The aim of this paper is to introduce a novel study of obtaining exact solutions to the (2+1) - dimensional conformable KdV equation modeling the amplitude of the shallow-water waves in fluids or electrostatic wave potential in plasmas. The reduction of the governing equation to a simpler ordinary differential equation by wave transformation is the first step of the procedure. By using the improved tan(φ/2)-expansion method (ITEM) and Jacobi elliptic function expansion method, exact solutions including the hyperbolic function solution, rational function solution, soliton solution, traveling wave solution, and periodic wave solution of the considered equation have been obtained. We achieve also a numerical solution corresponding to the initial value problem by conformable variational iteration method (C-VIM) and give comparative results in tables. Moreover, by using Maple, some graphical simulations are done to see the behavior of these solutions with choosing the suitable parameters.
This interdisciplinary study highlights the crucial role of mathematics
and physics in ocean engineering. In this study, the traveling wave
solutions of the general Drinfiel’d-Sokolov-Wilson (DSW)-system, which
was introduced as a model of water waves, were investigated. Converting
the DSW-system to a more straightforward system of ordinary differential
equation system with wave transform is the first step in the process.
The solutions of the system were obtained using five different methods.
These methods are effective methods for generating periodic solutions.
It has also been seen that the periodic solutions we got using the
Jacobi elliptic function expansions containing different Jacobi elliptic
functions might be different, and that we can get some new periodic
solutions. Given 3-dimensional simulations using Maple
were made to see the behaviour of the solutions
obtained for the appropriate different values of the parameters. This
study is very important as it is the unique study in the literature in
which five different Jacobi elliptic function expansion methods are
discussed together. Jacobi elliptic functions are valuable mathematical
tools that can be applied to various aspects of ocean engineering. Their
use helps engineers better understand and predict the behaviour of
waves, tidal forces, and other phenomena, ultimately leading to safer
and more efficient structures and systems. The stability property of the
obtained solutions was tested to demonstrate the ability of the obtained
solutions.
In this paper, we study on Moufang–Klingenberg planes [Formula: see text] defined over a local alternative ring [Formula: see text] of dual numbers. We get relations between some special 6-figures and its ratios. Also, we construct a relation between the solvability of the special quadric equation in [Formula: see text] and the existence of the special 6-figure in [Formula: see text]. Moreover, we give a construction which provides an algorithm for "squaring" and "cubing" a cross-ratio of points in [Formula: see text].
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