Various wave solutions to the nonlinear fractional Korteweg-de Vries Zakharov-Kuznetsov equation by a new approach to the two-variable expansion scheme
Аннотация
Abstract Solving nonlinear differential equations is crucial for the design, optimization, and characterization of engineering systems. However, investigating such equations poses significant challenges and requires the development of new mathematical and computational techniques. In this article, we extend the two-variable <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> <mml:mrow> <mml:mo accent="true">′</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>/</mml:mo> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo>/</mml:mo> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:math> -expansion approach and establish scores of general analytical solutions, consisting of different functions and arbitrary parameters, to the time-fractional Korteweg–de Vries Zakharov-Kuznetsov (KdV-ZK) equation. The KdV-ZK equation provides a powerful model for modulating nonlinear waves in an assortment of physical systems, including plasma physics, atmospheric science, fluid dynamics, soliton theory, and optical fibers. The proposed technique allows extracting several novel solutions to the KdV-ZK equation, which is essential for gaining new insights into physical problems. In addition, the suggested extension shortens the calculation process and enhances computational efficiency. The obtained solitons have potential applications in several scientific and technological fields. The beta fractional derivative and the consistent wave variable are considered to restructure the stated fractional nonlinear equation. This study investigates the merits and demerits of the fractional-order derivative β on the dynamics of the system through the representation of two-dimensional graphs, which vary according to different values of β . The enhanced computational efficiency might assist the researchers in exploring a wider range of phenomena, facilitating further extensive investigations, and contributing to overall progress in the fields of science and engineering.