New exact solutions for the loaded Korteweg-de Vries and the loaded modified Korteweg-de Vries by the functional variable method

CC BY f
103-105
0
0
Поделиться
Бабажанов, Б., & Майлиева, С. (2024). New exact solutions for the loaded Korteweg-de Vries and the loaded modified Korteweg-de Vries by the functional variable method . Современные тенденции в развитии науки: перспективы и практика, 1(1), 103–105. извлечено от https://inlibrary.uz/index.php/trends-development-science/article/view/30132
Базар Бабажанов, Ургенчский государственный университет
доктор физико-математических наук, кафедра прикладной математики и математической физики
Садокат Майлиева, Urganch davlat universiteti
Студент магистратуры
Crossref
Сrossref
Scopus
Scopus

Аннотация

In this paper, we construct exact traveling wave solutions of the loaded Korteweg-de Vries and the loaded modified Korteweg-de Vries by the functional variable method. The performance of this method is reliable and effective and gives the exact solitary and periodic wave solutions. All solutions to these equations have been examined and 3D graphics of the obtained solutions have been drawn using the MATLAB program. We get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions, and solitary wave solutions of kink type. Our results reveal that the method is a very effective and straightforward way offormulating the exact traveling wave solutions of non-linear wave equations arising in mathematical physics and engineering.


background image

«Современные тенденции в развитии науки:

перспективы и практика»

103

NEW EXACT SOLUTIONS FOR THE LOADED KORTEWEG-DE VRIES

AND THE LOADED MODIFIED KORTEWEG-DE VRIES BY THE

FUNCTIONAL VARIABLE METHOD

Bazar Babajanov

Doctor of Sciences in Physical and Mathematical Sciences, Department of Applied

Mathematics and Mathematical Physics, Urgench State University

Sadokat Maylieva

Master degree student, Urgench State University

Abstract:

In this paper, we construct exact traveling wave solutions of the

loaded Korteweg-de Vries and the loaded modified Korteweg-de Vries by the

functional variable method. The performance of this method is reliable and effective

and gives the exact solitary and periodic wave solutions. All solutions to these

equations have been examined and 3D graphics of the obtained solutions have been

drawn using the MATLAB program. We get some traveling wave solutions, which

are expressed by the hyperbolic functions and trigonometric functions. The
graphical representations of some obtained solutions are demonstrated to better

understand their physical features, including bell-shaped solitary wave solutions,

singular soliton solutions, and solitary wave solutions of kink type. Our results reveal

that the method is a very effective and straightforward way of formulating the exact

traveling wave solutions of non-linear wave equations arising in mathematical

physics and engineering.

Key words:

the loaded Korteweg-de Vries equation, the loaded modified

Korteweg-de Vries equation, periodic wave solutions, soliton wave solutions,

functional variable method.

The investigation of exact traveling wave solutions to non-linear evolution

equations plays an important role in the study of non-linear physical phenomena.

These equations arise in several fields of science, such as fluid dynamics, physics

of plasmas, biological models, non-linear optics, chemical kinetics, quantum

mechanics, ecological systems, electricity, ocean, and sea. One of the most

important non-linear evolution equations is the Korteweg De Vries (KdV)

equation.

The KdV equation was first observed by John Scott Russell in experiments,

and then Lord Rayleigh and Joseph Boussinesq studied it theoretically. Finally, in

1895, Korteweg and De Vries formulated a model equation to describe the

aforementioned water wave, which helped to prove the existence of solitary
waves. In the mid-1960s, Zabusky and Kruskal discovered the remarkably stable

particle-like behavior of solitary waves. The KdV equation is especially


background image

"Современные тенденции в развитии науки:

перспективы и практика"

104

important due to the potential application of different properties of electrostatic

waves in the development of new theories of chemical physics, space

environments, plasma physics, fluid dynamics, astrophysics, optical physics,

nuclear physics, geophysics, dusty plasma, fluid mechanics, and different other

fields of applied physics [1, 2, 3, 4].

In recent years, studying electrostatic waves specifically to discuss different

properties of solitary waves in the field of soliton dynamics has played a

significant role for many researchers and has received considerable attention

from them. The ion acoustic solitary wave is one of the fundamental non-linear
wave phenomena appearing in plasma physics. In 1973, Hans Schamel studied a

modified Korteweg-de Vries equation for ion-acoustic waves. The modified KdV

equation has been applied widely in the molecular chain model, the generalized

elastic solid, and so on [5, 6, 7]. Non-linear interactions between low-hybrid

waves and plasmas can be described well by using the modified KdV equation [8].

In arterial mechanics, a model is widely used in which the artery is

considered as a thin-walled prestressed elastic tube with a variable radius (or

with stenosis), and blood as an ideal fluid [9]. The governing equation that models

weakly nonlinear waves in such fluid-filled elastic tubes is the modified KdV

equation

2

6

( )

0,

t

x

xxx

x

u

u u

u

h t u

+

=

where

t

- is a scaled coordinate along the axis of the vessel after static

deformation characterizing axisymmetric stenosis on the surface of the arterial

wall.

x

- is a variable that depends on time and coordinates along the axis of the

vessel.

( )

h t

- is a form of stenosis and characterizes the average axial velocity of

the fluid.

We suppose that a form of stenosis

( )

h t

proportional to

(0, )

u

t

and we

consider the loaded KdV and the loaded modified KdV equation

1

6

( ) (0, )

0,

t

x

xxx

x

u

uu

u

t u

t u

+

+

=

2

2

12

( ) (0, )

0,

t

x

xxx

x

u

u u

u

t u

t u

+

+

=

where

( , )

u x t

is an unknown function,

x

R

,

0

t

,

and

are any constants,

1

( )

t

and

2

( )

t

are the given real continuous functions.

We establish exact traveling wave solutions of the loaded KdV and the

loaded modified KdV by the functional variable method. The performance of this

method is reliable and effective and gives the exact solitary wave solutions and

periodic wave solutions. The traveling wave solutions obtained via this method

are expressed by hyperbolic functions and trigonometric functions. The graphical

representations of some obtained solutions are demonstrated to better

understand their physical features, including bell-shaped solitary wave solutions,


background image

«Современные тенденции в развитии науки:

перспективы и практика»

105

singular soliton solutions, and solitary wave solutions of kink type. This method

presents wider applicability for handling non-linear wave equations.

References:

1. Sagdeev R.Z. (1966). Cooperative Phenomena and Shock Waves in

Collision less Plasmas, Reviews of Plasma Physics, 4, 23-91.

2. Seadawy A.R., Cheemaa Nadia. (2020). Some new families of spiky

solitary waves of one-dimensional higherorder KdV equation with power law

nonlinearity in plasma physics, Indian Journal of Physics, 94(1), 117-126.
https://doi.org/10.1007/s12648-019-01442-6.

3. Seadawy A.R., Cheemaa Nadia. (2019). Propagation of nonlinear complex

waves for the coupled nonlinear Schr¨odinger Equations in two core optical fibers,

Physica A: Statistical Mechanics and its Applications, 529(12), 13-30.

https://doi.org/10.1016/j.physa.2019.121330.

4. Seadawy A.R., Cheemaa Nadia. (2019). Applications of extended modified

auxiliary equation mapping method for high order dispersive extended nonlinear

Schr¨odinger equation in nonlinear, Modern Physics Letters B, Volume 33(18), 1

-

11. https://doi.org/10.1142/S0217984919502038.

5. Gorbacheva, O.B. and Ostrovsky, L.A. (1983). Nonlinear vector waves in a

mechanical model of a molecular chain, Physica D: Nonlinear Phenomena, 8(1-2),

223-228. https://doi.org/10.1016/0167-2789(83)90319-6.

6. Erbay, S. and Suhubi, E.S. (1989). Nonlinear wave propagation in

micropolar media. II: Special cases, solitary waves and Painlev´e analysis,

International

Journal

of

Engineering

Science,

27(8),

915-919.

https://doi.org/10.1016/0020-7225(89)90032-3.

7. Zha, Q.L. and Li, Z.B. (2008). Darboux transformation and multi-solitons

for complex mKdV equation, Chinese Physics Letters, Volume 25(1), 8.

https://doi.org/10.1088/0256-307X/25/1/003.

8. Karney, C.F.F., Sen, A. and Chu, F.Y.F. (1979). Nonlinear evolution of lower

hybrid

waves,

The

Physics

of

Fluids,

22(5),

940-952.

https://doi.org/10.1063/1.862688.

9. Demiray H. (2009). Variable coefficient modified KdV equation in fluid-

filled elastic tubes with stenosis: Solitary waves, Chaos, Solitons and Fractals, 42,

358-364. https://doi.org/10.1016/j.chaos.2008.12.014.


Библиографические ссылки

Sagdeev R.Z. (1966). Cooperative Phenomena and Shock Waves in Collision less Plasmas, Reviews of Plasma Physics, 4, 23-91.

Seadawy A.R., Cheemaa Nadia. (2020). Some new families of spiky solitary waves of one-dimensional higherorder KdV equation with power law nonlinearity in plasma physics, Indian Journal of Physics, 94(1), 117-126. https://doi.org/10.1007/sl2648-019-01442-6.

Seadawy A.R., Cheemaa Nadia. (2019). Propagation of nonlinear complex waves for the coupled nonlinear Schr odinger Equations in two core optical fibers, Physica A: Statistical Mechanics and its Applications, 529(12), 13-30. https://doi.Org/10.1016/j.physa.2019.121330.

Seadawy A.R., Cheemaa Nadia. (2019). Applications of extended modified auxiliary equation mapping method for high order dispersive extended nonlinear Schr odinger equation in nonlinear, Modern Physics Letters B, Volume 33(18), 1-11. https://doi.org/10.1142/S0217984919502038.

Gorbacheva, O.B. and Ostrovsky, L.A. (1983). Nonlinear vector waves in a mechanical model of a molecular chain, Physica D: Nonlinear Phenomena, 8(1-2), 223-228. https://doi.org/10.1016/0167-2789(83)90319-6.

Erbay, S. and Suhubi, E.S. (1989). Nonlinear wave propagation in micropolar media. II: Special cases, solitary waves and Painlev'e analysis, International Journal of Engineering Science, 27(8), 915-919. https://doi.org/10.1016/0020-7225(89)90032-3.

Zha, Q.L. and Li, Z.B. (2008). Darboux transformation and multi-solitons for complex mKdV equation, Chinese Physics Letters, Volume 25(1), 8. https://doi.Org/10.1088/0256-307X/25/l/003.

Karney, C.F.F., Sen, A. and Chu, F.Y.F. (1979). Nonlinear evolution of lower hybrid waves, The Physics of Fluids, 22(5), 940-952. https://doi.Org/10.1063/l.862688.

Demiray H. (2009). Variable coefficient modified KdV equation in fluid-filled elastic tubes with stenosis: Solitary waves, Chaos, Solitons and Fractals, 42, 358-364. https://doi.Org/10.1016/j.chaos.2008.12.014.

inLibrary — это научная электронная библиотека inConference - научно-практические конференции inScience - Журнал Общество и инновации UACD - Антикоррупционный дайджест Узбекистана UZDA - Ассоциации стоматологов Узбекистана АСТ - Архитектура, строительство, транспорт Open Journal System - Престиж вашего журнала в международных базах данных inDesigner - Разработка сайта - создание сайтов под ключ в веб студии Iqtisodiy taraqqiyot va tahlil - ilmiy elektron jurnali yuridik va jismoniy shaxslarning in-Academy - Innovative Academy RSC MENC LEGIS - Адвокатское бюро SPORT-SCIENCE - Актуальные проблемы спортивной науки GLOTEC - Внедрение цифровых технологий в организации MuviPoisk - Смотрите фильмы онлайн, большая коллекция, новинки кинопроката Megatorg - Доска объявлений Megatorg.net: сайт бесплатных частных объявлений Skinormil - Космецевтика активного действия Pils - Мультибрендовый онлайн шоп METAMED - Фармацевтическая компания с полным спектром услуг Dexaflu - от симптомов гриппа и простуды SMARTY - Увеличение продаж вашей компании ELECARS - Электромобили в Ташкенте, Узбекистане CHINA MOTORS - Купи автомобиль своей мечты! PROKAT24 - Прокат и аренда строительных инструментов