Research Article
Explicit Solutions to the (3+1)-Dimensional
Kudryashov-Sinelshchikov Equations in Bubbly Flow Dynamics
Y. B. Chukkol , M. N. B. Mohamad, and Mukhiddin Muminov
Department of Mathematical Sciences, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor, Malaysia
Correspondence should be addressed to Y. B. Chukkol; bcyusuf2@live.utm.my
Received 26 August 2018; Accepted 2 October 2018; Published 1 November 2018
Academic Editor: Mehmet Sezer
Copyright © 2018 Y. B. Chukkol et al. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A modifed tanh-coth method with Riccati equation is used to construct several explicit solutions of (3+1)-dimensional
Kudryashov-Sinelshchikov equations in bubble gas liquid fow. Te solutions include solitons and periodic solutions. Te method
applied can be used in further works to obtain entirely new solutions to many other nonlinear evolution equations.
1. Introduction
Many complex phenomena in nature are describe in various
scientifc and industrial felds and especially in areas of
physics such as fuid mechanics [1], optical fbres [2], plasma
physics [3], and so on. Nonlinear evolution equations are used
to describe these nonlinear phenomena, and that has led to
the development of methods to look for exact solutions of
nonlinear partial diferential equations [4].
Recently, Kudryashov and Sinelshchikov [5] developed a
(3+1)-dimensional nonlinear evolution equation in a model
of wave propagation in bubbly fuid fow. Te model equation
has gained a lot of attention where B
˜
A¤cklund transformation
and conservation laws [6], bifurcation [7], and density-
fuctuation [8] analysis were carried out the important evo-
lution equation.
Many methods have been developed to fnd the explicit
solutions of nonlinear evolution equations; example of such
methods are the frst integral method [9], Jacobi elliptic
function method [10], Hirota bilinear method [11], Wron-
skian determinant technique [12], F-expansion method [13],
Darboux Transformations [14], Backlund transformation
method [6], Miura transformation [15], homotopy pertur-
bation method [16], and Adomian decomposition method
[17]. Many algebraic methods were proposed so far, such as
tanh method which was proposed by Malfie [18]. Fan [19]
extended the tanh method and obtained new exact solution
that cannot be obtained by using the conventional tanh
method. Further extension called tanh-coth was proposed
by Wazwaz [4] which provides a wider applicability for
solving nonlinear evolution equations. A modifcation was
also proposed by El-Wakil [20] and Soliman [21, 22].
In this paper, we present two equations (3+1)-dimen-
sional Kudryashov and Sinelshchikov equation written as
(
+
+
)
+
+
=0,
(
+
−
+
)
+
+
z
=0
(1)
where , , and represent the nonlinearity, dissipation
dispersion terms, while and stand for transverse variation
of wave in and directions; we assume all the coefcient
to be constant parameters. We shall use modifed tanh-coth
to obtain many explicit exact solutions for (3+1)-dimensional
Kudryashov-Sinelshchikov equations. Te travelling wave
solution of a special case of (1) is given in [7]. Note that when
=0, (1) reduces to their two-dimensional counterparts
(
+
+
)
+
=0,
(
+
−
+
)
+
=0,
(2)
which were widely studied in [16, 17, 23, 24]. Furthermore
when ==0, (1) reduces to one-dimensional Korteweg-
de-Vries equations (KdV) and one-dimensional Korteweg-
de-Vries-Burgers equations (KdVB). Tese equations have
many applications in fuid dynamics [25], plasma physics
Hindawi
Journal of Applied Mathematics
Volume 2018, Article ID 7452786, 9 pages
https://doi.org/10.1155/2018/7452786