Page 149 - IJOCTA-15-2

A. Yoku¸s, H. Durur, M.H. Ekici / IJOCTA, Vol.15, No.2, pp.343-353 (2025)
solutions of this equation will provide a better studied. 12 The first equations describing the con-
understanding of the physical mechanisms under- vective motion of a viscous fluid were introduced
lying natural processes and the dynamical struc- experimentally and theoretically, the Oberbeck-
tures of these processes. The BE is one of the Boussinesq equations. Using these equations, the
NLPDE s that model shallow water waves. BEs linear dependence of density on temperature was
examined. 13 There are analyses and discussions
were first formulated by Boussinesq in 1872 and
regarding fluid dynamics and convective heat
their applicability is mainly limited to relatively 14
1
shallow water bodies. Boussinesq-type equations, transfer. They included a third-order additional
term in the Boussinesq equation, which is based
which include time dependence, dispersion prop-
on long wave equations and can be neglected in
erties and weak nonlinearity, have become signifi- 15
shallow water conditions. Full traveling wave so-
cant in predicting wave transformations in coastal lutions were proposed for the (2+1) dimensional
2
areas. The standard BEs, which are effective for 16
Boussinesq equation. Based on Lie group anal-
shallow water, were derived by Peregrine from the ysis, the generator of the new Boussinesq equa-
3
first set of equations for variable water depth. In tion and its single-parameter invariant group were
the analysis stated by McCowman, the relation- obtained. 17 The Hirota bilinear method has been
ship between the depth of the medium in which proposed as an effective tool for attaining soliton
the wave propagates and the wavelength is dis- solutions of nonlinear equations, based on the bi-
cussed in the discussion of the phase velocity es- linear derivative method. 18 Relationships between
timates obtained with these equations. 4 periodic wave solutions and soliton solutions were
In this study, we examine the singularly per- established, and the asymptotic behavior of quasi-
turbed (sixth-order) Boussinesq equation (SPBE) periodic waves has been analyzed using a con-
presented by. 5,6 SPBE straint procedure. 19 Solutions for the sixth-order
BE describing Rossby waves were produced us-
ing the Jacobi elliptic function expansion method

u tt = u xx + u 2 + u xxxx + `ou xxxxxx , (1) 20
xx along with symmetry and conservation laws.
The “good” and “bad” versions of the BE rep-
resent the two most commonly utilized formula-
where |`o| ≤ 1 is a small parameter. First, tions of this equation in scientific research. The
7
by Daripa and Hua, this equation is presented ”good” BE is a well-known form with specified
as a regularization of the classical (ill-posed) BE, assumptions and boundary conditions that is ap-
corresponding to `o = 0 in equation (1). plied in many scientific domains. 21,22 Conversely,
We derive equation (1) from the two- the “bad” BE is seen to be less sophisticated
dimensional potential flow equations that govern and is linked to being ill-posed, which reduces
small-amplitude long capillary gravity waves at its dependability in some situations. 23,24 Scholars
the shallow water surface for bond numbers very have directed their attention toward examining
close to but less than 1/3. 5 both types to comprehend their characteristics,
There are many studies on the BE in the actions, and uses in various situations, such as
literature. Researchers have been interested in simulating wave and groundwater flow. The two
the analytical solution of this equation. Among types differ in that they are appropriate for differ-
these solutions, non-symmetrical bending solu- ent modeling scenarios and have different mathe-
tions, symmetrical, single pattern solutions, Ja- matical features. In the study conducted for the
cobi and Weierstrass elliptic function and trian- BE, it was shown that the correct cut of the series
2
gular function solutions were obtained. A fully representing the distribution is after the sixth de-
nonlinear model has been studied for shallow wa- rivative. Therefore, this derived nonlinear equa-
8
ter waves using the BE. Local well-locality for tion is known as the sixth-order generalized BE.
the nonlinear sixth-order BE was investigated. 9 It is also shown that the sixth-order generalized
The tanh-coth method was used to attain soliton BE, laws of conservation of energy and mass, and
25
solutions and the Hirota binaural methods were a balance law for momentum are valid. In light
used to determine multiple soliton solutions. 10 of this scientific reality, this study focuses on the
The fully nonlinear variant of the Boussinesq sixth-order BE.
model was found to predict wave heights, phase There are many studies in the liter-
velocities, and particle kinematics more accu- ature on analytical and numerical meth-
rately than the standard approach. 11 The analyt- ods such as Laplace transform, 26 collocation
ical solution of the thermal diffusion equations for method, 27 the Laplace homotopy transformation
steady-state shear flows of a binary fluid has been method, 28 the reduction perturbation method, 29
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