Novel Exact Traveling Wave Solutions for Nonlinear Wave Equations
with Beta-Derivatives via the sine-Gordon Expansion Method
THITTHITA IATKLIANG1..
ID , SUPAPORN KAEWTA1..
ID , NGUYEN MINH TUAN1..
ID and
SEKSON SIRISUBTAWEE1,2..
ID
1Department of Mathematics, Faculty of Applied Science
King Mongkut's University of Technology North Bangkok, Bangkok 10800,
THAILAND
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400,
THAILAND
Abstract: The main objectives of this research are to use the sine-Gordon expansion method (SGEM) along with
the use of appropriate traveling transformations to extract new exact solitary wave solutions of the (2 + 1)-
dimensional breaking soliton equation and the generalized Hirota-Satsuma coupled Korteweg de Vries (KdV)
system equipped with beta partial derivatives. Using the chain rule, we convert the proposed nonlinear problems
into nonlinear ordinary differential equations with integer orders. There is then no further demand for any nor-
malization or discretization in the calculation process. The exact explicit solutions to the problems obtained with
the SGEM are written in terms of hyperbolic functions. The exact solutions are new and published here for the
first time. The effects of varying the fractional order of the beta-derivatives are studied through numerical simu-
lations. 3D, 2D, and contour plots of solutions are shown for a range of values of fractional orders. As parameter
values are changed, we can identify a kink-type solution, a bell-shaped solitary wave solution, and an anti-bell
shaped soliton solution. All of the solutions have been carefully checked for correctness and could be very im-
portant for understanding nonlinear phenomena in beta partial differential equation models for systems involving
the interaction of a Riemann wave with a long wave and interactions of two long waves with distinct dispersion
relations.
Key-Words: - sine-Gordon expansion method, Exact traveling wave solutions, Beta partial derivatives, Breaking
soliton equation, Generalized Hirota-Satsuma coupled KdV
Received: October 26, 2022. Revised: April 28, 2023. Accepted: May 19, 2023. Published: June 2, 2023.
1 Introduction
Exact solutions are of great importance for describing
several nonlinear wave phenomena arising in nature
such as the propagation of shallow water waves, [1],
nonlinear optics, [2], [3], plasma physics, [4],
quantum mechanics, [5], [6], fluid dynamics, [7],
hydrodynamics, [8], and acoustics, [9]. Mathematical
models for most of the mentioned phenomena can
be developed using nonlinear partial differential
equations (NLPDEs). Unlike numerical methods for
NLPDEs, the powerful analytic methods for obtaining
exact solutions which are now available and which
use the chain rule do not require any normalization or
discretization, [10]. Also, with the fast development
of computer science and computerized symbolic
computation, the direct search for exact solitary wave
solutions of NLPDEs has become practicable and is
now attracting much attention from research scholars
all over the world. Many techniques have now been
developed to obtain exact analytical solutions for
NLPDEs equipped with classical, conformable and
beta-derivatives, [11], [12], [13], [14], [15], [16].
Useful techniques that have been successfully
applied to obtain exact solutions of NLPDEs in-
clude the modified Kudryashov method, [17],
the (G′/G, 1/G)-expansion method, [18], the
(G′/G2)-expansion method, [19], [20], the mod-
ified simple equation (MSE) method, [21], the
Hirota bilinear approach, [22], the Jacobi elliptic
equation method, [23], the exp(−φ(ξ))-expansion
approach, [24], the Ricatti-Bernoulli sub-ODE
method, [25], the Bäcklund transformation, [26], the
auxiliary equation method, [27], the trial equation
technique, [28], the new extended direct algebraic
method, [29], [30], and the Sardar sub-equation
method, [31].
Recently, many researchers have given new def-
initions of derivatives with a fractional order which
have been used for NLPDEs instead of the clas-
sical partial derivative. These derivatives include
the conformable derivative, [32], the Atangana con-
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formable derivative, [33], and the truncated Ω-
fractional derivative, [34].
In this article, we concentrate on the use of
the sine-Gordon expansion method, [35], to gener-
ate exact traveling wave solutions of the (2 + 1)-
dimensional breaking soliton equation and the gen-
eralized Hirota-Satsuma coupled Korteweg de Vries
(KdV) system for beta partial derivatives. The defi-
nitions of the two systems are as follows.
1. The (2+1)-dimensional breaking soliton equa-
tion with beta space-time derivatives is defined by:
∂α
∂tα∂βu
∂xβ−4∂βu
∂xβ∂β
∂yβ∂βu
∂xβ
−2∂β
∂xβ∂βu
∂xβ∂βu
∂yβ
+∂β
∂yβ∂β
∂xβ∂β
∂xβ∂βu
∂xβ= 0,(1)
where ∂α
∂tα(·),∂β
∂xβ(·)and ∂β
∂yβ(·)denote the beta par-
tial derivatives with respect to t of order 0 < α ≤ 1, to
x of order 0 < β ≤ 1 and to y of order 0 < β ≤ 1, re-
spectively. If α = β = 1, then equation (1) reduces to
the original breaking soliton equation, [36], initially
proposed by [37], [38]. The equation explains the
(2 + 1)-dimensional interaction of a Riemann wave
propagated along the y-axis with a long wave prop-
agated along the x-axis, [39], [40], [41]. If we set
y = x in the original breaking soliton equation and
integrate the resulting equation, then we obtain the po-
tential KdV equation. This KdV equation is an impor-
tant mathematical model for the special waves called
solitons on shallow water surfaces. The significant
characteristic feature of breaking soliton equations is
that the spectral parameter used in the Lax representa-
tions possesses so-called breaking behavior, [42]. De-
tails of some existing methods for finding exact solu-
tions of the (2+1)-dimensional breaking soliton equa-
tion of integer-orders are given in [43], [44], [45].
2. The generalized Hirota-Satsuma coupled Ko-
rteweg de Vries (KdV) system with beta time deriva-
tive is defined by:
∂ηu
∂tη=1
4uxxx + 3uux+ 3(−v2+w)x,
∂ηv
∂tη=−1
2vxxx −3uvx,(2)
∂ηw
∂tη=−1
2wxxx −3uwx,
where ∂η
∂tη(·)represents the beta partial derivative
with respect to tof order 0< η ≤1. If η= 1,
then equation (2) becomes the first-order generalized
Hirota-Satsuma coupled KdV system, which was first
introduced by Satsuma and Hirota in 1982. This sys-
tem can be reduced to the Hirota-Satsuma system
by using the four-reduction Kadomtsev-Petviashvili
(KP) hierarchy and introducing the new dependent
variables u, v and w, [46]. The first-order gen-
eralized Hirota-Satsuma coupled KdV equation can
then be reduced further to the Hirota-Satsuma cou-
pled KdV (cKdV) equation by taking w = 0 and
scaling the variables, [46], [47]. The integer-order
system, [48], [49], [50], [51], for equation (2) has
been extensively studied by researchers because the
system is used as a model for dispersive long waves
in shallow water which appear in many applications
of fluid mechanics and related fields. These applica-
tions include shallow-water undulations with weakly
nonlinear retrieve vigor, ion-acoustic undulations in
a plasma and the interaction of neighboring parti-
cles of equal mass in a crystal lattice, [51], [52].
Further interesting applications of the generalized
Hirota-Satsuma coupled KdV system can be found
in [53], [54].
Since the (2 + 1)-dimensional breaking soliton
equation and the generalized Hirota-Satsuma cou-
pled KdV system especially play an important role in
shallow-water phenomena, we believe that the gen-
eralized models in equations (1) and (2) could give
some very useful insights into traveling wave behav-
ior in fluids and related fields. To the best of the
authors’ knowledge, there are no researchers who
have obtained exact traveling wave solutions for the
NLPDEs with beta partial derivatives in equations (1)
and (2) using the sine-Gordon expansion method.
Therefore, some novel exact solutions to these two
problems will be reported here for the first time.
The paper is arranged as follows. In Section 2,
the definition and some useful features of the beta-
derivative are presented. Then, in Section 3, the im-
portant steps of the sine-Gordon expansion method
are briefly explained. In Section 4, we use the
sine_Gordon method to derive new exact solutions of
equations (1) and (2). Also, in this section, we use nu-
merical simulations to obtain graphs of the solutions
for a range of fractional orders and to give physical
interpretations of some selected exact solutions. Fi-
nally, in Section 5, we give a discussion and conclu-
sions.
2 Beta-Derivative and Its Properties
A significant advantage of some fractional deriva-
tives such as the Caputo fractional derivative, [55],
the Riemann Liouville fractional derivative, [55],
the conformable derivative, [56], and the beta-
derivative, [57], is that, unlike integer-order deriva-
tives, they can describe memory properties of various
materials and processes. In this section, we define the
beta-derivative, which was initially proposed by [58],
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and give some of its properties. The beta-derivative
can be considered as a natural generalization of the
classical derivative as it obeys most of the fundamen-
tal properties of the classical derivative.
Definition 2.1 Let f : [0, ∞) → R be a function.
Then, the beta-derivative of f of order β, where 0 <
β ≤ 1, is defined as [57], [58], [59], [60].
Dβ
tf(t) = lim
ε→0
ft+εt+1
Γ(β)1−β−f(t)
ε.(3)
Some useful properties of the beta-derivative are
as follows, [57], [58], [59], [60]. Let f(t), g(t)be β-
differentiable functions for all t > 0and β∈(0,1].
Then
(1) Dβ
t(λ) = 0,∀λ∈R.
(2) Dβ
t(af(t) + bg(t)) = aDβ
tf(t) + bDβ
tg(t),for
all values a, b ∈R.
(3) Dβ
t(f(t)g(t)) = f(t)Dβ
tg(t) + g(t)Dβ
tf(t).
(4) Dβ
tf(t)
g(t)=g(t)Dβ
tf(t)−f(t)Dβ
tg(t)
(g(t))2,
where g(t)= 0.
(5) If fis differentiable, then Dβ
t(f(t)) =
t+1
Γ(β)1−βdf(t)
dt .
Theorem 2.1 Suppose f, g : (0, ∞) → R are dif-
ferentiable and also β-differentiable. Further assume
that g is a function defined in the range of f. Then,
the beta-derivative of a composite function f ◦ g can
be expressed as [57], [58], [59], [60].
Dβ
t(f◦g)(t) = t+1
Γ(β)1−β
f′(g(t))g′(t),(4)
where the prime symbol (′)denotes the classical
derivative.
By equation (3), the beta partial derivative of, for ex-
ample, a function u=u(x, t)with respect to tof
order β∈(0,1] can be defined by
∂β
tu(x, t) = ∂β
∂tβu(x, t) =
lim
ε→0
ux, t +εt+1
Γ(β)1−β−u(x, t)
ε, t > 0.
(5)
A review of recent literature on applications of the
beta-derivative in NLPDEs is as follows. In [57] the
authors studied magnetic solitons and periodic wave
propagation in a Heisenberg ferromagnetic spin chain
using the (2 + 1)-dimensional nonlinear Schrödinger
equation (NLSE) with beta-derivative. They found
that the beta-derivative parameter significantly af-
fected the rogue wave phenomena in which the am-
plitudes and widths of such waves are enlarged with
an increase of β. The results for this system can be
very helpful in analyzing the wave dynamics arising
in any non-local and non-conservative/conservative
physical system. In [61], the authors studied the space-
time fractional modified equal width (FMEW) equa-
tion with beta-derivative. This equation is related to
the regularized long wave (RLW) equation which has
solitary wave solutions with both positive and neg-
ative amplitudes but the same width. In this paper,
new traveling wave solutions for the equation were
constructed using the unified method and the effects
of varying the fractional order were studied. These
new traveling wave solutions were expressed in both
polynomial and rational forms. Further applications
of the beta-derivative in physical applications such as
group velocity dispersion, unidirectional propagation
of long waves and transmission in monomode optical
fibers can be found in [59], [60], [62].
3 The sine-Gordon Expansion
Method
When compared with other existing methods, the
sine-Gordon expansion method has been found to
be a simple, direct and powerful mathematical tool
for obtaining exact solutions of NLPDEs arising in
the fields of science, engineering and mathematical
physics, [63], [64]. The solutions obtained by the
SGEM show many new behavioral aspects such as
wave solutions which are expressed in terms of hyper-
bolic, exponential and complex function structures.
In particular, it has been found that solutions of some
real physical models are related to hyperbolic func-
tions, [65]. For example, hyperbolic sine and co-
sine functions occur in the gravitational potential of
a cylinder and in the shape of a hanging cable, re-
spectively. The hyperbolic tangent function occurs
in some applications involving special relativity and
the hyperbolic secant and cotangent functions occur
in the profile of a laminar jet and in the Langevin func-
tion for magnetic polarization, respectively.
A review of recent literature in which the sine-
Gordon expansion method has been used to obtain ex-
act solutions of NLPDEs is as follows. In [66], the
authors applied the sine-Gordon expansion method
to obtain exact solutions of some conformable time
fractional equations in the Regularized Long Wave
(RLW)-class. They found some real-valued and
complex-valued solutions which were combinations
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of powers of hyperbolic tangent and hyperbolic se-
cant functions. A recent study also used their
exact solutions and numerical simulations to study
the ef-fects of changing the fractional time order. In
[67], the authors used the sine-Gordon expansion
method to solve Kudryashov's equation. They
obtained new so-lutions which included singular and
bright-dark opti-cal soliton solutions. Also, in [68],
the authors used the sine-Gordon expansion method
to derive exact soli-ton solutions for the higher
dimensional generalized Boussinesq equation and
the Klein-Gordon equation. They found breather,
rogue, bell-shaped, bright-dark, kink and periodic
solitons. In [69], the authors applied the sine-Gordon
expansion method to obtain exact optical soliton
solutions of the Fokas-Lenells equa-tion. Further
papers on the applications of the sine-Gordon
expansion method to obtain exact solutions for
NLPDEs can be found in [70], [71], [72], [73].
We will now apply the sine-Gordon expansion
method to obtain exact solutions of the following
sine-Gordon equation with beta partial derivatives.
∂β
x∂β
xu−∂β
t∂β
tu=m2sin(u), β ∈(0,1],
(6)
where u=u(x, t)and mis a nonzero real constant.
Substituting the wave variable
u=u(x, t) = U(ξ),
ξ=µ
x+1
Γ(β)β
β−
ct+1
Γ(β)β
β
,(7)
into equation (6) and using the chain rule mentioned
above, we obtain the following nonlinear ODE
U′′ =m2
µ2(1 −c2)sin(U),(8)
where U=U(ξ), ξ is the amplitude of the traveling
wave, µis a nonzero real constant and cis the velocity
of the traveling wave. After applying integration by
parts and some trigonometric identities to equation (8)
and then simplifying the result, we get
U
2′2
=C2sin2U
2+K, (9)
where C2=m2
µ2(1−c2), c =±1and Kis a constant of
integration. Then, replacing ω(ξ) = U
2and K= 0 in
equation (9) and simplifying, we obtain
ω′=Csin(ω).(10)
By choosing C= 1 for equation (10), we have
ω′= sin(ω).(11)
Using the method of separation of variables to solve
equation (11), we obtain two important relations as
follows, [35]:
sin(ω) = sin(ω(ξ)) = 2peξ
p2e2ξ+ 1p=1 = sech(ξ),(12)
and
cos(ω) = cos(ω(ξ)) = p2e2ξ−1
p2e2ξ+ 1p=1 = tanh(ξ),(13)
where pis a constant of integration.
We will now give a summary of the sine-Gordon
expansion method (SGEM) for NLDEs. In or-
der to apply the sine-Gordon expansion method to
nonlinear space-time partial differential equations
with beta partial derivatives, we must first trans-
form the original problem into an ordinary differen-
tial equation (ODE) in a new variable ξ. Consider
the following nonlinear partial differential equation
with beta partial derivatives of a dependent variable
u=u(x1, x2, ..., xn, t)and independent variables
x1, x2, ..., xnand t:
F1u, ∂β
tu, ∂β1
x1u, ..., ∂βn
xnu, utt, ux1x1, ...,
uxnxn, ∂β
t∂β1
x1u, ...= 0,(14)
where 0< β, β1, β2, ..., βn≤1. Further, ∂γ
vu=
∂γ
∂vγuis a generic term for the beta partial derivative
of the dependent variable uwith respect to the inde-
pendent variable vof order γ∈(0,1]. Finally, the
terms of the form utt and uxixjrepresent classical
integer-order partial derivatives. We also assume that
the function F1in equation (14) is a polynomial of u
and its various partial derivatives.
Then, applying the following fractional complex
traveling wave transformation in ξto (14)
u(x1, x2, ..., xn, t) = U(ξ),
ξ=
k1x1+1
Γ(β1)β1
β1
+
k2x2+1
Γ(β2)β2
β2
+... +
knxn+1
Γ(βn)βn
βn
+
ct+1
Γ(β)β
β,(15)
where k1, k2, ..., kn, c are nonzero constants which
will be found at a later step, and then integrating the
resulting equation with respect to ξas many times as
possible, we obtain an ODE in U=U(ξ)as
F2(U, U′, U′′, U ′′′, ...) = 0,(16)
where F2is a polynomial function of Uand its var-
ious integer-order derivatives and the prime notation
(′) denotes the ordinary derivative with respect to ξ.
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The next major steps of the sine-Gordon expansion
method, [35], [68], [74], [75], are as follows.
Step 1: Suppose that the trial solution of equation
(16) is of the form
U(ξ) =A0+
N
i=1
tanhi−1(ξ) (Aitanh (ξ)
+Bisech (ξ)) ,(17)
where the coefficients A0, Ai, Bi(i= 1,2, ..., N)
will be determined later. Then equation (17) can be
rewritten in terms of the sine and cosine functions via
equations (12) and (13) as:
U(ω(ξ)) =A0+
N
i=1
cosi−1(ω(ξ)) (Aicos(ω(ξ))
+Bisin(ω(ξ))) .(18)
Step 2: The positive integer Nin equation (17)
(or equation (18)) can be found using the homoge-
neous balance principle, in other words, by substitut-
ing equation (17) (or equation (18)) into equation (16)
and balancing the nonlinear terms and the highest-
order derivatives appearing in the resulting equation.
More precisely, if the degree of U(ξ)is Deg[U(ξ)] =
N, then the degree of some specific terms can be cal-
culated using the following formulas, [13]:
Deg dqU(ξ)
dξq=N+q,
Deg (U(ξ))pdqU(ξ)
dξqs=N p +s(N+q).
(19)
Step 3: Substituting solution form (18) with
the known value Nobtained from Step 2 into
the ODE (16) and setting the summation of co-
efficients of sinr(ω(ξ)) coss(ω(ξ)) with the same
power to zero, we obtain a system of nonlinear al-
gebraic equations for the unknowns A0, Ai, Bi(i=
1,2, ..., N), kj(j= 1,2, ..., n)and c. With the help
of symbolic software packages such as Maple, it is
assumed that the resulting algebraic system can be
solved for the unknown constants.
Step 4: Substituting the unknown values obtained
from Step 3 and the wave transformation ξin equa-
tion (15) into solution (17), we finally derive the exact
traveling wave solutions of equation (14).
In the following section, we justify the perfor-
mance of the method by applying it to the problems
(1) and (2).
4 Applications of the Method and
Graphical Simulations
In this section, we use the sine-Gordon expansion
method to obtain exact traveling wave solutions of
the PDEs with beta partial derivatives in equations (1)
and (2).
4.1 Exact solutions for the
(2 + 1)-dimensional breaking soliton
equation with beta space-time
derivatives
We first convert equation (1) to an ODE using the
chain rule (4) and the following transformation
u(x, y, t) = U(ξ),
ξ=k1x+1
Γ(β)β
β+
k2y+1
Γ(β)β
β
−
ct+1
Γ(α)α
α,(20)
where k1, k2are care nonzero constants which will
be determined at a later step. The resulting ODE in
the variable U=U(ξ)is then
k4
1k2U(4) −6k3
1k2U′U′′ −ck2
1U′′ = 0,(21)
where the prime notation (′)denotes the ordinary
derivative with respect to ξand U(4) denotes the
fourth derivative. Integrating equation (21) with re-
spect to ξand letting a constant of integration be zero,
we eventually get the following ODE
k4
1k2U′′′ −3k3
1k2(U′)2−ck2
1U′= 0.(22)
Utilizing the solution form (18) and balancing the
highest-order derivative U′′′ with the nonlinear term
of the highest-order (U′)2via the formulas in equa-
tion (19), we get N= 1. Then, using equation (17),
we can write the solution of equation (22) as
U(ξ) = A0+A1tanh(ξ) + B1sech(ξ),(23)
or equivalently from equation (18), we obtain
U(ω(ξ)) = A0+A1cos(ω(ξ))+B1sin(ω(ξ)),(24)
where A0, A1, B1are unknown constants with A2
1+
B2
1= 0 which will be determined at a later step.
Then, by differentiating equation (24), we can obtain
the following expressions for U′and U′′′.
U′(ω(ξ)) = −A1sin2(w(ξ))
+B1sin(w(ξ)) cos(w(ξ)),
U′′′(ω(ξ)) = −4A1sin2(w(ξ)) cos2(w(ξ))
+ 2A1sin4(w(ξ)) (25)
+B1sin(w(ξ)) cos3(w(ξ))
−5B1sin3(w(ξ)) cos(w(ξ)).
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Inserting equation (25) into equation (22), we obtain
3k3
1k2A2
1sin2(w(ξ)) cos2(w(ξ))
−6k4
1k2A1sin2(w(ξ)) cos2(w(ξ))
−3k3
1k2B2
1sin2(w(ξ)) cos2(w(ξ))
−6k3
1k2A1B1sin(w(ξ)) cos3(w(ξ))
+ 6k4
1k2B1sin(w(ξ)) cos3(w(ξ))
−3k3
1k2A2
1sin2(w(ξ)) + 2k4
1k2A1sin2(w(ξ))
+ 6k3
1k2A1B1sin(w(ξ)) cos(w(ξ))
−5k4
1k2B1sin(w(ξ)) cos(w(ξ))
+ck2
1A1sin2(w(ξ))
−ck2
1B1sin(w(ξ)) cos(w(ξ)) = 0.
(26)
Next, setting the summation of the coefficients of
each power of sinr(ω(ξ)) coss(ω(ξ)) to zero, we ob-
tain the following set of nonlinear algebraic equa-
tions:
sin(ω(ξ)) cos(ω(ξ)) : 6k3
1k2A1B1−5k4
1k2B1
−ck2
1B1= 0,
sin(ω(ξ)) cos3(ω(ξ)) : −6k3
1k2A1B1
+ 6k4
1k2B1= 0,
sin2(ω(ξ)) : −3k3
1k2A2
1+ 2k4
1k2A1
+ck2
1A1= 0,
sin2(ω(ξ)) cos2(ω(ξ)) : 3k3
1k2A2
1−6k4
1k2A1
−3k3
1k2B2
1= 0.
By solving the algebraic set of equations with the aid
of symbolic computation software such as Maple 17,
we get the following two cases of the unknown con-
stants c, k1, k2, A0, A1, B1listed in Table 1, where
c, k2, A0are arbitrary nonzero constants such that
ck2>0.
Table 1. Values of the unknown coefficients and pa-
rameters for the solutions of equation (1)
Case 1 Case 2
k1=±1
2c
k2, k2= 0 k1=±c
k2
A1= 2k1A1=k1
B1= 0 B1=±k1i, where i=√−1
We therefore get two independent solutions.
Case 1: After substituting the values of the unknown
constants in Case 1 of Table 1 and the transforma-
tion (20) into equation (23), we obtain one exact trav-
eling wave solution of equation (1) as
u1(x, y, t) = A0+ 2k1tanh(ξ),(27)
where
ξ=k1(x+1
Γ(β))β
β+k2(y+1
Γ(β))β
β−c(t+1
Γ(α))α
α.
Case 2: After substituting the values of the un-
known constants in Case 2 of Table 1 and the trans-
formation (20) into equation (23), we obtain a second
exact traveling wave solution of equation (1) as
u2(x, y, t) = A0+k1tanh(ξ)±ik1sech(ξ),(28)
where
ξ=k1(x+1
Γ(β))β
β+k2(y+1
Γ(β))β
β−c(t+1
Γ(α))α
α.
Next, with the help of the Maple program, we will
plot graphs of the exact solutions of (27) and (28)
in which both real-valued and complex-valued solu-
tions consisting of the hyperbolic functions are con-
structed via the SGEM. In the graphs, we will show
how the solutions change as the temporal and spatial
fractional- orders αand βof equation (1) are var-
ied. In particular, α=β= 1,α=β= 0.8and
α=β= 0.4are used for the plots. The solutions
(27) and (28) are shown as 3D, 2D, and contour plots
according to the values of αand β. All of the 3D solu-
tion graphs are plotted on the domain {(x, y, t)|0≤
x≤10, y = 1 and 0≤t≤10}. All 2D solu-
tion graphs, showing relationships between u(x)and
x, are plotted on {0≤x≤10, y = 1 and t= 7}.
The variables yand tare fixed for all 2D plots since a
variation of the fractional-order βof the spatial beta-
derivative for xis being studied. In the contour plots,
the contours of u(x, t)for y= 1 are plotted on the
x−tplane. The physical meanings of all plotted so-
lutions will also be described. All of the graphs men-
tioned in this section are shown in the Appendix.
In Fig. 1, graphs of the exact solution u1(x, y, t)
in equation (27) are plotted on the specified domains
using the parameter values c=k2= 1, A0= 1. In
particular, Fig. 1 (a)-(c), Fig. 1 (d)-(f) and Fig. 1 (g)-
(i) show the 3D, 2D and contour plots for the exact
solution (27) evaluated at α=β= 1,α=β=
0.8and α=β= 0.4, respectively. As can be seen
from the 3D plots of Fig. 1, the solution (27) can be
characterized as a kink-type solution.
Fig. 2 and Fig. 3 show the plots of the real and
imaginary parts of the solution u2(x, y, t)in equa-
tion (28), respectively, when plotted on the domains
defined above using the parameter values c=k2=
1, A0= 1. In particular, Fig. 2 (a)-(c), Fig. 2 (d)-(f)
and Fig. 2 (g)-(i) show the 3D, 2D and contour plots
for the real part of equation (28), i.e., Re(u2(x, y, t))
computed at α=β= 1,α=β= 0.8and α=
β= 0.4, respectively. We can see from the 3D plots
of Fig. 2 that Re(u2(x, y, t)) represents a kink-type
solution. In a similar manner, the 3D, 2D and con-
tour plots for the imaginary part of equation (28), i.e.,
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Im(u2(x, y, t)) evaluated at α = β = 1, α = β = 0.8
and α = β = 0.4 are plotted in Fig. 3 (a)-(c), Fig. 3
(d)-(f) and Fig. 3 (g)-(i), respectively. By classifying
the shape of the 3D graphs in Fig. 3, the imaginary
solution Im(u2(x, y, t)) can be identified as a bell-
shaped solitary wave solution. Finally, Fig. 4 (a)-(c),
Fig. 4 (d)-(f) and Fig. 4 (g)-(i) show the 3D surface
plots for y = 1, the 2D curves for y = 1, t = 7,
and the contour plots when y = 1 for the magnitudes
|u2(x, y, t)| for the fractional orders α = β = 1,
α = β = 0.8 and α = β = 0.4, respectively.
To conclude this section, we summarize results re-
ported by previous authors for the (2+1)-dimensional
breaking soliton equation. In [76], the authors used
the generalized exponential rational function (GERF)
method to solve the integer-order (2+1)-dimensional
breaking soliton equation. They obtained exact solu-
tions including hyperbolic, trigonometric, exponen-
tial and rational functions and identified the dynamics
of some solutions as a singular kink wave and a soli-
ton. In [77], the author solved the classical (2 + 1)-
dimensional breaking soliton equation to obtain exact
non-traveling wave solutions using the idea of merg-
ing the generalized variable separation method with
the extended homoclinic test approach. By combin-
ing the trigonometric, exponential, hyperbolic func-
tions and three arbitrary functions, Shang identified
non-traveling wave solutions as a periodic solitary
wave, a cross soliton-like wave, a periodic cross-kink
wave and a period two solitary wave. In [78], the au-
thors obtained analytical solutions for the space-time
conformable fractional (2 + 1)-dimensional break-
ing soliton equation using the simplified tan(ϕ(ξ)
2)-
expansion method. They obtained traveling wave so-
lutions in terms of trigonometric, exponential and ra-
tional functions with arbitrary parameters. Inserting
specific values for the parameters, they identified the
physical behaviors of the solutions as periodic and
anti-kink wave solutions. They also investigated how
the graphical behaviors of the solutions changed as
the fractional-orders of the equation were varied.
We have found it quite difficult to directly com-
pare our obtained solutions with those described in
the abovementioned references except by comparing
them in terms of their mathematical expressions and
graphical structures. Using the SGEM, we have ob-
tained results for equation (1), including the hyper-
bolic tangent and hyperbolic secant functions with the
special transformation ξin equation (20), which are
novel.
4.2 Exact solutions for the generalized
Hirota-Satsuma coupled KdV system
with beta time derivative
Using the traveling wave transformation,
u(x, t) = U(ξ), v(x, t) = V(ξ), w(x, t) = W(ξ),
ξ=k
x−
ct+1
Γ(η)η
η
,(29)
where kand care nonzero constants, we can reduce
the generalized Hirota-Satsuma coupled KdV system
with beta time derivative in (2) to the following sys-
tem of nonlinear ODEs:
−ckU′=1
4k3U′′′ + 3kUU′+ 3k(−V2+W)′,
(30)
−ckV ′=−1
2k3V′′′ −3kUV ′,(31)
−ckW ′=−1
2k3W′′′ −3kUW ′,(32)
where the prime notation (′)represents the ordinary
derivative with respect to ξ.
Let, [12], [79], [80],
U=αV 2+βV +γ, W =AV +B, (33)
where α, β, γ, A and Bare real constants to be de-
termined later. Substituting equation (33) into equa-
tions (31) and (32), and then integrating once, we find
that equations (31) and (32) can be converted to the
following equation:
k2V′′ =−2αV 3−3βV 2+ 2(c−3γ)V+c1,(34)
where c1is a constant of integration. Multiplying
equation (34) by V′and then integrating the resulting
equation with respect to ξ, we obtain
k2(V′)2=−αV 4−2βV 3+ 2(c−3γ)V2
+ 2c1V+c2,(35)
where c2is another constant of integration.
Then, differentiating equation (33) with respect to
ξand using equations (34) and (35), we get
k2U′′ = 2αk2(V′)2+k2(2αV +β)V′′,
= 2α[−αV 4−2βV 3+ 2(c−3γ)V2
+ 2c1V+c2]
+ (2αV +β)[ −2αV 3−3βV 2
+ 2(c−3γ)V+c1].(36)
Integrating equation (30) with respect to ξonce, we
obtain
1
4k2U′′ +3
2U2+cU + 3(−V2+W) + c3= 0,(37)
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where c3is a constant of integration. Substituting
equations (33) and (36) into equation (37), we find
that the coefficients of the resulting polynomial must
be zero as shown below:
3αc −3αγ +3
4β2−3 = 0,
1
2(αc1+βc +γβ) + A= 0,
1
4(2αc2+βc1) + 3
2γ2+cγ + 3B+c3= 0.
(38)
Let
c1=1
2α2(β3+ 2cαβ −6αβγ),
V(ξ) = aP (ξ)−β
2α,
(39)
where a= 0 is a constant. From (38) and after some
algebraic manipulations, we have
α=β2−4
4(γ−c), A =4β(c−γ)
β2−4, c =γ, β =±2,
B=1
6 (c−γ) (β2−4)216 c3c β2
−2c3c β4−16 c3γ β2+ 2 c3γ β4
+ 56 γc2β2−48 cγ2β2−16 c2+1
4c2β6
−3c2β4+ 12 c2β2−16 cγ2
−32 γc2−8β2c3+γ3β4−2c3β4
+ 32 c3γ−32 c3c+ 48 γ3+cγ2β4.
(40)
Then, from equation (34), we have
k2P′′ −2c−6γ+3β2
2αP+ 2αa2P3= 0.(41)
Applying the homogeneous balance principle and
the formulas in equation (19) to the terms P′′ and P3
of the above equation, we obtain
Deg P′′=N+ 2 = Deg P3= 3N, (42)
which leads to N= 1. Therefore, the solution form
of the ODE (41) using equation (17) is
P(ξ) = A0+A1tanh(ξ) + B1sech(ξ),(43)
or equivalently, by using equation (18), the solution
form is
P(ω(ξ)) = A0+A1cos(ω(ξ))+B1sin(ω(ξ)),(44)
where the constant coefficients A0, A1and B1are de-
termined at a later step, provided that A2
1+B2
1= 0.
After performing the appropriate algebraic manipula-
tions for equation (44), we have
P3(ω(ξ)) = −2A1sin2(ω(ξ)) cos (ω(ξ))
+B1sin (ω(ξ)) cos2(ω(ξ))
−B1sin3(ω(ξ)) ,
P′′(ω(ξ)) = A3
0+ 3A2
0A1cos (ω(ξ))
+ 3A2
0B1sin (ω(ξ))
+ 3A0A2
1cos2(ω(ξ))
+A3
1cos3(ω(ξ))
+ 6A0A1B1cos (ω(ξ)) sin (ω(ξ))
+ 3A0B2
1sin2(ω(ξ))
+ 3A2
1B1cos2(ω(ξ)) sin (ω(ξ))
+ 3A1B2
1cos (ω(ξ)) sin2(ω(ξ))
+B3
1sin3(ω(ξ)) .
(45)
Then, substituting equations (44) and (45) into (41),
we obtain
2k2B1sin (ω(ξ)) cos2(ω(ξ)) + 2B3
1αa2sin (ω(ξ))
+ 6a2αA0B2
1+ 2αa2A3
1cos3(ω(ξ))
−3β2B1sin (ω(ξ))
2α−3β2A1cos (ω(ξ))
2α
+ 6αa2A12B1sin (ω(ξ)) cos2(ω(ξ))
−3β2A0
2α+ 12αa2A0A1B1sin (ω(ξ)) cos (ω(ξ))
−2cA0+ 6γA0+ 6αa2A0A2
1cos2(ω(ξ))
+ 6αa2A2
0B1sin (ω(ξ)) + 6αa2A2
0A1cos (ω(ξ))
−6αa2A1B2
1cos3(ω(ξ))
−6αa2A0B2
1cos2(ω(ξ))
−2αa2B3
1sin (ω(ξ)) cos2(ω(ξ))
+ 6αa2A1B2
1cos (ω(ξ))
−2cA1cos (ω(ξ)) + 6γB1sin (ω(ξ))
+ 6γA1cos (ω(ξ)) −2k2A1cos (ω(ξ)) + 2αa2A3
0
−k2B1sin (ω(ξ)) + 2k2A1cos3(ω(ξ))
−2cB1sin (ω(ξ)) = 0.
(46)
The following algebraic system of equations can then
be derived by summing up the coefficients of the
trigonometric functions with the same power and
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equating each sum to zero:
constant :−2cA0+ 2αa2A3
0+ 6γA0
−3β2A0
2α+ 6a2αA0B2
1= 0,
sin(ω(ξ)) : 6γB1−k2B1−2cB1
+ 6αa2A2
0B1
−3β2B1
2α+ 2B3
1αa2= 0,
cos(ω(ξ)) : 6αa2A1B2
1+ 6αa2A2
0A1
+ 6γA1−2cA1−2k2A1
−3β2A1
2α= 0,
sin(ω(ξ)) cos(ω(ξ)) : 12αa2A0A1B1= 0,
cos2(ω(ξ)) : 6a2αA0A2
1−6a2αA0B2
1= 0,
cos3(ω(ξ)) : 2αa2A3
1−6αa2A1B2
1
+ 2k2A1= 0,
sin(ω(ξ)) cos2(ω(ξ)) : 6α a2A2
1B1
−2αa2B3
1+ 2k2B1= 0.
Solving the above system with the help of Maple,
we obtain the three cases of the unknown coefficients
shown in Table 2 in which c, a, β and γare arbitrary
nonzero constants and α=β2−4
4(γ−c)= 0 with γ=c.
Case 1: Substituting the values of the unknown co-
efficients and parameters in Case 1 of Table 2 and the
transformation (29) into equations (43), (39) and (33),
respectively, we obtain the following exact solution of
equation (2):
v1(x, t) = ±k
√αsech (ξ)−β
2α,
u1(x, t) =α(v1(x, t))2+βv1(x, t) + γ,
w1(x, t) =Av1(x, t) + B,
(47)
where ξ=kx−c(t+1
Γ(η))η
η.
Case 2: Substituting the values of the unknown
coefficients and parameters in Case 2 of Table 2 and
the transformation (29) into equations (43), (39) and
(33), respectively, we get the following exact travel-
ing wave solution of equation (2):
v2(x, t) = ±k
√−αtanh (ξ)−β
2α,
u2(x, t) =α(v2(x, t))2+βv2(x, t) + γ,
w2(x, t) =Av2(x, t) + B,
(48)
where ξ=kx−c(t+1
Γ(η))η
η.
Case 3: Substituting the values of the unknown
coefficients and parameters in Case 3 of Table 2 and
the transformation (29) into equations (43), (39) and
(33), respectively, we obtain the following exact so-
lution of equation (2):
v3(x, t) = ±k
2√−αtanh (ξ)±k
2√αsech (ξ)−β
2α,
u3(x, t) = α(v3(x, t))2+βv3(x, t) + γ, (49)
w3(x, t) = Av3(x, t) + B,
where ξ=kx−c(t+1
Γ(η))η
η.
Next, with the aid of the Maple software, we will
plot graphs of the case 1 solution v1(x, t)in equa-
tion (47) and the case 3 solution v3(x, t)in equa-
tion (49) of system (2). As shown in the Appendix,
these graphs will be plotted in Figs. 5-7 for a range
of values of the temporal fractional-order ηof equa-
tion (2), namely η= 1,η= 0.8and η= 0.4. The so-
lutions v1(x, t)in equation (47) and v3(x, t)in equa-
tion (49) are plotted as 3D, 2D, and contour plots ac-
cording to the values of η. All 3D graphs are plotted
on the domain {(x, t)|0≤x≤10 and 0≤t≤
10}. All 2D graphs which show the relationships be-
tween v(t)and t, are plotted on {0≤t≤10 and x=
5}. The variable tis varied but x= 5 is fixed for
all of the 2D plots because the effects of changing η
in equation (2) on the solutions will be considered.
The contour plots show the contours of v1(x, t)and
v3(x, t)on the x−tplane. As for the breaking soli-
ton equation solutions, the physical meanings of the
selected solutions will be discussed.
In Fig. 5, the graphs of the exact solution v1(x, t)
in equation (47) are plotted on the domains described
in the previous paragraph using the parameter values
c= 1, β = 3, γ = 2, a = 1 and α=5
4in equa-
tion (40). In particular, the 3D, 2D and contour graphs
for the exact solution (47) are plotted for η= 1,
η= 0.8and η= 0.4in Fig. 5 (a)-(c), Fig. 5 (d)-(f)
and Fig. 5 (g)-(i), respectively. From the 3D plots of
Fig. 5, the solution of equation (47) can be identified
as a bell-shaped solitary wave.
In Fig. 6 and Fig. 7, the graphs of the solutions of
the real and imaginary parts of the solution v3(x, t)in
equation (49), respectively, are plotted. The plots are
for the domains specified above for parameter values
c= 1, β = 1, γ = 2, a = 1 and α=−3
4in equa-
tion (40). In particular, Fig. 6 (a)-(c), Fig. 6 (d)-(f)
and Fig. 6 (g)-(i) show the 3D, 2D and contour plots
for the real part of equation (49), i.e., Re(v3(x, t))
evaluated at η= 1,η= 0.8and η= 0.4, re-
spectively. From the 3D plots of Fig. 6 the solution
of Re(v3(x, t)) can be identified as a kink-type solu-
tion. The graphs in Fig. 7 (a)-(c), Fig. 7 (d)-(f) and
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Table 2. Values of the unknown coefficients and parameters for the solutions of equation (2)
Case 1 Case 2 Case 3
∆ = 3β2+ 4αc −12αγ ∆ = 3β2+ 4αc −12αγ ∆ = 3β2+ 4αc −12αγ
k=±∆
2αk=±1
2−∆
αk=±−∆
α
A0= 0 A0= 0 A0= 0
A1= 0 A1=±k
a√−αA1=±k
2a√−α
B1=±k
a√αB1= 0 B1=±k
2a√α
Fig. 7 (g)-(i) show the 3D, 2D and contour plots for
the imaginary part of equation (49), i.e., Im(v3(x, t))
evaluated at η = 1, η = 0.8 and η = 0.4, respec-
tively. From the 3D graphs in Fig. 7, the physical be-
havior of Im(v3(x, t)) can be identified as an anti-bell
shaped soliton solution. Finally, Fig. 8 (a)-(c), Fig. 8
(d)-(f) and Fig. 8 (g)-(i) show the 3D, 2D and contour
plots of the magnitude |v3(x, t)| for the fractional or-
ders η = 1, η = 0.8 and η = 0.4, respectively.
To conclude this section, we summarize results
reported by previous authors for the generalized
Hirota-Satsuma coupled KdV system. In [81], the
authors derived exact solutions of the generalized
Hirota-Satsuma coupled KdV model with classi-
cal derivatives using the generalized Kudryashov
method. Some new singular and kink soliton wave
solutions, expressed in terms of the exponential func-
tion solutions, were obtained. They also plotted
some graphs to show the interactions of two long
waves with different dispersion relations. In [82],
the authors used the auxiliary equation method to
obtain exact traveling wave solutions for the
generalized Hirota-Satsuma coupled KdV system
with the commensurate-order beta time derivative
for a range of fractional orders. All of the
solutions they obtained were expressed in terms of
exponential functions and they characterized the real
parts as a kink-type solution. Also, in [83],
the authors derived exact solutions for the
generalized Hirota-Satsuma coupled KdV
system for a commensurate-order conformable
derivative using the sub-equation method.
They obtained their solutions as hyperbolic
tangent and cotangent functions, tangent and
cotangent functions and rational functions by
using the solutions of the Riccati equation. The
effects of variation of the time-fractional order on
the 3D solution graphs were also studied.
We again found that only mathematical expres-
sions and graphical representations between our so-
lutions and the solutions described in the mentioned
references could be usefully compared. Utilizing the
SGEM, we found exact solutions for equation (2) ex-
pressed in equations (47)-(49) which contained the
hyperbolic tangent and hyperbolic secant functions.
These solutions are new due to the special transfor-
mation for ξin equation (29).
5 Discussions and Conclusions
In this paper, the sine-Gordon expansion method
has been successfully used to derive exact traveling
wave solutions for the (2 + 1)-dimensional breaking
soliton equation with beta space-time derivatives
in equation (1) and the generalized Hirota-Satsuma
coupled KdV system with beta time derivative in
equation (2). The exact explicit solutions of the
proposed problems derived and plotted in Section 4,
are novel and have not been reported in any previous
literature. As seen from the results, the sine-Gordon
method combined with Maple is a powerful method
for obtaining real-valued and complex-valued so-
lutions expressed in terms of sums of hyperbolic
tangent and hyperbolic secant functions. In order
to check the results, all of the solutions obtained
were substituted back into the partial differential
equations (1) and (2) using the chain rule of the
beta-derivative and the Maple software package.
Numerical simulations were carried out for a range
of parameter values. From the 3D, 2D and contour
plots of selected solutions and their absolute values,
it was possible to identify physical behaviors of the
solutions which included a kink-type solution, a bell-
shaped solitary wave solution and an anti-bell shaped
soliton solution. In physical science and engineering,
the kink-shaped solitary wave localized traveling
structure is maintained by different balances, for
instance, nonlinearity is balanced by dissipation or
nonlinearity is simultaneously balanced by disper-
sion and dissipation. Moreover, the bell-shaped
solitary wave solution normally occurs as a result
of the balance between nonlinearity and dispersion.
These two main types of traveling solitary waves
are of great significance for various applications of
nonlinear physical phenomena described by NLPDEs
such as the generalized Benjamin-Bona-Mahony-
Burgers (BBMB) equation, [84], and the SIdV
equation, [85]. In particular, the variation of the
temporal fractional order αand the spatial fractional
order βin equation (1) and the temporal fractional
order ηin equation (2) were found to slightly affect
the amplitude and translation of the obtained exact
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solutions via the transformations (20) and (29).
However, as shown in Figs. 1-8, the fractional orders
did not change the type of solution structure when
compared with those for the integer-order cases.
The results shown in this research may be important
in diverse fields of nonlinear sciences including
physics, applied mathematics and engineering. In
conclusion, the SGEM could be a powerful and
reliable tool for studying physical applications of
a wide range of nonlinear phenomena modeled by
NLPDEs with emphasis on problems which can be
solved by analytical methods or admit certain special
kinds of explicit solutions. Because of the advantages
of analytical solutions, many researchers have now
developed a large number of different methods for
finding these solutions. Finally, one interesting
future development for research would be to apply
the SGEM to equations (1) and (2) with a truncated
M-fractional derivative which is a recent definition
of a fractional-order derivative.
Acknowledgments:
The authors are grateful to anonymous referees for
their valuable comments and several constructive
suggestions, which have significantly improved this
manuscript.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Thitthita Iatkliang, Supaporn Kaewta, Nguyen
Minh Tuan: Conceptualization, data curation, in-
vestigation, methodology, software, visualization,
writing-original draft and writing-review and editing.
Sekson Sirisubtawee: Conceptualization, data
curation, formal analysis, funding acquisition, in-
vestigation, methodology, project administration,
resources, supervision, validation, visualization,
writing-original draft and writing-review and editing.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The first author would like to acknowledge the partial
support from the Graduate College, King Mongkut’s
University of Technology North Bangkok. The last
author appreciates the financial support of the Fac-
ulty of Applied Science, King Mongkut’s University
of Technology North Bangkok (Grant no. 652110).
Conflicts of Interest
The authors declare that they have no conflicts of in-
terest.
Creative Commons Attribution License 4.0
(Attribution 4.0 International , CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 1: Plots for solution u1(x, y, t)in equation (27) constructed using the SGEM: (a)-(c) when α=β= 1; (d)-(f)
when α=β= 0.8; (g)-(i) when α=β= 0.4.
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 2: Plots of real part Re(u2(x, y, t)) of solution u2(x, y, t)in equation (28) constructed using the SGEM:
(a)-(c) when α=β= 1; (d)-(f) when α=β= 0.8; (g)-(i) when α=β= 0.4.
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Appendix
Figures described in sections 4.1-4.2.
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 3: Plots of imaginary part Im(u2(x, y, t)) of solution u2(x, y, t)in (28) constructed using the SGEM: (a)-(c)
when α=β= 1; (d)-(f) when α=β= 0.8; (g)-(i) when α=β= 0.4.
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 4: Plots of magnitude |u2(x, y, t))|of solution u2(x, y, t)in equation (28) constructed using the SGEM:
(a)-(c) when α=β= 1; (d)-(f) when α=β= 0.8; (g)-(i) when α=β= 0.4.
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(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 5: Plots of solution v1(x, t)for equation (47) constructed using the SGEM: (a)-(c) when η= 1; (d)-(f) when
η= 0.8; (g)-(i) when η= 0.4.
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 6: Plots of real part Re(v3(x, t)) of solution v3(x, t)in equation (49) constructed using the SGEM: (a)-(c)
when η= 1; (d)-(f) when η= 0.8; (g)-(i) when η= 0.4.
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(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 7: Plots of imaginary part Im(v3(x, t)) of solution v3(x, t)in equation (49) constructed using the SGEM:
(a)-(c) when η= 1; (d)-(f) when η= 0.8; (g)-(i) when η= 0.4.
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 8: Plots of magnitude |v3(x, t)|of solution v3(x, t)in equation (49) constructed using the SGEM: (a)-(c)
when η= 1; (d)-(f) when η= 0.8; (g)-(i) when η= 0.4.
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