuracy, new computational and approximation methods have been made possible by recent improvements in analytical and numerical methods. Numeric information is represented visually in the tables. All simulation results are shown in two and three dimensions to show both the numerical and fundamental behavior of the single soliton. Recent research shows that this method is the best way to solve nonlinear equations that are common in mathematical physics., Competing Interests: The authors declare no competing interests., (© 2023 The Author(s).)

This paper investigates the exact traveling wave solutions of the fractional model of the human immunodeficiency virus (HIV-1) infection for CD4 + T-cells. This model also treats with the effect of antiviral drug therapy. These solutions calculate both the boundary and initial conditions that allow employing the septic-B-spline scheme which is one of the most recent schemes in the numerical field. We use the obtained computational solutions via the modified Khater, the extended simplest equation, and sech-tanh methods through Atangana-Baleanu derivative operator. The comparison between the exact and numerical evaluated solutions is illustrated by some distinct sketches. The functioning of our numerical method is tested under three computational obtained solutions., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2020 Elsevier Ltd. All rights reserved.)

This study investigates the solitary wave solutions of the nonlinear fractional Jimbo-Miwa (JM) equation by using the conformable fractional derivative and some other distinct analytical techniques. The JM equation describes the certain interesting (3+1)-dimensional waves in physics. Moreover, it is considered as a second equation of the famous Painlev'e hierarchy of integrable systems. The fractional conformable derivatives properties were employed to convert it into an ordinary differential equation with an integer order to obtain many novel exact solutions of this model. The conformable fractional derivative is equivalent to the ordinary derivative for the functions that has continuous derivatives up to some desired order over some domain (smooth functions). The obtained solutions for each technique were characterized and compared to illustrate the similarities and differences between them. Profound solutions were concluded to be powerful, easy and effective on the nonlinear partial differential equation.