Download New Complex Hyperbolic Function Solutions for the (2+1

Mathematical
and Computational
Applications
Article
New Complex Hyperbolic Function Solutions for the
(2+1)-Dimensional Dispersive Long
Water–Wave System
Hasan Bulut 1, *,† and Haci Mehmet Baskonus 2,†
1
2
*
†
Department of Mathematics, Firat University, Elazig 23119, Turkey
Department of Computer Engineering, Tunceli University, Tunceli 62100, Turkey; [email protected]
Correspondence: [email protected]; Tel.: +90-424-237-0000
These authors contributed equally to this work.
Academic Editor: Mehmet Pakdemirli
Received: 18 February 2016; Accepted: 18 March 2016; Published: 24 March 2016
Abstract: In this paper, new algorithms called the “Modified exp(´Ω)-expansion function method”
and “Improved Bernoulli sub-equation function method” have been proposed. The first algorithm
is based on the exp(´Ω(ξ))-expansion method; the latter is based on the Bernoulli sub-Ordinary
Differential Equation method. The methods proposed have been expressed comprehensively in this
manuscript. The analytical solutions and application results are presented by drawing the two- and
three-dimensional surfaces of solutions such as hyperbolic, complex, trigonometric and exponential
solutions for the (2+1)-dimensional dispersive long water–wave system. Finally, a conclusion has
been presented by mentioning the important discoveries in this manuscript.
Keywords: modified exp(´Ω(ξ))-expansion function method; improved Bernoulli sub-equation
function method; the (2+1)-dimensional dispersive long water–wave system; hyperbolic function
solution; trigonometric function solution; complex function solution; exponential function solution
1. Introduction
In recent years, studies on the nonlinear differential equations (NDEs) and systems (NDESs) have
become a significant field among scientists. This is why many engineering problems can be represented
by using NDEs and with the rapid development of computational algorithms. Therefore, many
mathematical models have been proposed to understand of such problems in scientific and engineering
such as the optical fiber communications, coastal and oceans engineering, fluid dynamics, plasma
physics, chemical physics and many other scientific applications. Yong-Sik Cho, Dae-Hee Sohn
and Seung Oh Lee have conducted a study on shallow-water equations for distant propagation of
tsunamis [1]. Furthermore, even some important diseases such as dengue epidemics, tuberculosis,
HIV, AIDS, tsunamis, malaria, and cholera have been investigated by many scientists from all over
the word [2–7]. One of them is to obtain various solutions of coastal, oceans and fluid problems
such as approximate, numerical, analytical and traveling wave solutions. Some important traveling
wave solutions for nonlinear differential equations have been investigated by different authors [8–15].
In the rest of this manuscript; we have explained the fundamental properties of the modified
exp(´Ω(ξ))-expansion function method (MEFM) and Improved Bernoulli sub-equation function
method (IBSEFM) in Section 2. We have studied to obtain some new analytical solutions such as
hyperbolic, exponential, and complex hyperbolic function solutions by applying MEFM and IBSEFM
to the (2+1)-dimensional dispersive long water–wave (DLW) system defined by [16]:
` ˘
uyt ` u xxy ´ 2v xx ´ u2 xy “ 0,
vt ´ v xx ´ 2 puvqx “ 0,
Math. Comput. Appl. 2016, 21, 6; doi:10.3390/mca21020006
(1)
www.mdpi.com/journal/mca
Math. Comput. Appl. 2016, 21, 6
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where u = u(x, y, t) represents the horizontal velocity of water and v = v (x, y, t) gives the deviation
height from the equilibrium position of the liquid [16]. The solutions of (1) are very helpful for coastal
scientists and engineers to apply the nonlinear water model to coastal and harbor design [17–19].
Equation (1) was used to model nonlinear and dispersive long gravity waves traveling in two horizontal
directions on shallow waters of uniform depth. Equation (1) appears in many scientific applications
such as nonlinear fiber optics, plasma physics, fluid dynamics, and coastal engineering [16].
2. Fundamental Properties of Methods
2.1.The Modified Exp(´Ω(ξ))-Expansion Function Method
Let us consider the following nonlinear partial differential equations systems for functions
u, v [20,21]:
`
˘
P1 u x , v x , ut , vt , uy , vy , ¨ ¨ ¨ “ 0,
`
˘
(2)
P2 u x , v x , ut , vt , uy , vy , ¨ ¨ ¨ “ 0,
where u = u(x, y, t), v = v (x, y, t) are unknown functions, and Pi (i = 0, 1) are polynomials in u = u(x, y, t),
v = v (x, y, t)
Step 1: Combine the real variables x, y and t with a compound variable ξ:
u px, y, tq “ U pξq , ξ “ kx ` wy ´ ct,
v px, y, tq “ V pξq , ξ “ kx ` wy ´ ct,
(3)
Bu
Bu
Bu
“ kU 1 pξq , uy “
“ wU 1 pξq , ut “
“ ´cU 1 pξq ,
Bx
By
Bt
Bv
Bv
Bv
vx “
“ kV 1 pξq , vy “
“ wV 1 pξq , vt “
“ ´cV 1 pξq ,
Bx
By
Bt
..
.
(4)
ux “
where ω, c, k are both constants and non-zero. By travelling wave transformation, Equation (2) under
the terms of Equations (3) and (4) transforms Equation (2) system in a nonlinear ordinary differential
equation (NODE) as the following:
`
˘
NODE U, U 1 , U 2 , U 3 , ¨ ¨ ¨ “ 0,
where NODE is a polynomial of U and U 1 “
(5)
dU
d2 U
d3 U
3 “
, U2 “
,
U
.
dξ
dξ 2
dξ 3
Step 2: Conjecture of the travelling wave solutions for Equation (5) can be stated in the following form:
N
ř
U pξq “
i
Ai rexp p´Ω pξqqs
i “0
M
ř
j
Bj rexp p´Ω pξqqs
“
A0 ` A1 exp p´Ω pξqq ` ¨ ¨ ¨ ` A N exp p´NΩ pξqq
,
B0 ` B1 exp p´Ω pξqq ` ¨ ¨ ¨ ` B M exp p´MΩ pξqq
(6)
j “0
where Ai (0 ď i ď N) and B j (0 ď j ď M) are constants to be determined, such that A N ­“ 0, B M ­“ 0, and
Ω = Ω(ξ) satisfies the following ordinary differential equation:
Ω1 “ µexp pΩq ` exp p´Ωq ` λ.
The following exact analytical solutions can be written from Equation (7) [22–24]:
(7)
Math. Comput. Appl. 2016, 21, 6
3 of 19
Family-1: If µ ‰ 0, λ2 ´ 4µ ą 0,
¸
¸
˜ a
˜a
´ λ2 ´ 4µ
λ2 ´ 4µ
λ
pξ ` εq ´
.
Ω pξq “ ln
tanh
2µ
2
2µ
(8)
Family-2: When µ ‰ 0, λ2 ´ 4µ ă 0,
˜a
¸
¸
˜a
´λ2 ` 4µ
´λ2 ` 4µ
λ
pξ ` εq ´
Ω pξq “ ln
.
tan
2µ
2
2µ
(9)
Family-3: When µ “ 0, λ ‰ 0, and λ2 ´ 4µ ą 0,
ˆ
Ω pξq “ ´ln
λ
exp pλ pξ ` εqq ´ 1
˙
.
(10)
Family-4: When µ ‰ 0, λ ‰ 0, and λ2 ´ 4µ “ 0,
ˆ
˙
2λ pξ ` εq ` 4
Ω pξq “ ln ´
.
λ2 pξ ` εq
(11)
Family-5: When µ “ 0, λ “ 0, and λ2 ´ 4µ “ 0,
Ω pξq “ ln pξ ` εq ,
(12)
where A0 ,A1 , . . . ,A N , B0 ,B1 , . . . ,B M , ε, λ, µ are real constants to be identified later. ε is constant of
integration. The positive integer N and M can be identified by taking the balance rule between the
highest order derivatives and the nonlinear terms occurring in Equation (5).
Step 3: According to different values of N and M, we can find various forms of Equation (6). By putting
Equations (6) in (5), we can obtain a polynomial of exp(´Ω(ξ)). If we equal to zero all the coefficients
of same power of exp(´Ω(ξ)), we can obtain a system of equations. When we solve this system with
the aid of Wolfram Mathematica 9, this process gives different values of coefficients A0 ,A1 , . . . ,A N ,
and B0 ,B1 , . . . ,B M , ε. If we put these coefficients in Equation (6) by considering family conditions, we
obtain many analytical solutions for Equation (2).
2.1. Improved Bernoulli Sub-Equation Function Method (IBSEFM)
IBSEFM formed by improving the Bernoulli sub-equation function method [25] will be given in
this sub-section. Therefore, we consider the following steps.
Step 1. Let us consider the nonlinear partial differential equations system as the following:
`
˘
P1 u x , v x , ut , vt , uy , vy , ¨ ¨ ¨ “ 0,
`
˘
P2 u x , v x , ut , vt , uy , vy , ¨ ¨ ¨ “ 0,
(13)
u px, y, tq “ U pξq , ξ “ kx ` wy ´ ct,
v px, y, tq “ V pξq , ξ “ kx ` wy ´ ct,
(14)
and take the wave transformation
where w, k, c are both constants and non-zero and will be determined later. Substituting Equations (14)
into (13) converts a nonlinear ordinary differential equation (NODE) as the following:
`
˘
NODE U, U 1 , U 2 , U 3 , ¨ ¨ ¨ “ 0,
(15)
Math. Comput. Appl. 2016, 21, 6
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where NODE is a polynomial of U, and, U 1 “
dU
d2 U
d3 U
, U2 “
, U3 “
.
2
dξ
dξ
dξ 3
Step 2. Considering the trial equation of the solution in Equation (15), it can be written as the following:
n
ř
U “ U pξq “
i “0
m
ř
ai F i
“
bj F j
a0 ` a1 F ` a2 F 2 ` ¨ ¨ ¨ ` a n F n
,
b0 ` b1 F ` b2 F2 ` ¨ ¨ ¨ ` bm F m
(16)
j “0
and, according to the Bernoulli theory,
F1 “ bF ` dF M , b ‰ 0, d ‰ 0, M P R ´ t0, 1, 2u ,
(17)
dF
dF pξq
“
where F “ F pξq is Bernoulli differential polynomial and F1 “
. Substituting the above
dξ
dξ
relations in Equation (15) yields an equation of polynomial Ω pFq of F “ F pξq
Ω pFq “ ρs F s ` ¨ ¨ ¨ ` ρ1 F ` ρ0 “ 0.
(18)
According to the balance principle, we can get some values of n, m and M.
Step 3. Letting the coefficients of Ω(F) all be zero will yield an algebraic equations system
ρi “ 0, i “ 0, ¨ ¨ ¨ , s.
(19)
Solving this system, we will determine the values of k, w, c, a0 ,a1 ,a2 , . . . an , b0 ,b1 ,b2 , . . . ,bm .
Step 4. When we solve nonlinear Bernoulli differential Equation (17), we obtain, according to b and d,
two situations as the following:
 1
´d
ε
F pξq “
` bp M´1qξ 1 ´ M , b ‰ d,
b
e
„
ˆ
b p1 ´ Mq ξ
— pε ´ 1q ` pε ` 1q tanh
2
˙
ˆ
F pξq “ —
–
b p1 ´ Mq ξ
1 ´ tanh
2
»
˙ fi
(20)
1
1´M
ffi
ffi
fl
, b “ d, ε P R,
(21)
where ε is both an integration constant and non-zero. Using a complete discrimination system for
polynomial of F(ξ), we solve Equation (16) with the help of using Wolfram Mathematica 9 and classify
the exact solutions to Equation (16). For a better interpretations of results obtained in this way, we plot
two- and three-dimensional surfaces of the solutions obtained by taking suitable parameters.
3. The Implementations of Techniques
In this subsection of the paper, we have obtained new analytical solutions to the DLW system (1)
by using MEFM and IBSEFM.
3.1. Application of MEFM
Let us consider the travelling wave solutions of the DLW system (1) and we perform the
transformation as the following;
u px, y, tq “ U pξq , ξ “ kx ` wy ´ ct,
v px, y, tq “ V pξq , ξ “ kx ` wy ´ ct.
(22)
Math. Comput. Appl. 2016, 21, 6
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First of all, if we consider the following transformations for the DLW system (1),
v px, y, tq “ uy px, y, tq “
Bu px, y, tq
,
By
(23)
we can obtain following partial differential equations:
´ ¯
uyt ` u xxy ´ 2uy xx ´ u2
xy
“ 0,
`
˘
uyt ´ uy xx ´ 2 uuy x “ 0.
(24)
(25)
When we use the travelling wave transformations Equation (22),
uyt “ ´cwU 2 , u xxy “ wk2 U 3 , uyxx “ wk2 U 3 , u xy “ wkU 2 , u x “ kU 1 , uy “ wU 1 ,
` 2˘
`
˘
`
˘1
u xy “ 2 puu x qy “ 2 uy u x ` uu xy “ 2wk UU 1 ,
(26)
and using Equations (26) in (24), we obtain the following NODE;
`
˘1
´ cU 2 ´ k2 U 3 ´ 2k UU 1 “ 0.
(27)
If we integrate Equation (27) twice by getting zero the integration constants, we obtain the
following NODE
´ cU ´ k2 U 1 ´ kU 2 “ 0.
(28)
When we use Equations (26) in (25), after integrating twice by getting zero the integration
constants we obtain the same NODE as Equation (28) below:
´ cU ´ k2 U 1 ´ kU 2 “ 0.
(29)
When we rearrange to Equation (6), with the help of balance principle between U’ and U2 , we
obtain the term for suitability:
N “ M ` 1.
(30)
This relationship gives rise to various analytical solutions to the DLW system (1) as the following:
Case-1
If we choose M = 1 and N = 2, we can write the following equations for Equation (6):
U“
A0 ` A1 exp p´Ωq ` A2 exp p´2Ωq
Υ
“ ,
B0 ` B1 exp p´Ωq
Ψ
and
U1 “
..
.
Υ1 Ψ ´ Ψ1 Υ
,
Ψ2
(31)
(32)
where A2 ­“ 0, B1 ­“ 0 and Ω = Ω (ξ). When we use Equations (31,32) in the (29), we get a system of
equations for Equation (29) from the coefficients of polynomial of exp(´Ω(ξ)). Solving this system
with the help of Wolfram Mathematica 9 yields the following coefficients:
Case 1.1:
ˆ
˙
2A0
2kB0
2B0
A0 p´A0 ` kλB0 q
2A0
A1 “
` kB0 , A2 “
, B1 “
,µ “
, c “ k kλ ´
.
λ
λ
λ
B0
k2 B02
(33)
Math. Comput. Appl. 2016, 21, 6
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Case 1.2:
a
?
´i 2A2 pA1 ´ λA2 q
A2 pA1 ´ λA2 q
i cA2
?
?
,k “ ?
, B1 “
,
c
2A1 ´ 2λA2
2c
´A1 pA1 ´ 2λA2 q
.
µ“
4A22
(34)
a
a
´iλ A2 pA1 ´ λA2 q
´i 2A2 pA1 ´ λA2 q
´λ
?
?
p´2A1 ` λA2 q , B0 “
A0 “
, B1 “
,
4
c
2c
?
2
3λ
i cA2
A pA ´ 2λA2 q
,µ “ ´
.
k“ ?
´ 1 1 2
4
2A1 ´ 2λA2
A2
(35)
A2
´iλ
A0 “ 1 , B0 “
4A2
a
Case 1.3:
Case 1.4:
A0 “
A21
iλ
, B0 “
4A2
Case 1.5:
A0 “
a
a
?
i 2A2 pA1 ´ λA2 q
A2 pA1 ´ λA2 q
´i cA2
?
?
, B1 “
,k “ ?
,
c
2A1 ´ 2λA2
2c
A pA ´ 2λA2 q
.
µ“´ 1 1 2
4A2
k2 B02
3k2 B02
A2
2k3 B0
2kB0
, A1 “ kB0 , B1 “
,µ “
,c “
.
,λ “
2
4A2
k
A2
A2
4A2
(36)
(37)
Therefore, when we substitute coefficients Equations (33) along with (8) in (31) for u(x, y, t) and in
Equation (23) for v(x, y, t), we can obtain hyperbolic function solution to the DLW system (1) as the
following, under the condition of Family-1; λ2 ´ 4µ “ p´2A0 ` kλB0 q2 {k2 B02 ą 0 :
A0
2A0 pA0 ´ kλB0 q
,
`
2
B0
kλB0 ` 2kβB02 tanh pβ f px, y, tqq
wA0 p´2A0 ` kλB0 q2 p´A0 ` kλB0 q sech2 pβ f px, y, tqq
”
ı ,
v1 px, y, tq “
kB02 ` kλB0 ` p´2A0 ` kλB0 q tanh2 pβ f px, y, tqq
u1 px, y, tq “
where f px, y, tq “ ε ` wy ` k px ´ λktq `
(38)
2kA0
´2A0 ` kλB0
t, β “
, and, A0 , B0 , λ, k, w, ε are both
B0
2kB0
constants and non-zero.
For Equation (34), if we use coefficients Equations (34) along with (8) in (31) for u px, y, tq and
in Equation (23) for v px, y, tq, we can obtain new complex hyperbolic function solution to the DLW
system(1) as the following, under the condition of Family-1; λ2 ´ 4µ “ pA1 ´ λA2 q2 {A22 ą 0 :
u2 px, y, tq
“
v2 px, y, tq
“
`
?
iA21 cω r1 ` tanh rβ f px, y, tqss2
?
,
2 2A2 rA1 ` λA2 r´1 ` tanh pβ f px, y, tqqss r´λA2 ´ ωtanh pβ f px, y, tqqs
γ px, y, tq A21 ω 3{2 p´ωq p1 ` σ px, y, tqq2 γ px, y, tq A21 ω 3{2 p1 ` σ px, y, tqq
´
? 3{2
? 3{2
4 2A2 ` px, y, tq g px, y, tq2
2 2A2 ` px, y, tq g px, y, tq
γ px, y, tq A21 ω 3{2 p1 ` σ px, y, tqq2
? ?
,
4 2 A2 ` px, y, tq g px, y, tq
(39)
?
A1 ´ λA2
i cA2 x
where β “
, f px, y, tq “ ε ´ ct ` wy ` a
,ω “
A1 ´ λA2 ,
2A2
2 pA1 ´ λA2 q
„
2
„

?
ω f px, y, tq
ω f px, y, tq
γ px, y, tq “ i cwSech
, σ px, y, tq “ tanh
, ` px, y, tq “ A1 `
2A2
2A2
λA2 p´1 ` σ px, y, tqq , g px, y, tq “ ´λA2 ´ ωσ px, y, tq and, A0 , B0 , λ, k, w, ε are constants and not zero.
Math. Comput. Appl. 2016, 21, 6
7 of 19
Considering Equations (35) along with (8) in (31) for u px, y, tq and in Equation (23) for v px, y, tq, we
can obtain another new complex hyperbolic function solution to the DLW system (1) as the following,
under the condition of Family-1; λ2 ´ 4µ “ 4 pA1 ´ λA2 q2 {A22 ą 0 :
?
i cω p2A1 ´ λA2 q p1 ` tanh rω f px, y, tqsq
ˆ
˙ ,
? 3{2
2ω
2A2
λ`
tanh rω f px, y, tqs
A2
?
´i cw p2A1 ´ 3λA2 q ω 3{2 p2A1 ´ λA2 q
v3 px, y, tq “
ˆ
„

„
˙2 ,
? 3{2
ω
ω
2A2
λA2 Cosh
f px, y, tq ` 2ωSinh
f px, y, tq
A2
A2
u3 px, y, tq “
(40)
?
i cA2 x
being f px, y, tq “ ε ´ ct ` wy ` ?
, ω “ A1 ´ λA2 .
2ω
Substituting Equations (36) along with (8) in (31) for u px, y, tq and in Equation (23) for v px, y, tq, we
can obtain another new complex hyperbolic function solution to the DLW system (1) as the following,
under the condition of Family-1; λ2 ´ 4µ “ pA1 ´ λA2 q2 {A22 ą 0:
u4 px, y, tq
“
v4 px, y, tq
“
?
i cωA21 p1 ` σ px, y, tqq2
a
,
2 2A2 pA1 ` λA2 p´1 ` σ px, y, tqqq p´λA2 ´ ωσ px, y, tqq
γ px, y, tq A21 ω 5{2 p1 ` σ px, y, tqq2 γ px, y, tq A21 ω 3{2 p1 ` σ px, y, tqq
´
¯ `
? 3{2
? 3{2
2 2A2 ` px, y, tq g px, y, tq
4 2A2 ` px, y, tq g px, y, tq2
´
(41)
γ px, y, tq A21 ω 3{2 p1 ` σ px, y, tqq2
?
,
4 2A2 `2 px, y, tq g px, y, tq
?
„

?
ω
i cA2
where f px, y, tq “ ε ´ ct ` wy ´ ?
f px, y, tq , σ px, y, tq “
x, γ px, y, tq “ i cwSech2
2A2
2ω
„

ω
tanh
f px, y, tq , ω “ A1 ´ λA2 , ` px, y, tq “ A1 ` λA2 p´1 ` σ px, y, tqq , g px, y, tq “ ´λA2 ´
2A2
ωσ px, y, tq.
Taking Equations (37) along with (8) in (31) for u px, y, tq and in Equation (23) for v px, y, tq, we can
obtain another new hyperbolic function solution to the DLW system(1) as the following, under the
condition of Family-1; λ2 ´ 4µ “ k2 B02 {A22 ą 0 :
k2 B0
u5 px, y, tq “
4A2
v5 px, y, tq “ ´
where h px, y, tq “
ˆ
6
3
1´
`
2 ` tanh rh px, y, tqs 1 ` 2tanh rh px, y, tqs
9k3 wsech4 rh px, y, tqs B02
˙
,
(42)
,
4A22 p2 ` tanh rh px, y, tqsq2 p1 ` 2tanh rh px, y, tqsq2
˘
kB0 `
pε ` kx ` wyq A2 ´ 2k3 B0 t .
2
2A2
Case-2
If we choose M “ 2 and N “ 3, we can write following equations for Equation (6):
U“
A0 ` A1 exp p´Ωq ` A2 exp p´2Ωq ` A3 exp p´3Ωq
Υ
“ ,
B0 ` B1 exp p´Ωq ` B2 exp p´2Ωq
Ψ
and
U1 “
..
.
Υ1 Ψ ´ Ψ1 Υ
,
Ψ2
(43)
(44)
Math. Comput. Appl. 2016, 21, 6
8 of 19
where A3 ‰ 0, B2 ‰ 0 and Ω “ Ω pξq . When we use Equations (43) and (44) in the (29), we get a system
of equations for Equation (29) from the coefficients of polynomial of exp(´Ω(ξ)). Solving this system
with the help of Wolfram Mathematica 9 yields the following coefficients:
Case 2.1:
´
λ´
a
A0 “
¯
λ2 ´ 4µ A3 B0
´
´
¯ ¯
a
A3 2B0 ` λ ´ λ2 ´ 4µ B1
, A1 “
2B2
a
˙
ˆ2B2
2 ´ 4µA2
a
λ
1
2B
A3
1
3
A2 “ A3 λ ´ λ2 ´ 4µ `
,c “
,
,k “
2
B2
B2
B22
Case 2.2:
,
‰
1 “` 3
?
? ˘
λ ` λ2 p ´ 3λµ ´ pµ A0 ` µ p´ pλr ´ 2µq A1 ` rµA2 q ,
2µ3
rp1{4 A0
p1{4 p´ pλr ´ 2µq A0 ` rµA1 q
?
B0 “ ?
, B1 “
,
2 cµ
2 cµ2
`
˘
?
?
p1{4 λ3 ` λ2 p ´ 3λµ ´ pµ A0 ` µ p´ pλr ´ 2µq A1 ` rµA2 q
B2 “
,
A?0 ` µ p´ pλr ´ 2µq A1 ` rµA2 q
c
?
k “ 1{4 , p “ λ2 ´ 4µ, r “ λ ` p.
p
(45)
A3 “
(46)
When considering Equations (45) along with (8) in (31) for u px, y, tq and in Equation (23) for
v px, y, tq, another new hyperbolic function solution to the DLW system(1) can be obtained as the
following, under the condition of Family-1; λ2 ´ 4µ ą 0:
¨
˛
a
A3 ˚
˚λ ´ ψ ´
2B2 ˝
‹
4µ
‹
‚,
1a
λ ` ψtanh
ψ pz px, y, tqq
2
wψµA3
v6 px, y, tq “ ˆ
„

„
˙2 ,
a
1a
1a
λCosh
ψ pz px, y, tqq ` ψSinh
ψ pz px, y, tqq
2
2
u6 px, y, tq “
„
a
(47)
¯
A3 ´ a 2
λ
´
4µA
`
xB
´t
, and ψ “ λ2 ´ 4µ.
3
2
B22
Putting Equation (46) along with Equation (8) in Equation (31) for u px, y, tq and in Equation (23)
for v px, y, tq gives another new hyperbolic function solution to the DLW system (1) as the following,
under the condition of Family-1; λ2 ´ 4µ ą 0 :
where z px, y, tq “ ε ` wy `
„

¯ ¯
2 1
´λ ψ`2
µ sech
pθ px, y, tqq
2
„

u7 px, y, tq “
,
1
´8µ ´ 2ψsech2
pθ px, y, tqq
` a
˘ 2
˘
a `
?
cwψ1{4 µ 2 ψµ ` ψ λ2 ´ 2µ cosh rθ px, y, tqs ´ λψsinh rθ px, y, tqs
v7 px, y, tq “
,
`
˘2
λ2 ´ 2µ ` 2µcosh rθ px, y, tqs
?
cψ1{4
´
λ2
?
ˆ
where θ px, y, tq “
a
ψ ε ´ ct ` wy `
´
a
cx
ψ1{4
´1 ` e´pθ px,y,tqq
(48)
˙
, ψ “ λ2 ´ 4µ.
3.2. Application of IBSEFM
Let us consider the travelling wave solutions of the DLW system (1), and we perform the
transformation as the following:
u px, y, tq “ U pξq , ξ “ kx ` wy ´ ct,
v px, y, tq “ V pξq , ξ “ kx ` wy ´ ct.
(49)
Math. Comput. Appl. 2016, 21, 6
9 of 19
First of all, if we consider the above transformations for the DLW system (1), we can obtain the
same NODE as Equation (29):
´ cU ´ k2 U 1 ´ kU 2 “ 0.
(50)
When we rearrange to Equation (16), with the help of balance principle between U 1 and U 2 , we
obtain the term for suitability:
m ` M “ n ` 1.
(51)
This relationship allows us various analytical solution forms for the DLW system (1).
Case-1
If we choose M “ n “ 3 and m “ 1, we can write the following equations for Equation (6):
U pξq “
a0 ` a1 F ` a2 F 2 ` a3 F 3
Υ
“ ,
b0 ` b1 F
Ψ
and
U1 “
..
.
Υ1 Ψ ´ Ψ1 Υ
,
Ψ2
(52)
(53)
where a3 ‰ 0, b1 ‰ 0 and F “ F pξq .When we use Equations (52) and (53) in Equation (50), we get a
system of equations for Equation (50) from the coefficients of polynomial of F. Solving this system
with the help of Wolfram Mathematica 9 yields the following coefficients:
Case 1.1:
? ?
? ?
?
i 2 cdb1
i c
i 2 cdb0
?
?
, a3 “
, k “ ´? ? ,
a0 “ 0, a1 “ 0, a2 “
b
b
2 b
(54)
Case 1.2:
? ? ?
? ? ?
a0 “ 2 b cb0 , a1 “ 2 b cb1 , a2 “
Case 1.3:
? ?
? ?
?
2 cdb0
2 cdb1
c
?
?
, a3 “
, k “ ´? ? ,
b
b
2 b
? ?
?
i 2 cdb1
i c
?
a0 “ 0, a1 “ 0, a2 “ 0, a3 “ ´
, b0 “ 0, k “ ? ? .
b
2 b
(55)
(56)
If Equations (54) along with (8) is put into (51) for u px, y, tq and in Equation (23) for v px, y, tq, new
complex exponential function solution to the DLW system(1) can be obtained as the following:
? ?
i 2 cd
˙,
u8 px, y, tq “ ? ˆ
? ? ?
d
2bct
`
i
2
b
cx
´
2bwy
ε
b ´ `e
b
? 5{2 ? 2bct`i?2?b?cx´2bwy
εw
2i 2b
cde
v8 px, y, tq “
.
´
? ? ? ¯2
de2bwy ´ be2bct`i 2 b cx ε
(57)
When we use coefficients Equations (55) along with (8) in (51) for u px, y, tq and in Equation (23)
for v px, y, tq, we can obtain another new exponential function solution to the DLW system (1) as
the following:
? 3{2 ?
2b
cε
? ? ?
u9 px, y, tq “
,
´2bct´ 2 b cx`2bwy ` bε
? ? ?
?´de
?
(58)
2 2b5{2 cde2bct` 2 b cx`2bwy εw
v9 px, y, tq “
.
´
¯
? ? ?
2
de2bwy ´ be2bct` 2 b cx ε
Math. Comput. Appl. 2016, 21, 6
10 of 19
Substituting Equations (56) along with (8) in (51) for u px, y, tq and in Equation (23) for v px, y, tq, we
can obtain another new complex exponential function solution to the DLW system(1) as the following:
? ?
i 2 cd
˙,
u10 px, y, tq “ ´ ? ˆ
? ? ?
d
b ´ ` e2bct´i 2 b cx´2bwy ε
b
? ? ?
?
?
2i 2b5{2 cde2bct`i 2 b cx´2bwy εw
v10x px, y, tq “ ´ ´ ? ? ?
.
Math. Comput. Appl. 2016, 21,
¯2
dei 2 b cx`2bwy ´ be2bct ε
(59)
7 of 23
Case 1.5:
4. Conclusions
k 2 B02
3k 2 B02
2kB0
2k 3 B0
A
, A1  kB0 , B1  2 ,  
, 
,c 
.
(37)
2
4 A2have been applied
k to the (2+1)-dimensional
4 A2
A2
A2
The MEFM and IBSEFM
dispersive
long water–wave
A0 
system Equation
(1). Then, these methods which are newly submitted to literature in this paper have
Therefore, when we substitute coefficients Equations (33) along with Equation (8) in Equation
given new complex and exponential functional solutions such as Equations (39)–(41) and (57)–(59) for
(31) for u  x, y, t  and in Equation (23) for v  x, y, t  , we can obtain hyperbolic function solution to
the DLW system Equation (1). We have shown that these analytical solutions are verified in Equation (1).
the DLW system (1) as the following, under the condition of Family-1;
Furthermore, we have already2found the similar hyperbolic function solutions such as Equations (38),
 2 (48),
4 
A0 with
k  Banalytical
k 2 B02 
0:
 2(58)
0
(42), (47),
and
solutions
obtained by Wazwaz [16]. Moreover, complex function
solutions such as Equations (39)–(41), (57), and (59) obtained in this paper by MEFM and IBSEFM
2 A0  A0  k  B0 
A0
 compared
,
are new analyticalu1solutions
with analytical solutions
obtained by Wazwaz [16].
 x, y, t   when
2
B0 k  B0  2k  B02 tanh   f  x, y, t  
These results are very helpful for
coastal and civil engineers to apply the nonlinear water model to
(38)
2
coastal and harbor design [16].
1–15 have
been plotted by2 using Wolfram Mathematica 9
wAFigures
0  2 A0  k  B0    A0  k  B0  sech   f  x, y , t  
t 
, for applications
1  x, y , of
under the suitable vvalues
parameters.
It can be predicted that they are very useful
kB02   k  B0   2 A0  k  B0  tanh 2   f  x, y, t   
in engineering and science, especially, coastal and ocean engineering. To the best of our knowledge,
applications of MEFM and IBSEFM to the DLW
not
previously been submitted to
2kA0 system(1)
2 A0have
 k B
0
x, y, tthat
 wy newly
 k  x modified
 kt   methods
t ,  can also
, and,to A
whereWef think
are result
 , k , w, that
  these
0 , B0 ,models
literature.
be
applied
other
B0
2kB0
from engineering,
science, coastal and ocean engineering.
both constants and non-zero.
ux,y,t
vx,y,t
0.4
0.04
0.03
0.3
0.02
0.2
0.01
0.1
40
40
20
20
40
x
20
20
40
x
0.01
A0  2,AB
 3, k  5,
and2D
2Dsurfaces
surfaces ofofEquation
(38)(38)
by considering
the values
FigureFigure1.The
1. The 3D3D
and
Equation
by considering
the values
0 0= 2, B0 = 3, k = 5,
for 2D
 ω0.2,
 y 4,= y0.2,
 0.2,
50< x x<50,
50,´50
 50 < tt<50,
λ = 0.2,
= 6,wε=6,4,
´50
50,and
and tt 
= 0.2,
0.2, for
2D surfaces.
surfaces.
For Equation (34), if we use coefficients Equation (34) along with Equation (8) in Equation (31)
for
u  x, y, t  and in Equation (23) for v  x, y, t  , we can obtain new complex hyperbolic function
solution to the
DLW
system(1) as the following, under the condition of Family-1;
  4   A1   A2  A  0 :
2
2
2
2
where 

, f  x, y, t     ct  wy 
2 A2
2  A1   A2 
,   A1   A2 ,
  f  x, y , t  
  f  x, y , t  
  x, y, t   i cw Sech 
 ,   x, y, t   tanh 
,
2 A2
2 A2




  Appl.
x, y, t2016,
  A121, 6 A2  1    x, y, t   , g  x, y, t    A2    x, y, t  and, A0 , B0 , , k , w,  are
Math. Comput.
2
11 of 19
constants and not zero.
3D
Equation
(39)
by
considering
values
Figure 2. Figure2.The
The 3D surfaces
of surfaces
Equation of
(39) by
considering
the
values
A1 “ 2, Athe
2 “ 3, c “ 5, λ = 0.2,
A1  2, A2  3, c  5,   0.2, w  0.1,   3, y  0.2, 50  x  50,  25  t  25.
w “ 0.1,
ε Comput.
“ ´3,Appl.
y “2016,
0.2, 21,
´50
Math.
x ă x ă 50, ´25 ă t ă 25.
9 of 23
Imux,y,t
40
40
Reux,y,t
0.4
0.4
0.2
0.2
20
20
40
20
0.2
0.2
0.4
0.4
Imvx,y,t
Revx,y,t
0.04
0.04
0.02
0.02
20
Figure3.The
x
40
20
40
x
40
20
0.02
0.02
0.04
0.04
2D
surfaces
of
Equation
(39)
by
20
40
20
40
considering
the
x
x
values
Figure 3. The 2D surfaces of Equation (39) by considering the values A1 = 2, A2 = 3, c = 5, λ = 0.2,
A  2, A  3, c  5,   0.2, w  0.1,   3, y  t  0.2, 50  x  50.
ω = 0.1, ε = 1´3, y =2 t = 0.2, ´50 < x < 50.
Considering Equation (35) along with Equation (8) in Equation (31) for
Equation (23) for v
DLW
system
u  x, y, t  and in
 x, y, t  , we can obtain another new complex hyperbolic function solution to the
(1)
as
the
following,
under
the
 2  4  4  A1   A2  A22  0 :
2
u3  x, y, t  

i c  2 A1   A2  1  tanh  f  x, y, t  

2

,
condition
of
Family-1;
Math. Comput. Appl. 2016, 21, 6
12 of 19
Math. Comput. Appl. 2016, 21, x
Math. Comput. Appl. 2016, 21, x
10 of 23
10 of 23
The
3D
surfaces (40)
of by
Equation
(40) the
byvalues
considering
values
Figure 4. Figure4.
The 3D surfaces
of Equation
considering
A1 = 2, Athe
c = 5, λ = 0.2,
2 = 3,
Figure4.
The
3D
surfaces
of
Equation
(40)
by
considering
the
values
A

2,
A

3,
c

5,


0.2,
w

0.12,



3,
y

0.2,

50

x

50,

25

t

25.
2= 0.2, ´50 < x < 50, ´25 < t < 25.
ω = 0.12, εA=1 
´3,
y
2, A  3, c  5,   0.2, w  0.12,   3, y  0.2, 50  x  50,  25  t  25.
1
2
Reux,y,t
0.15
Reux,y,t
Imux,y,t
Imux,y,t
0.15
0.10
0.10
0.05
0.05
0.05
0.05
40
40
20
20
20
20
x
x
40
40
40
40
20
20
0.05
0.05
0.05
0.05
20
20
40
40
x
x
0.10
0.10
Math. Comput. Appl. 2016, 21, x
11 of 23
0.15
0.15
Revx,y,t
Imvx,y,t
0.02
0.015
0.010
0.01
0.005
40
20
20
40
x
40
20
20
40
x
0.005
0.01
0.010
0.015
0.02
The
2D
surfaces (40)
of by
Equation
(40)
by
considering
Figure 5. Figure
The 2D5. surfaces
of Equation
considering
the values
A1 “the2, Avalues
2 “ 3, c “ 5,
A

2,
A

3,
c

5,


0.2,
w

0.12,



3,
y

t

0.2,

50

x

50.
1
2
λ = 0.2, w “ 0.12, ε “ ´3, y “ t “ 0.2, ´50 ă x ă 50.
Substituting Equation (36) along with Equation (8) in Equation (31) for
u  x, y, t  and in
 x, y, t  , we can obtain another new complex hyperbolic function solution to the
2
2
2
DLW system (1) as the following, under the condition of Family-1;   4   A1   A2  A2  0 :
Equation (23) for v
u4  x, y , t  

i c A12 1    x, y, t  

2
2 2 A2 A1   A2  1    x, y, t     A2    x, y , t  
,
Math. Comput. Appl. 2016, 21, 6
13 of 19
Math. Comput. Appl. 2016, 21, x
Math. Comput. Appl. 2016, 21, x
Figure
6.
The
12 of 23
12 of 23
3D
surfaces
of
Equation
(41)
by
considering
the
values
The
3D
surfaces (41)
of by
Equation
(41) the
by values
considering
Figure 6. Figure
The 3D6. surfaces
of Equation
considering
A1 “the
2, Avalues
2 “ 3, c “ 5,
A1  2, A2  3, c  5,   0.2, w  0.12,   3, y  0.2, 50  x  50,  25  t  25.
A

2,
A

3,
c

5,


0.2,
w

0.12,



3,
y

0.2,

50

x

50,

25

t

25.
λ = 0.2, w “1 0.12, ε2 “ ´3, y “ 0.2, ´50 ă x ă 50, ´25 ă t ă 25.
Imu
Açıklama [m1]: The minus sig
Açıklama [m1]: The minus sig
before the numbers are not disp
before the numbers are not disp
properly in Figures 6-10, 12, 13
properly in Figures 6-10, 12, 13
Dear Editor;
Dear Editor;
They are much better now.
They are much better now.
Reu
Reu
Imu
0.4
0.4
0.4
0.4
0.3
0.3
0.2
0.2
0.2
0.2
0.1
0.1
40
40
20
20
0.1
0.1
20
20
x
x
40
40
40
40
20
20
0.4
0.4
x
x
13 of 23
Rev
Imv
0.04
0.04
0.02
0.02
20
40
40
0.2
0.2
0.2
0.2
Math. Comput. Appl. 2016, 21, x
0.3
0.3
40
20
20
20
40
x
40
20
20
0.02
0.02
0.04
0.04
40
x
2D of surfaces
(41)
by
considering
Figure 7. Figure
The 2D7.The
surfaces
Equationof(41)Equation
by considering
the values
A1 “the2, Avalues
2 “ 3, c “ 5,
A

2,
A

3,
c

5,


0.2,
w

0.12,



3,
y

t

0.2,

50

x

50.
1
2
λ = 0.2, w “ 0.12, ε “ ´3, y “ t “ 0.2, ´50 ă x ă 50.
Taking Equation (37) along with Equation (8) in Equation (31) for
(23) for
u  x, y, t  and in Equation
v  x, y, t  , we can obtain another new hyperbolic function solution to the DLW system(1) as
the following, under the condition of Family-1; 
u5  x, y, t  
2
 4  k 2 B02 A22  0 :

k 2 B0 
6
3

1 
,
4 A2  2  tanh  h  x, y, t   1  2 tanh  h  x, y, t   
v5  x, y, t   




9k w sech  h  x, y, t   B
3

4
2
0
4 A22 2  tanh  h  x, y, t  
 1  2 tanh h  x, y, t  
2
(42)
,
2
  kx  wy  A  2k B t  .
kB0
2 A2



2
h 2016,
x, yAppl.
, t21,
where
Math.
Comput.
2016,
Math. Comput.
Appl.
6 21,
2 x
3
2
15 of 23
0
14 of 19
Case2.2:


A3 
1  3
   2 p  3  p  A0       r  2  A1  r  A2   ,

2 3 
B0 
p1/4     r  2  A0  r  A1 
rp1/4 A0
, B1 
,
2 c
2 c 2
B2 


p1/4  3   2 p  3  p  A0       r  2  A1  r  A2 
A0       r  2  A1  r  A2 
(46)
,
c
, p   2  4 , r    p .
p1/4
k
Math. Comput. Appl. 2016, 21, x
14 of 23
When considering Equation (45) along with Equation (8) in Equation (31) for
Equation (23) for
u  x, y, t  and in
v  x3, y, t  , another new hyperbolic function solution to the DLW system(1) can bex
vx,y,t
ux,y,t
15
10
obtained as the following, under the condition of Family-1; 
5
2
5
 4  0 :
2
10
15
1



1
A3 
4
   
u6  x , y , t  
2  ,
2 B2 
1

  x tanh    z  x, y, t    
5
2
15
10
5
10
15



(47)
3
w A3
1
v6  x, y, t  
,
2

1

1

  Cosh  2   z  x, y, t      Sinh  2 4 z  x, y, t    





Figure 8.The 3D and 2D surfaces of Equation (42) by considering the values A0  2, B0  3, k  5,
Figure 8. The 3D and 2D surfaces
(42) by considering the values A0 “ 2, B0 “ 3, k “ 5,
A3 of Equation
and
 “z0.2,
, t6,4,y4,
 0.2,
50
xt  50,
and
0.2,
wy
´50
 2 504 tA 50,
 xB
, tand
where
 tfor
 22D
 surfaces.
4. 2D surfaces.
6,x,wεy“
2 50,
2 ă x ă 50, ´503 ă t ă
λ “ 0.2,
w
“y0.2,
“
0.2,for
B2


Case-2
If we choose M  2 and N
 3 , we can write following equations for Equation (6):
A0  A1 exp  -Ω   A2 exp  -2Ω   A3 exp  -3Ω  
 ,
B0  B1 exp  -Ω   B2 exp  -2Ω 

U
(43)
and
U 
  
,
2
(44)

where
A3  0, B2  0
and
Ω  Ω  ξ  . When we use Equations (43) and (44) in the Equation (29),
Math. Comput. Appl. 2016, 21, x
we get a system of equations for Equation (29) from the coefficients of polynomial of exp
     .
16 of 23
ux,y,t
Solving this system with the
help of Wolfram Mathematica 9 yields thevx,y,t
following coefficients:
10
0.8
10
20
x
Case 2.1:
0.6
A0 0.4


 
 2  4 A3 B0
2 B2
, A1 

0.05
 ,

A3 2 B0     2  4 B1
0.10
2 B2
(45)
 2 0.15
4 A32
A
2B 
1 
A2  A3     2  4  1  , c 
,k  3 ,
2 
B2 x 
B22
B2
0.2
10
10
Figure9.The
3D
and
0.20
20
2D
surfaces
of
Equation
(47)
by
considering
the
values
Figure 9. The
Equation
considering
A3 “t ´2,
A3 3D
2,and
B2 2D
3, surfaces
 2,  of0.6,
w  0.5,  (47)
 4, yby
 0.2,
25  x  25,the
 5 values
t  5, and
0.2,B2for“ 3, λ “ 2,
µ “ 0.6, w2D“surfaces.
0.5, ε “ 4, y “ 0.2, ´25 ă x ă 25, ´5 ă t ă 5, and t “ 0.2,for 2D surfaces.
Putting Equation (46) along with Equation (8) in Equation (31) for
(23) for
u  x, y, t  and in Equation
v  x, y, t  gives another new hyperbolic function solution to the DLW system (1) as the
following, under the condition of Family-1; 
u7  x , y , t  


2
 4  0 :
 
   x , y ,t  
1

c 1/4  2     2 1  e 
 sech 2    x, y, t   
2
,

2 1
8  2 sech    x, y, t   
2


cw 1/4  2       2  2  cosh   x, y, t     sinh   x, y, t 

(48)
v7  x, y, t  


 2


cw  2        2  cosh   x, y, t     sinh   x, y, t 
1/4
2

2
 2  2 cosh   x, y, t  

2
(48)
,

cx 
2
  ct  wy   1/4  ,     4.


Math. Comput.
Appl.
 2016,
x, y, 21,
t 6  
where
Math. Comput. Appl. 2016, 21, x
18 of 23
15 of 19
Case 1.1:
i 2 cdb0
i 2 cdb1
i c
, a3 
,k  
,
b
b
2 b
a0  0, a1  0, a2 
(54)
Case 1.2:
2 cdb0
2 cdb1
c
, a3 
,k  
,
b
b
2 b
a0  2 b cb0 , a1  2 b cb1 , a2 
Case
1.3: Appl. 2016, 21, x
Math. Comput.
17 of 23
ux,y,t
a0 0.2 0, a1  0, a2  0, a3  
vx,y,t
i 2 cdb1
i c
, b0  0, k  0.2
.
b
2 b
20
10
20
(56)
10
x
20
u  x, y, t  and in Equation
1
If Equation (54) along
with Equation (8) is put into Equation (51)0.5
for
(23) for
(55)
v  x, y, t  , new complex exponential x function solution to the DLW system(1) can be
10
10
20
1.0
obtained as the following:
1
2
u8  x, y, t  
v  x, y , t  
i 2 cd
d

b    e2bct i 2 b
 b
2i 2b5/2 cde 2bct i
cx  2 bwy
1.5
,


2.0
2 b cx  2 bwy
w
(57)
3D 2D
and surfaces
2D 8surfacesofofthe
the Equation
(48)
by considering
2the values c  2,   0.2,
Figure 10.Figure
The 10.The
3D and
Equation
(48)
the values c “ 2, λ “ 0.2,
2 bct by
i 2 considering
b cx
de52bwyt 5,beand
for 2D surfaces.



0.1,
w

0.5,


4,
y

0.2,

25

x

25,

t

0.2,
µ “ ´0.1, w “ 0.5, ε “ 4, y “ 0.2, ´25 ă x ă 25, ´5 ă t ă 5, and t “ 0.2,for 2D surfaces.


.
3.2. Application of IBSEFM
Let us consider the travelling wave solutions of the DLW system (1), and we perform the
transformation as the following:
u  x, y, t   U   ,   kx  wy  ct ,
(49)
v  x, y, t   V   ,   kx  wy  ct.
First of all, if we consider the above transformations for the DLW system (1), we can obtain the
same NODE as Equation (29):
cU  k 2U   kU 2  0.
(50)
Equation (16), with the help of balance principle between U  and U19 ,ofwe
23
obtain the term for suitability:
2
rearrange
Math. When
Comput.we
Appl.
2016, 21, x to
m  M  n  1.
(51)
This relationship allows us various analytical solution forms for the DLW system (1).
Case-1
If we choose
M  n  3 and m  1 , we can write the following equations for Equation (6):
U   
a0  a1F  a2 F 2  a3 F 3 
 ,
b0  b1F

(52)
and
U 
  
,
2
(53)
 (57) by considering the values
Figure11.The 3D surfaces of Equation
b  0.2, c  0.3,
Figure 11. The 3D surfaces of Equation (57) by considering the values b “ 0.2, c “ 0.3, d “ 0.4, w “ 0.12,
 0.4,
y  0.2,
50,  25we
 t use
 25.Equations (52) and (53) in Equation (50), we
 w0,b0.12,
 0 3,
F F50 ξ x.When
whered a
and
ε “ 3, y “ 0.2,3 ´50 ă1 x ă 50, ´25 ă tă 25.
get a system of equations for Equation (50) from the coefficients of polynomial of F . Solving this
Reu
Imu
system with the help of Wolfram Mathematica 9 yields the following coefficients:
0.4
0.6
0.2
0.4
40
0.2
20
20
0.2
40
20
20
40
x
40
x
Figure11.The 3D surfaces of Equation (57) by considering the values b  0.2, c  0.3,
Math. Comput. Appl.
2016, 21, 6
16 of 19
d  0.4, w  0.12,   3, y  0.2, 50  x  50,  25  t  25.
Reu
Imu
0.4
0.6
0.2
0.4
40
0.2
20
20
40
x
0.2
40
20
20
x
40
0.4
Imv
Rev
0.005
0.02
40
20
20
x
40
Math. Comput. Appl. 2016,
21, x
0.005
0.01
40
de2bct 
2 b cx  2bwy
2 2b5/2 cde 2bct 
20
x
40 20 of 23
0.01
2b3/2 c
u9  x, y, t  
0.010
20
 b
,
0.02
2 b cx  2 bwy
(58)
w
b  0.2, c  0.3,


Figure 12. The 2D surfaces of Equation (57) by considering
b “ 0.2, c “ 0.3, d “ 0.4, w “ 0.12,
2 bwy
2 bct the
2 bvalues
cx
v9 xof, y,Equation
t 
.
Figure12.The 2D surfaces
(57) by considering the
2 values


 be
de
d  0.4, w  0.12,   3, y  t  0.2, 50  x  50.
ε “ 3, y “ t “ 0.2, ´50 ă x ă 50.

When we use coefficients Equation (55) along with Equation (8) in Equation (51) for
and in Equation (23) for v
u  x, y, t 
 x, y, t  , we can obtain another new exponential function solution to the
DLW system (1) as the following:
v
0.2
u
0.2
0.8
0.0020
0.6
0.0015
0.4
0.2
40
20
0.0010
20
x
40
0.0005
0.2
0.4
40
20
20
Figure13.The 3D and 2D surfaces of Equation (58) by considering the values
40
x
b  0.2, c  0.3,
Figure 13. The 3D and 2D surfaces of Equation (58) by considering the values b “ 0.2, c “ 0.3,
d  0.4, w  0.12,   3, y  0.2, 50  x  50,  25  t  25, and t  0.2, for 2D surfaces.
d “ 0.4, w “ 0.12, ε “ 3, y “ 0.2, ´50 ă x ă 50, ´25 ă t ă 25, and t “ 0.2, for 2D surfaces.
Substituting Equation (56) along with Equation (8) in Equation (51) for
Equation (23) for
u  x, y, t  and in
v  x, y, t  , we can obtain another new complex exponential function solution to
the DLW system(1) as the following:
u10  x, y, t   
v10  x, y, t   
i 2 cd
 d 2bct i 2 b
b  e
 b
2i 2b5/2 cde 2bct i
 de
i 2 b cx  2bwy
cx  2 bwy



2 b cx  2 bwy
 be 2bct 

2
,
w
(59)
.
Math. Comput. Appl. 2016, 21, 6
17 of 19
Math. Comput. Appl. 2016, 21, x
Math. Comput. Appl. 2016, 21, x
21 of 23
21 of 23
Figure14.The 3D surfaces of Equation (59) by considering the values
b  0.2, c  0.3,
3D surfaces
of Equation
(59) by considering
the bvalues
 0.2,
Figure 14.Figure14.The
The 3D surfaces
of Equation
(59) by considering
the values
“ 0.2, cb “
0.3, cd “0.3,
0.4, w “ 0.12,
d

0.4,
w

0.12,


3,
y

0.2,

50

x

50,

25

t

25.
 0.4,
w ă0.12,
 50,
3, y´25
 0.2,ă
50t ăx 25.
50,  25  t  25.
ε “ 3, y “ d0.2,
´50
xă
Imu
Imu
0.2
0.2
40
40
20
20
Reu
Reu
0.2
0.2
0.4
0.4
20
20
40
40
x
x
0.2
0.2
0.2
0.2
40
40
20
20
0.4
0.4
20
20
40
40
x
x
0.2
0.2
0.6
Math. Comput. Appl. 2016, 0.6
21, x
22 of 23
0.4
0.4
Imv
0.2
Rev
0.2
0.02
0.010
0.01
0.005
40
40
20
20
40
x
20
20
40
x
0.01
0.02
0.005
2D surfaces
of Equation
(59) by considering
the bvalues
 0.2,
Figure 15.Figure15.The
The 2D surfaces
of Equation
(59) by considering
the values
“ 0.2, cb “
0.3, cd 
“0.3,
0.4, w “ 0.12,
d

0.4,
w

0.12,


3,
y

t

0.2,

50

x

50.
ε “ 3, y “ t “ 0.2, ´50 ă x ă 50.
4. Conclusions
The MEFM and IBSEFM have been applied to the (2 + 1)-dimensional dispersive long
water–wave system Equation (1). Then, these methods which are newly submitted to literature in
this paper have given new complex and exponential functional solutions such as Equations (39)–(41)
and (57)–(59) for the DLW system Equation (1). We have shown that these analytical solutions are
verified in Equation (1). Furthermore, we have already found the similar hyperbolic function
solutions such as Equations (38), (42), (47), (48), and (58) with analytical solutions obtained by
Wazwaz [16]. Moreover, complex function solutions such as Equations (39)–(41), (57), and (59)
obtained in this paper by MEFM and IBSEFM are new analytical solutions when compared with
analytical solutions obtained by Wazwaz [16]. These results are very helpful for coastal and civil
Math. Comput. Appl. 2016, 21, 6
18 of 19
Acknowledgments: The authors would like to thank the reviewers for their valuable comments and suggestions
to improve the present work.
Author Contributions: All authors have equally contributed to this paper. They have read and approved the final
version of the manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
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