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Effect of diffusion parameters on traveling wave solutions of singular perturbed boussinesq equation
Sumudu transform, 30 Adomian decomposition This method is particularly advantageous due to
method, 31,32 Fifth step block method, 33 Crank- its systematic approach, which reduces compu-
Nicolson method, 34 artificial neural network tational complexity and increases the accuracy
technique, 35 the cubic spline method, 36 the fi- of the solutions compared to traditional numer-
nite difference method, 37 expansion method. 38 ical or approximation techniques. Kudryashov’s
Numerical solutions were produced using the method is expected to contribute to the litera-
Runge-Kutta method for the one-dimensional BE ture by providing new traveling wave solutions
with sixth-order space derivative. 39 Discretiza- and improving the understanding of the underly-
tion of the sixth-order Boussinesq-Type problem ing physics in various applications such as fluid
was carried out using the Septic B-spline sort- dynamics, wave propagation and biological sys-
ing technique, and numerical solutions were pro- tems. There is also the idea that the BE can
duced for this problem by non-linear least squares help us understand its mathematical and physi-
optimization using the Tikhonov Regularization cal properties in a broader context. Additionally,
function. 40 It also obtains weakly nonlocal single- one of the most significant goals of this study
wave solutions of the BE and provides estimates is to analyze the diffusion phenomenon of the
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of the amplitude of the oscillations. In her study, wave. To perform this analysis, a coefficient can
Arslan obtained an approximate solution of the be added to the u xx term in equation (1), which
sixth-order BE using the hybrid method as a dif- represents the diffusion phenomenon of the wave
ferent alternative method. 41 Song et al. discussed in shallow waters. In this case, equation (1) can
the lack of a global solution to the initial bound- be expanded as follows.
ary value problem for the BE and two examples
were given. 42 The SPBE exact solutions were ana- u tt = κu xx + u 2 + u xxxx + `ou xxxxxx (2)
lyzed via Lie approximation symmetry analysis. 6 xx
The solution was analyzed with the standard per-
turbation method and physical discussions were
held. 43 Numerical solutions were produced with where κ is a constant and is the diffusion coeffi-
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the finite difference method. Existing techniques cient.
Moreover, |`o| ≤ 1 is a small parameter gov-
for solving NLPDEs often involve restrictive as-
erning the effect of sixth-order dispersion on the
sumptions, limited applicability or computation-
waveform. 45 It is a physically important constant
ally intensive procedures. There is a clear need
and is related to the phenomenon of long wave
for efficient methods that can systematically gen-
propagation in shallow water. 46
erate exact solutions without relying on approxi-
In this study, the effect of the diffusion
mations. The Kudryashov method has proven to coefficient on wave diffusion will be discussed.
be a powerful analytical tool for solving various
NLPDEs. 44 In this work, we explore the full po- In the second section, the methodology of the
tential for BE, a certain class of equations relevant Kudryashov method will be explained. In the
to modern applications such as fluid dynamics third section, the Kudryashov method will be
and wave theory, and provide practical insights used to obtain traveling wave solutions for equa-
into the behavior of the modeled systems. We tion (2). In the fourth section, the effects of physi-
also contribute to the theoretical development of cal parameters on the obtained solution and phys-
ical discussions will be given and in the last sec-
analytical techniques. Solutions of the BE, which
tion, the obtained results will be presented and
plays the leading role in this study, are important
general conclusions will be given.
for analyzing wave behavior and understanding
the movement of fluids. However, while the BE is 2. The methodology of the Kudryashov
generally solved by numerical methods, analytical method
solutions appear to be limited. For this reason,
the analytical method Kudryashov method will Assume you have an NLPDE of the below form
be used as a tool for the exact solution. The
reason for choosing the proposed method lies Ω (u, u t , u x , u xx , u tt , ...) = 0, (3)
in its ability to generate exact analytical solu-
where u = u (x, t). We present the basic steps of
tions for nonlinear partial differential equations 44
the method.
(NLPDEs), which are fundamental in modeling
Step 1. By using the wave transformation
complex physical phenomena. By making use of
Kudryashov’s method, exact solutions can be ef- u (x, t) = U (ξ) , ξ = kx ± vt, (4)
ficiently generated that provide valuable insights
where k is the wave number and v is the angular
into the dynamic behavior of nonlinear systems.
frequency. Additionally, the angular frequency v
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