New analytic solutions of the space-time fractional Broer-Kaup and approximate long water wave equations

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JournalofOceanEngineeringandScience3(2018)295–302

www.elsevier.com/locate/joes

Original

Article

New

analytic

solutions

of

the

space-time

fractional

Broer–Kaup

and

approximate

long

water

wave

equations

H. Çerdik

Yaslan

Department of Mathematics, Pamukkale University, Denizli 20070, Turkey

Received29April2018;receivedinrevisedform20October2018;accepted20October2018 Availableonline26October2018

Abstract

In the present paper, the exp(−φ(ξ )) expansion method is applied to the fractional Broer–Kaup and approximate long water wave equations.Theexplicitapproximatetravelingwavesolutionsareobtainedbyusingthismethod.Here,fractionalderivativesaredefinedinthe conformablesense.Theobtainedtravelingwavesolutionsareexpressedbythehyperbolic,trigonometric,exponentialandrationalfunctions. Simulations oftheobtained solutionsare givenatthe endof thepaper.

Thisisanopenaccess articleunderthe CCBY-NC-NDlicense.(http://creativecommons.org/licenses/by-nc-nd/4.0/)

Keywords: ThefractionalBroer–Kaupequations;Thefractionalapproximatelongwaterwaveequations;Conformablederivative;exp(−φ(ξ )) expansion method;Travelingwavesolutions.

1. Introduction

Nonlinearpartial differentialequations are importanttools used tomodelednonlinear dynamicalphenomenaindifferent fields such as mathematical biology, plasma physics, solid state physics, andfluid dynamics [1].The travelingwave so-lutions of nonlinear partial differentialequations play an im-portant role in the study of nonlinear physical phenomena such as fluid dynamics, water wavemechanics, meteorology, electromagnetic theory, plasma physics and nonlinear optics etc. In the recent decade, many methods have been devel-oped for finding the travelingwave solutions such as the Ja-cobi ellipticfunction method [2],the ansatz method [3], the exp-(φ(η))) method [4], exp-function method [5], consistent Riccati expansion method [6], the (G/G)-expansion method

[7].

Waves havea majorinfluence on the marine environment and ultimately on the planet climate. One of the most im-portant andapplication classificationsof marine wavesis the shallow water wave. The shallow water equations describe the motion of water bodies wherein the depth is short rela-tive to the scale of the wavespropagating on that body and

E-mail address: hcerdik@pau.edu.tr.

arederivedfrom thedepth-averagedNavier–Stokesequations

[8]. These equations are used to describe flow in vertically well-mixed water bodies where the horizontal length scales are much greater than the fluid depth (i.e., long wavelength phenomena)andtomodelthehydrodynamicsoflakes, estuar-ies,tidalflatsandcoastalregions,aswellasdeepoceantides. Theequationsalso,areused tostudymanyphysical phenom-enasuchforcesactingonoff-shorestructuresandinmodeling the transport of chemicalspecies such as storm surges, tidal fluctuationsandtsunami waves [9].

Inthepresentpaper, weconsiderspace-timefractional ap-proximatelongwater wave equations andBroer–Kaup equa-tions which are used to model the bidirectional propagation of long waves in shallow water. The space-time fractional approximate long water wave equations (see, for example,

[10–12]) are giveninthe form

Ttαu− uTxβu− Txβv+γ TxβTxβu=0,

Ttαv− Txβ(uv)− γ TxβTxβv=0, t >0, 0<α,β ≤ 1,

(1) andthe space-timefractional Broer–Kaupequations (see,for example, [13]) are givenas follows

https://doi.org/10.1016/j.joes.2018.10.004

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296 H. Çerdik Yaslan / Journal of Ocean Engineering and Science 3 (2018) 295–302

Ttαu+uTxβu+Txβv=0,

Ttαv+Txβu+Txβ(uv)+TxβTxβTxβu=0,

t >0, 0<α,β ≤ 1,

(2) Here T_tα and T_xβ denote conformable fractional derivative with respect to t and x, respectively. These equations have been investigated in [14–17]. New exact solutions for frac-tional DR equation and fractional approximate long water wave equation with the modified Riemann–Liouville deriva-tive have been obtained by using G/G-expansion method in [14]. The time fractional coupled Boussinesq–Burger and timefractional approximate long water wave equations with conformable derivative by using the generalized Kudryashov methodhavebeen solvedin [15].The analyticalapproximate traveling wave solutions of time fractional Whitham–Broer– Kaupequations,timefractionalcoupled modifiedBoussinesq and time fractional approximate long wave equations have been obtained by using the coupled fractional reduced dif-ferentialtransform methodin [16]. Herefractional derivative is defined by the Caputo sense. The fractional sub-equation methodhasbeen appliedtothe fractional variantBoussinesq equation and fractional approximate long water wave equa-tion with Jumarie’s modified Riemann–Liouville derivatives in [17].

2. Description of theconformable fractional derivative andits properties

For a function f: (0, ∞)→R, the conformable fractional derivative of f of order 0<α <1 is defined as (see, for ex-ample, [18])

Ttαf(t )=lim

ε→0

f(t+εt1−α₎_{− f}_{(t )}

ε . (3)

Some important properties of the conformable fractional derivative areas follows:

T_tα(a f+bg)(t )=aT_tαf(t )+bT_tαg(t ), ∀a,b∈R, (4)

T_tα(tμ)=μtμ−α, (5)

Ttα( f(g(t ))=t1−αg

(t )f(g(t )). (6)

3. Analytic solutions to thespace-time fractional approximatelong water wave equations

Let us considerthe following transformation

u(x,t)=U(ξ ), ξ =at

α

α +b

xβ

β , (7)

where a, b are constants. Substituting (7) into (1) we have thefollowing ordinarydifferential equations

aU− bUU− bV+γ b2U =0, (8)

aV− b(UV+VU)− γ b2V=0. (9) Integrating (8) withrespect to ξ,thenwe have

V = a bU− C1 b − U2 2 +γ bU . (10)

Substituting (10)into (9) yields −γ2_b3_U₊b 2U 3₋3a 2 U 2₊ C1+ a2 b U− C2 =0. (11)

Here,C1 andC2areintegrationconstants.Letussupposethat

thesolutionof (11)canbeexpressedinthefollowingform:

U(ξ )=

N

i₌₀

ai(exp(−Q(ξ )))i, (12)

where ai are constantstobe determinedlaterandQ(ξ)

satis-fiesthe following auxiliaryordinary differentialequation:

Q(ξ )=exp(−Q(ξ ))+μ exp(Q(ξ ))+λ. (13)

Inserting (12) into (11) then by balancing the highest or-derderivative term andnonlinear term inresultequation, the value of N can be determined as 1. Collecting all the terms withthe samepower of exp(−φ(ξ )),we canobtain aset of algebraicequations for the unknowns a0,a1, C1,C2,a, b:

−2a2_a 0− 3aa20b− a 3 0b 2₊_2C 1a0b− 2a1λμb4γ2− 2C2b=0; 2a1a2− 6a1aa0b+3a1a20b2− 2a1b4λ2γ2 − 4a1μb4γ2+2C1a1b=0; 3a0a21b2− 3aa12b− 6λa1b4γ2=0; a₁3b2− 4a1b4γ2=0.

SolvingthealgebraicequationsintheMathematica,weobtain the followingset of solutions:

a1=2bγ , C1= b 2(a 2 0− 2a0bγ λ + 4b2γ2μ), C2=− b 2(−a0+bγ λ)(a 2 0− 2a0bγ λ +4b2γ2μ), a=−b(−a0+bγ λ).

The solutions of Eq. (1) are givenas follows:

ui(x,t)=a0+2bγ Ri(x,t), (14) vi(x,t)=(a0− bγ λ)ui(x,t)− u2i(x,t) 2 −1 2(a 2 0− 2a0bγ λ + 4b2γ2μ) − 2b2_γ2_(R2 i(x,t)+μ+λRi(x,t)), i=1,2,3,4,5. (15) Here Ri(x,t),i=1,2,3,4,5, isdefined as follows:

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Fig.1. 3Dplotofthesolitarywavesolution u 1(x, t )ofEq.(1) for a 0=10,b =1,μ =1,C =10,λ =3,γ =10, α =0.75, β =0.5. R₁(x,t) = 2μ −λ −λ2 _{− 4μ tanh}₍ √ λ2 −4μ 2 (b(a 0 − bγ λ)t α α + bx β β + C)) , (16) Whenλ2_{− 4μ < 0,} _μ_{= 0,} R 2 (x,t) = 2μ −λ +4μ − λ2 tan √ 4μ−λ2 2 (b(a 0 − bγ λ)t α α + bx β β + C) (17) When λ2 − 4μ > 0, μ = 0, λ = 0, R3(x,t)= λ cosh(b(a0− bγ λ)t α α + bx β β + C) +sinh(b(a0− bγ λ)t α α + bx β β + C) − 1 . When λ2 − 4μ = 0, μ = 0, λ = 0, R 4 (x,t)=− λ 2 _(b(a 0 − bγ λ)tαα +bx β β +C) 2λ(b(a₀− bγ λ)t_αα +bx_ββ +C)+4. (18) When λ2 − 4μ = 0, μ = 0, λ = 0, R₅(x,t)= 1 (b(a 0 − bγ λ)t_αα + bx_ββ + C). (19)

Here C isthe integrationconstant.

Figs. 1–4 represent the change of amplitude and na-ture of the solitary waves for each obtained solitary wave solutions. The solutions u1(x, t), u2(x, t) and v1(x, t)

of Eq. (1) are simulated as traveling wave solutions for various values of the physical parameters in Figs. 1–4.

Figs. 1 and 2 show solitary wave solutions of Eq. (1). 3D plots of the obtained solutions u1(x, t) andv1(x, t) are given

in Fig. 1 and Fig. 2 for parameters a0=10,b=1,μ =

1,C=10,λ =3,γ =10, α =0.75, β =0.5, respectively.

Figs. 3 and 4 are kink-type periodic wave solutions of Eq. (1). 3D plot of the obtained solution u2(x, t) is given for

parametersa0 =0.5,b=1,μ =1,C=5,λ =1,γ =1 α =

0.75, β =0.5 in Fig. 3. Fig. 4demonstrates the same solu-tion with2D plot for 0≤ x≤ 50 att =1.

4. Analytic solutions to the space-timefractional Broer–Kaup equations

Applying the transformation (7) into (2) we have the fol-lowing ordinarydifferentialequations

aU+bUU+bV =0, (20)

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Fig.2. 3Dplotofthesolitarywavesolution v 1(x, t )ofEq.(1) for a 0=10,b =1,μ =1,C =10,λ =3,γ =10, α =0.75, β =0.5.

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Fig.4. 2Dplotoftheperiodicwavesolution u 2(x ,1)ofEq.(1) for a 0=0.5,b =1,μ =1,C =5,λ =1,γ =1, α =0.75, β =0.5.

Integrating (20) withrespect toξ, thenwe have

V =C1 b − U2 2 − a bU. (22)

Substituting (22) into (21)yields

b3U−b 2U 3₋3a 2U 2₊ C1+b− a2 b U− C2 =0. (23)

HereC1 andC2 areintegrationconstants.Letussupposethat

thesolutionof (23)canbeexpressedintheform (12). Insert-ing (12) into (23) and balancing the highest order derivative term and nonlinear term in result equation, the value of N

can be determined as 1. Collecting all the terms with the samepowerof exp(−φ(ξ )),wecanobtain asetofalgebraic equations for the unknowns a0, a1, C2,C2,a, b:

− 2a2_a 0− 3aa20b− a30b2+2a0b2+2C1a0b +2a1λμb4− 2C2b=0; −2a1a2− 6a1aa0b− 3a1a20b2+2a1b4λ2 +4a1μb4+2a1b2+2C1a1b=0; −3a0a12b2− 3aa21b+6λa1b4=0; −a3 1b2+4a1b4=0;

SolvingthealgebraicequationsintheMathematica,weobtain the following setof solutions:

a1 =2b, C1=− b 2(2+a 2 0− 2a0bλ +4b2μ), C2 = b 2(−a0+bλ)(a 2 0− 2a0bλ +4b2μ), a=b(−a0+bλ).

The solutionsof Eq. (1) are given as follows:

ui(x,t)=a0+2bRi(x,t), (24) vi(x,t)= −1 2 (2+a 2 0− 2a0bλ +4b2μ)− u2 i(x,t) 2 +(a0− bλ)ui(x,t) i=1,2,3,4,5. (25)

HereRi(x,t),i=1,2,3,4,5, isdefined as follows:

Whenλ2_{− 4μ >}₀_, _μ₌₀_, R₁(x,t) = 2μ −λ −λ2 _{− 4μ tanh}₍√λ2 −4μ 2 (b(−a 0 +bλ)t α α +bx β β +C)) , (26) Whenλ2 − 4μ <0, μ =0, R₂(x,t) = 2μ −λ +4μ − λ2 tan( √ 4μ−λ2 2 (b(−a 0 +bλ)t α α +bx β β +C)) (27) Whenλ2 − 4μ >0, μ =0, λ =0, R3(x,t)= λ cosh(b(−a0+bλ)t α α +bx β β +C)+sinh(b(−a0+bλ)t α α +bx β β +C)− 1 . Whenλ2 − 4μ =0, μ =0, λ =0, R₄(x,t)=− λ 2 _(b(−a₀₊_b_λ)tα α +bx β β +C) 2λ(b(−a₀+bλ)tα α +bx β β +C)+4 . (28)

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Fig.5. 3Dplotofthesolitarywavesolution u 1(x, t )ofEq.(2) for a 0=0.5,b =0.7,μ =1,C =1,λ =3, α =0.75, β =0.5.

Fig.6. 2Dplotofthesolitarywavesolution u 1(x ,1)ofEq.(2) for a 0=0.5,b =0.7,μ =1,C =1,λ =3, α =0.75, β =0.5.

Whenλ2 − 4μ =0, μ =0, λ =0, R₅(x,t)= 1 (b(−a 0 +bλ)tαα +bx β β +C) . (29)

The solutions u1(x, t), v2(x, t) and v3(x, t) of Eq. (2) are

simulated as traveling wave solutions for various values of the physical parameters in Figs. 5–9. Figs. 5 and 6 show solitary wave solutions of Eq. (2). 3D plot of the obtained solution u1(x, t) is given for a0=0.5,b=0.7,μ =1,C=

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Fig.7. 3Dplotoftheperiodicwavesolution v 2(x, t )ofEq.(2) for a 0=0.5,b =0.7,μ =2,C =1,λ =1, α =0.75, β =0.5.

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Fig.9. 3Dplotofthesolitarywavesolution v 3(x, t )ofEq.(2) for a 0=0.5,b =0.7,μ =0,C =1,λ =0.1, α =0.75, β =0.5.

1,λ =3, α =0.75, β =0.5. Fig.6alsoillustratesthesame solution with 2D plot for 0≤ x≤ 10 at t =1. Figs. 7 and

8 show periodicwave solutions of Eq. (2).3D and2D plots of the obtained solution v2(x, t) and v2(x, 1) are given for

a0=0.5,b=0.7,μ =2,C=1,λ =1, α =0.75, β =0.5,

respectively. From Fig. 8, we can see that the wave ampli-tudes go to infinity and the wavelengths increase when x approaches to infinity. Fig. 9 shows solitary wave solution

v3(x, t) of Eq. (2). 3D plot of the obtained solution v3(x, t)

is given for a0=0.5,b=0.7,μ =0,C=1,λ =0.1, α =

0.75, β =0.5.Note that the 3D graphs describethe behav-iorofuandvinspacexattimet,whichrepresentsthechange ofamplitudeandshapefor each obtainedsolitarywave solu-tions.2D graphsdescribethe behavior of u andvinspace x

atfixedtimet =1. All graphicsinfigures are drawn bythe aidof Mathematica10.

5. Conclusion

In the present paper, the space andtimefractional Broer– Kaup and approximate long water wave equations with the conformablefractionalderivativeareconsidered.Byusingthe

exp(−φ(ξ )) expansionmethod newapproximateanalytic

so-lutions are obtained. The new analytical solutions obtained inthis paper have not been reported in the literature so far.

Thismethodisusefulinsolvingwideclasses ofconformable nonlinear fractional differential equations.

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