Page 148 - IJOCTA-15-2

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.2, pp.343-353 (2025)
https://doi.org/10.36922/ijocta.1715

RESEARCH ARTICLE

Effect of diffusion parameters on traveling wave solutions of singular
perturbed Boussinesq equation

1
2*
Asıf Yoku¸s , H¨ulya Durur , and Mehmet Hakan Ekici 3
1
Department of Mathematics, Faculty of Science, Firat University, Elazig, T¨urkiye
2
Department of Computer Engineering, Faculty of Engineering, Ardahan University, Ardahan, T¨urkiye
3
Institute of Graduate Education, Department of Advanced Technologies, Ardahan University, Ardahan, T¨urkiye
asfyokus@firat.edu.tr, hulyadurur@ardahan.edu.tr, m.hakanekici@hotmail.com
ARTICLE INFO ABSTRACT
Article History: This paper focuses on the nature of the traveling wave solutions of the singular
Received: October 29, 2024 perturbed (sixth-order) Boussinesq equation, which is important for the phys-
Accepted: February 19, 2025 ical relationships of water waves and shows strong interactions. Most studies
Published Online: April 11, 2025 of this equation have been analyzed by numerical methods and it has been
Keywords: observed that analytical solutions are limited. This has formed the main moti-
The Singular Perturbed (sixth-order) vation for an analytical method, the Kudryashov method, for the generation of
Boussinesq Equation traveling wave solutions. One of the important goals of this work is to analyze
The Kudryashov method the wave propagation phenomena in detail. For this purpose, the model has
Traveling wave solution been extended by adding parameters to the diffusion term, which can illumi-
Nonlinear wave propagation nate shallow fluid layers and nonlinear atomic phenomena. This model aims
to provide a deeper understanding of the motion of waves and the behavior of
AMS Classification 2010:
fluids, as well as its solution by the analytical method. To deepen the physical
35C07, 35Qxx, 35A20
discussion, the responses of the solution are simulated for different values of
the diffusion coefficient and the parameter associated with the wave velocity.

1. Introduction applications in various scientific and engineering
disciplines. Especially in subjects such as fluid
There are countless intricate and dynamic pro- dynamics and water waves, the use of NLPDEs
cesses found in nature. Since these processes are plays an important role in understanding the com-
frequently controlled by nonlinear physical fac- plexity of nature and predicting future events. In
tors, they are frequently outside the scope of sim- this context, the Boussinesq equation is a striking
ple mathematical models. It is possible to ex- example for modeling water waves. The Boussi-
amine phenomena more complex than an input- nesq equation is a nonlinear partial differential
output relationship thanks to nonlinear physical equation used to describe wave motions in shal-
principles. low water. This equation includes both nonlin-

An essential tool for simulating intricate and ear effects and dispersion terms to successfully
realistic natural processes is the nonlinear par- model the propagation, dispersion and refraction
tial differential equation (NLPDE). It has numer- of waves at the water surface. In this study, the
ous scientific and engineering applications. We Boussinesq equation (BE), which has an impor-
can forecast future events and comprehend the tant reputation in the field of fluid mechanics and
complexity of nature by finding solutions to these is a fundamental equation used in wave analy-
equations. These equations have a wide range of sis, will be discussed. It is envisaged that the
*Corresponding Author
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