Applied Mathematics and Computation A modification to the first integral method and its applications

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Applied Mathematics and Computation

journalhomepage:www.elsevier.com/locate/amc

A modification to the first integral method and its applications

Hong-Zhun Liu

Zhijiang College, Zhejiang University of Technology, Shaoxing 312030, PR China

a rt i c l e i nf o

Article history:

Received 30 January 2019 Revised 3 December 2020 Accepted 3 December 2021 Available online 18 December 2021 Keywords:

First integral method

Conformable fractional diffusion-reaction equation

Duffing-van der Pol oscillator Complex cubic-quintic Ginzburg–Landau equation

Equation for surface waves KdV–Burgers–Fisher equation

a b s t ra c t

In thisarticle,we modifyFeng’s firstintegral method(FIM)for the purposeof enlarg- ingitsapplications.ComparedwithoriginalFIM,ourmodifiedFIMismorestraightand canalsobeemployedtofindfirstintegralofhigher-orderordinarydifferentialequations (ODEs).WeemployourmodifiedFIMintofivedifferentialequations,namely,thedensity- dependentconformablefractionaldiffusion-reactionequation,theDuffing-vanderPolos- cillator,the complexcubic-quintic Ginzburg–Landauequation,thewell-knownnonlinear evolutionequationfordescriptionofsurfacewavesinaconvectingliquid,andtheKdV–

Burgers–Fisherequation.Consequently, wegetthesamefirstintegralobtainedbyFeng’s FIM forthe firstequation; for the secondequation,we reobtaincertainimportantfirst integral reportedpreviously;forthe thirdequation, weconstructanewfirstintegralof complexcubic-quinticGinzburg–Landauequation;andforthefourthandfifthequations, weshowtheeffectivenessofourapproachtothird-orderODEsandreobtainthesamefirst integralrecentlypresentedbyKudryashovforthefourthequation,andforthefifthequa- tion, twonewfirstintegralsarepresented.Alltheabovefullyrevealtheeffectivenessof ourmodification.

© 2021ElsevierInc.Allrightsreserved.

1. Introduction

In2002,forconstructingtravelingwavesolutionsofBurgers–KdVequation,Feng[1,2]proposedapopularmethodcalled firstintegralmethod(FIM)whichmakesuseofthealgebraictheoryofcommutativeringandhasattractedmuchattention fromresearcherstoinvestigatefirstintegralandthereaftertravelingwavesolutionsofpartialdifferentialequations(PDE)in nonlinearscience,seereferences[3–31]andthereferencestherein.Forconvenience,we firstintroduceFeng’sfirstintegral methodinbrief.

1.1. OutlineofFIM

SupposethatweinvestigateanonlinearPDEwhichreads

E

(

u,ux,ut,uxx,...

)

=0, (1)

whereu=u(^x,t)^{isadependentvariabletobe}determined,E isapolynomial ofuandits partialderivativesinwhichthe highestorderderivativesandnonlineartermsareinvolved.

E-mail addresses: mathlhz@163.com , mathlhz@zjut.edu.cn https://doi.org/10.1016/j.amc.2021.126855

0 096-30 03/© 2021 Elsevier Inc. All rights reserved.

• Step1

Carryingoutthefollowingtravelingwavetransformation

u

(

x,t

)

=u

( ξ )

,

ξ

=kx+ct, (2)

wegetanonlinearordinarydifferentialequation(ODE)

Q

(

u,u,u,· · ·

)

=0, (3)

wheretheprimemeansthederivativewithrespecttoindependentvariable

ξ

^.

• Step2

Wesupposethat(3)admitsthefollowingparticularsolution

u

( ξ )

=X

( ξ )

, (4)

then(3)canbechangedtoanonlinearplaneautonomousequationsas

X

( ξ )

=Y

( ξ )

,

Y

( ξ )

=H

(

^X

( ξ )

,Y

( ξ ))

, (5)

• Step3

AccordingtothequalitativetheoryofODEs[32],wecandirectlyconstructthegeneralsolutionsto(3)bymakinguseof its twoindependentfirstintegrals.Nevertheless,thereisneitheralogicalwaynorsystematictheoriestoconstructitsfirst integralsforagivenplaneautonomoussystem,soitisnoteasyforustogetevenonefirstintegral.Inreference[1],Feng employedtheDivisionTheoremtosearchforonefirstintegralfor(3)whichreduces(3)toafirst-orderODE.Subsequently, exacttravelingwavesolutionsto(1)wereconstructedbysolvingthefirst-orderODEdirectly.Forthesakeofcompleteness, wepresentthefamousDivisionTheoreminwhatfollows.

DivisionTheorem. SupposethatM(

ω

,z)^isanirreduciblepolynomialinC[

ω

,z],andN(

ω

,z)^{isalsoapolynomialin}C[

ω

,z]; IfthepolynomialN(

ω

,z)^{vanishesatallzeropointofM}(

ω

,z)^{,thenthereexistsapolynomialG}(

ω

,z)ⁱⁿC[

ω

,z]suchthat

N

( ω

^,z

)

=G

( ω

^,z

)

·M

( ω

^,z

)

.

Indetail,wesupposethat p(^X(

ξ

),Y(

ξ

))=0isthefirstintegralfor(5)andp(^X(

ξ

),Y(

ξ

))^isalsoanirreduciblepolyno- mialinC[X,Y]suchthat

p

(

X,Y

)

= m

i=0

ai

(

^X

)

^Yi. (6)

Accordingtothedivisiontheorem,theremustexistW(^X,Y)=h(^X)^Y+g(^X)ⁱⁿC[X,Y],suchthat

dp

(

X,Y

)

d

ξ

⁼

⁽

^h

⁽

^X

⁾

^Y+g

⁽

^X

⁾⁾

^·

m

i=0

ai

(

^X

)

^Yi, (7)

from which we determine each a_i(^X),i=1...m. Subsequently, we obtain the desired first integral p(^X(

ξ

),Y(

ξ

))=0 for plane autonomous system (5). As a result, exact traveling wave solutions to (1) can be constructed by solving p(^X(

ξ

),Y(

ξ

))=0directly.

1.2. AnobservationofFIM

Wenoticethefollowingfact,namely

Observation1. Becauseofthedivisiontheorem,thesupposedirreduciblefunction p(^X,Y)^isabinarypolynomial.Thus,the finalemployedODE(3)isexactlyasecond-orderODE,namelyQ(^u,u,u)=0.Asaresult,wecannotapplyFIMtoamore higher-orderODE,sayathird-orderODE.

So, howtomodify FIMto letit beapplied tothe higher-orderODEisstill an open problem.In thisarticle,wetry to answerthisquestion.

Rest ofthe articleisarranged asfollow:inSection2,we givethedescriptionofourmodificationto FIM.In Section3, the firstintegrals ofdensity-dependentconformable fractional diffusion-reaction(cfDR) equation, the Duffing-vander Pol (DVDP) oscillator,the(1+1)dimensionalcomplexcubic-quinticGinzburg–Landau(cqGL)equation,thenonlinearequation forsurfacewavesarisingina convectingfluid,andtheKdV–Burgers–Fisherequation areillustrated asapplications ofour modifiedFIM,respectively.Section4concludesourarticle.

2. AmodificationtoFIM

ThefirststepisthesameasoriginalFIM.Werewrite(3)intheform

Q

(

u,u,u,...,u⁽ⁿ⁾

)

=0, (8)

wherethehighestorderofderivativesncanbegreaterthan2.

Withoutlossofgenerality,inwhatfollows,weonlyconsiderthecaseofn=3,namely

Q

(

^u,u,u,u

)

=0, (9)

• Step2

Wesupposethat(9)admitsthefollowingparticularsolution

u

( ξ )

^=X

( ξ )

^{, (10)}

then(9)canbechangedtoasystemofnonlinearODEsas X

( ξ )

=Y

( ξ )

,

Y

( ξ )

^=Z

( ξ )

^,

Z

( ξ )

=H

(

^X

( ξ )

,Y

( ξ )

,Z

( ξ ))

, (11)

whichisnotaplaneautonomousinthiscase.

• Step3

Wesupposethat(9)admitsafirstintegralintheform p=0, where p

( ξ

^,X,Y,Z

)

^{de f}=

m

i=0

ai

( ξ

^,X,Y

)

^Zi, (12)

wherea_i(

ξ

,X,Y)(ⁱ=1,...,m−1)^{arearbitraryfunctionstobe}determined.

Thenbymakinguseof(11),wehave dp

( ξ

,X,Y,Z

)

d

ξ

^=R

⁽ ξ

,X,Y,Z

)

, (13)

whichmustbeequalto0sincep(

ξ

,X,Y,Z)=0isthefirstintegralfor(11). Inordertospecifya_i(

ξ

,X,Y)(ⁱ=1,...,m)^,wesuppose

R

( ξ

^,X,Y,Z

)

=f

(

^p

( ξ

^,X,Y,Z

))

, (14)

where f(·)^{isafunctionadmittingthefollowingconditionf}(⁰)=0,andhereby f(^p(

ξ

,X,Y,Z))^{isexactlyequalto0.}

In order to eliminate termsof Z,we can take f(^p(

ξ

,X,Y,Z))^{such that both ofR}(

ξ

,X,Y,Z) ^{and f}(^p(

ξ

,X,Y,Z))^have termswhichinvolveZincommon.

Thusfrom(14),wecangetanODE

G

( ξ

^,X,Y

)

=0. (15)

From(15),wedeterminea_i(

ξ

,X,Y)(ⁱ=1,...,m)^.

Finally,wegettherequiredfirstintegral p(

ξ

,X,Y,Z)=0withdetermineda_i(

ξ

,X,Y)(ⁱ=1,...,m)^. 3. Applications

3.1. Thedensity-dependentcfDRequation

Thedensity-dependentcfDRequationreads

∂

^αu

(

^t,x

)

∂

^{tα +ku}

⁽

^t,x

⁾ ∂

^u

(

t,x

)

∂

^{x =D}

∂

^2u

(

^t,x

)

∂

^{x2 +au}

⁽

^t,x

⁾

^−bu2

⁽

^t,x

⁾

^{, (16)}

where0<

α

≤1,t>0anda,b,kandDarepositiveparameters[24].Weshouldmentionthat(16)canbeservedasamodel arisinginthenonlinearscienceandhasattractedmuchgreatefforts[24,33,34].

Assumingthat fisacontinuousfunction f:(⁰,∞)−>,thentheconformablefractionalderivativeoforder

α

^isintro-

ducedas[35]

D_αf

(

^t

)

=lim →0

f

(

^t+

^t1−α

)

−f

(

^t

)

^{, (17)}

whichbearsfollowingproperties[24,35]

(1) D_α(^{a f}+bg)=aD_α(^f)+bD_α(^g),

∀

^a,b∈. (2) D_α(^tμ)=

μ

^tμ−α,

∀ μ

∈.

(3) D_α(^{f g})= f D_α(^g)+gD_α(^f)^. (4) D_α(g^f)= ^gDαf_g⁻2^fDαg.

Furthermore,ifgisaarbitrarydifferentialfunctiondefinedintherangeof f,and fisdifferentiableand

α

-differentiable, thenwehave

D_αf

(

^t

)

=t^1−αdf

dt, (18)

D_α

(

^f◦g

)(

^t

)

=t^1−αg

(

^t

)

^f

(

^g

(

^t

))

. (19)

Nowlet

u

(

x,t

)

=U

( ξ )

,

ξ

=mx− c

α

^{tα, (20)}

wecanchange(16)tothefollowingODE[24]

Dm²U+cU−kmUU+aU−bU²=0. (21)

Wesupposethat(21)admitsaparticularsolutionintheform

U

( ξ )

=X

( ξ )

, (22)

then(21)canbechangedtoaplaneautonomoussystemofnonlinearODEsas X

( ξ )

=Y

( ξ )

,

Y

( ξ )

= 1 m²D

−cY+kmXY−aX+bX²

, (23)

In2018,Rezazadeetal.[24]investigated(23)byusingFIMandobtainedthefollowingfirstintegral Y=− ak

2mbDX+ k

2mDX², (24)

alongwiththefollowingcondition c=

(

^4b2D+ak²

)

^m

2kb . (25)

Inwhatfollows,weemployourmodificationmethodto(21).Forthesake ofsimplicity,wesupposethat(23)admitsa firstintegralintheform

p=0, where p^{de f}=A

(

^X

)

^{+Y, (26)}

whereA(^X)^{isafunctiontobe}determined.

Thenwehave dp

d

ξ

^=AY+Y=AY+

1 m²D

−cY+kmXY−aX+bX²

=

A− c m²D+ k

mDX

Y− a

m²DX+ b

m²DX². (27) Accordingtoourmodification,wetake

A− c m²D+ k

mDX

Y− a

m²DX+ b

m²DX²= f

(

^p

)

=

A− c m²D+ k

mDX

p=

A− c m²D+ k

mDX

(

^A

(

^X

)

+Y

)

, (28) whichcanbereducedto

AA+ k

mDXA− c

m²DA+ a

m²DX− b

m²DX²=0. (29)

Byobservation,wetake

A=k0+k1X+k2X², (30)

wherek_isarearbitraryconstantstobespecified.

Substituting theansatz (30)into (29)andequating thecoefficientsofpowers X withzero,we canobtain anonlinear over-determined systemof algebraicequations. Then by solving the over-determinedsystem, we get the followingthree cases

(a) k0=−_mk^a ,k1= _mk^b ,k2=0,underconditionofc=^mbD_k .

(b) k₀=k₂=0,k₁=_mk^b ,underconditionofc= ^(b2D+_bk^ak2)m.

(c)k₀=0,k₁=_2mbD^ak ,k₂=−_2mD^k ,underconditionofc= ^(4b2D₂⁺_bk^ak2)m. Inaword,wehaveconstructedthreefirstintegralsto(21),namely

U+ b mkU− a

km=0, (31)

U+ b

mkU=0, (32)

U+ ak

2mbDU− k

2mdU²=0. (33)

Remark 1. The third first integral (33)is the same as (24)obtained by Rezazade etal, butthe formertwo along with correspondingconstraintconditionshavenotbeenpresentedbyresearcherstothebestofourknowledge.

3.2. TheDVDPoscillator

TheDVDPoscillatorbeingconsideredreads

x+

( α

⁺

β

^x2

)

^x−

γ

^{x+x3=0, (34)}

wherethe primedenotesderivative withrespecttotimet and

α

,

β

^and

γ

^arerealparameters.In2010, bymeans ofthe Prelle–Singermethod[36],Fengetal.[37]obtainedthefollowingfirstintegral

x+

α

⁻

β

³

x+

β

3x³=Iexp

−3t

β

, (35)

whereIisaconstant,alongwiththefollowingcondition

α

= 3

β

⁻

βγ

3 . (36)

Inwhatfollows,weemployourmodificationmethodto(34). Atthebeginning,wesupposethat(34)hasasolutionintheform

x

(

^t

)

=X

(

^t

)

. (37)

Then(34)ischangedtoanonlinearplaneautonomoussystemas X=Y,

Y=−

( α

+

β

^X2

)

^Y+

γ

^X−X³, (38)

Forthesakeofsimplicity,wesupposethat(38)admitsafirstintegralintheform

p=0, where p^{de f}=G

(

^t

)

+A

(

^X

)

+Y. (39)

Sowehave dp

dt =G+AY+Y=G+AY−

( α

+

β

^X2

)

^Y+

γ

^X−X³=G+

γ

^X−X³+

(

^A−

α

−

β

^X2

)

Y. (40) Accordingtoourmodification,wetake

G+

γ

^X−X3+

(

^A−

α

⁻

β

^X2

)

^Y=

(

^A−

α

⁻

β

^X2

)

^p=

(

^A−

α

⁻

β

^X2

)(

^G

(

^t

)

+A

(

^X

)

+Y

)

, (41) whichcanbereducedto

G−

(

^A−

α

−

β

^X2

)

^G+

γ

^X−X³+

( α

+

β

^X2

)

^A−AA=0. (42) Tosolve(42),wetake

γ

^X−X³+

( α

+

β

^X2

)

^A−AA=0, (43)

G−

(

^A−

α

⁻

β

^X2

)

^{G=0. (44)}

Tosolve(43),forthesakeofsimplicity,wetake

A=aX+bX³. (45)

Substituting(45)into(43),wehave

γ

^X−X3+

( α

⁺

β

^X2

)

^A−AA=

(

^b

β

^−3b2

)

^X5+

(

^a

β

^+b

α

^−1−4ab

)

^X3+

( γ

^+a

α

^−a2

)

^X=0. (46) Equating the coefficientsof powers X with zero,we obtain thefollowing nonlinear over-determined systemof algebraic equations

b

β

−3b²=0,

a

β

^+b

α

^{−1−4ab=0,}

γ

^+a

α

^{−a2=0. (47)}

Bysolving(47),weobtainthefollowingnontrivialcase a=−

γ β

3 ,b=

β

3, (48)

alongwiththefollowingcondition

α

=9−

γ β

²

3

β

^{, (49)}

whichisthesameas(36).

Inthiscase,(44)issimplifiedas G+

α

+

βγ

3

G=0. (50)

Solving(50),weobtain G=Cexp

−

α

+

βγ

3

t

, (51)

whereCisanintegrationconstant.

Inaword,wegetafirstintegralto(34)intheform x−

γ β

3 x+

β

3x³+Cexp

−

α

+

βγ

3

t

=0. (52)

Remark2. Thefirstintegral(52)isexactlythesameas(35).ComparedwiththePrelle–Singermethod,ourmethodismore straightforwardandeffective.

3.3. The(1+1)dimensionalcomplexcqGLequation The(1+1)dimensionalcomplexcqGLequationreads

iux+D

2utt+

|

^u

|

^2u=i

δ

^u+i

|

^u

|

^2u+i

β

^utt+i

α|

^u

|

^4u−

γ |

^u

|

⁴u, (53) where,inoptics,u(^x,t)^{iscomplexamplitude,tistheretardedtime,xisthe}propagationdistance,theparameter

δ

^islinear

lossorgain,

β

^{describesdiffusioncoefficient,D}=±1isthedispersioncoefficientandparameters

α

,

γ

^and

^determineits

shapeofnonlinear[23].

Introducingthefollowingtravelingwavetransformation

u=

v ₍ ψ )

^exp

(

ⁱ

η )

^,

η

^=px+st,

ψ

^{=kx+ct, (54)}

wherek,c,pandsarearbitraryconstantstobespecified,alongwithconstraintconditions[23]

r=−4

β

^s

Dc =k+Dsc

β

^{c2 ,l=}

2p+Ds²

Dc² =−

β

^s2−

δ

β

^{c2 , (55)}

m=− 2

Dc² =−

β

^c2,n=−

2

γ

Dc² =−

α

β

^{c2, (56)}

wecanreduce(53)intothefollowingODE

v

−r

v

−l

v

−m

v

³−n

v

⁵=0. (57)

Inreference[23],G.Akrametal.constructed(57)’sexactsolutions byFeng’sfirstintegralmethod[1].Inwhatfollows,we employourmethodtoobtainitsfirstintegral.

Atthebeginning,wesupposethat(57)hasasolutionintheform

v ( ψ )

=X

( ψ )

. (58)

Then(57)canbechangedtoaplaneautonomoussystemofnonlinearODEsintheformof X=Y,

Y=

γ

^{Y+lX+mX3+nX5, (59)}

Wesupposethat(59)admitsafirstintegralintheform

p=0, where p^{de f}=G

( ψ )

^+A

(

^X

)

^{+Y. (60)}

Sowehave dp

d

ψ

^{=G+AY+Y=G+AY+}

γ

^{Y+lX+mX3+nX5=G+lX+mX3+nX5+}

(

^A+

γ )

Y. (61) Accordingtoourmodification,wetake

G+lX+mX³+nX⁵+

(

^A+

γ )

^Y=

(

^A+

γ )

^p=

(

^A+

γ )(

^G

(

^t

)

+A

(

^X

)

+Y

)

, (62) whichcanbesimplifiedas

G−

(

^A+

γ )

^{G+lX+mX3+nX5−}

γ

^{A−AA=0. (63)}

Tosolve(63),wetake

lX+mX³+nX⁵−

γ

^A−AA=0, (64)

G−

(

^A+

γ )

^{G=0. (65)}

From(64),AisacubicpolynomialofX,soweassume

A=aX+bX³, (66)

whereaandbarearbitraryconstantstobespecified.

Nowsubstituting(66)into(64),weobtain

(

ⁿ−3b²

)

^X5+

(

^m−rb−4ab

)

^X3+

(

^l−a²−ra

)

^X=0. (67)

Thus,equatingthecoefficientsofpowersXwithzero,weobtain n−3b²=0,

m−rb−4ab=0,

l−a²−ra=0. (68)

Solving(68),wehave a=−r

4±m 4

3

n,b=± n

3, (69)

alongwiththefollowingconstraintcondition l=3m²

16n ±mr 8

3 n−3r²

16. (70)

ThusweobtainthevalueofAas A=

−r 4±m

4 3 n

X± n

3X³. (71)

Substituting(71)into(65),andsolvingG(

ψ

)^,weget G

( ψ )

=Cexp

(

³₄^r

ψ )

^exp

±√ 3n

(

^X2+ m 4n

)

^d

ψ

. (72)

Inaword,undertheconstraintcondition(70),wefindthefirstintegralof(57)intheform

v

+

−r 4±m

4 3 n

v

± n

3

v

³+Cexp

_3r

4

ψ

^±√3n

( v

²+ m 4n

)

^d

ψ

=0. (73)

IfwetakeC=0,(73)isreducedtothefollowingBernoulliequation

v

−

r 4∓m

4 3 n

v

=∓ n

3

v

³, (74)

whosegeneralsolutioncanbeobtainedintheform

v

²= 1

C₁exp

(

⁻¹2

(

^r∓m

₃

n

) ψ )

^±_r∓⁴_m^√√n33 n

, (75)

whereC₁isanarbitraryintegrationconstant.

Remark3. Tothebestofourknowledge,firstintegral(73)isnew.

Remark4. Weshouldpointoutthat(57)admitsgeneralsolution(75),whichisalsonovel.

3.4. Theequationforsurfacewavesinaconvectingfluid

In this subsection,we will considera higher-orderdifferential equation, namely,the equation forsurface wavesin a convectingfluidwhichreads[38]

ut+

μ

^uxx+

σ

^uxxx+

δ

^uxxxx+

α (

^u2

)

x+

χ (

^u2

)

xx=0. (76) Inaconvectingfluid,theaboveequationdepictsalongshallowwaveevolutionasandwhentheRayleighnumberfraction- allysurpassitscriticalvalueandhasbeeninvestigatedmanytimes,seereference[38]andthereferencestherein.At

χ

=0, (76)isreducedtotheknownKuramoto–Sivashinskyequation.

Takingintoaccounttravelingwave solutionu=^μ_χy(^z),z=

_μ

δx−^C0_δ^μ2t alongwithtworedefinedparameters

σ

^and

α

^,

wehavethenonlinearODEintheform[38]

yzzz+9

σ

^yzz+yz+2yyz+

α

^y2−C₀y+C₁=0. (77) In orderto constructthe auxiliaryequation, Kudryashov [38] usedthe Fuchsindex which exists inthe Painlevé test and foundthat(77)admitsafirstintegralintheform

yzz+

(

⁹

σ

⁻

α )

^yz+y2+

(

¹⁺

α

²⁻⁹

ασ )

^y+C

α

^{1 +4C2exp}

⁽

⁻

α

^z

)

^{=0. (78)}

Inwhatfollows,weuseourapproachtoreobtainthisfirstintegral.

Wesupposethat(77)hasasolutionintheform

y

(

^z

)

=X

(

^z

)

. (79)

Then(77)canbechangedtoasystemofnonlinearODEsas X=Y,

Y=Z,

Z=−9

σ

^Z−Y+2XY−

α

^{X2+C0X−C1, (80)}

Forthesakeofsimplicity,wesupposethat(80)admitsafirstintegralintheform

p=0, where p^{de f}=G

(

^z

)

+A

(

X,Y

)

+Z. (81)

Sowehave dp

dz =G+A₁Y+A₂Z+Z=G+A₁Y+A₂Z−9

σ

^Z−Y+2XY−

α

^{X2+C0X−C1=G+A}1Y−Y

+2XY−

α

^X2+C0X−C1+

(

^A2−9

σ )

Z. (82) Accordingtoourmodification,wetake

G+A₁Y−Y+2XY−

α

^X2+C0X−C1+

(

^A2−9

σ )

^Z=

(

^A2−9

σ )

^p=

(

^A2−9

σ )(

^G

(

^z

)

+A

(

X,Y

)

+Z

)

, (83) whichcanbesimplifiedas

G−

(

^A2−9

σ )

^G+A1Y−Y−2XY−

α

^{X2+C0X−C1−AA}2+9

σ

^{A=0. (84)}

Tosolve(84),wetake

A₁Y−Y−2XY−

α

^{X2+C0X−C1−AA}2+9

σ

^{A=0, (85)}

G−

(

^A2−9

σ )

^G=0. (86)

ByobservingthedegreeofY inthelefthandsideof(85),weassume

A

(

X,Y

)

=M

(

^X

)

+kY, (87)

whereM(^X)^{isafunctionofX tobe}determined,andkisaconstant.

Substituting(87)into(85),wehave

[M−1−2X+9

σ

^k−k2]Y−

α

^{X2+C0X−C1−kM+9}

σ

^{M=0 (88)}

Therefore,wetake

M−1−2X+9

σ

^k−k²=0 (89)

−

α

^{X2+C0X−C1−kM+9}

σ

^{M=0. (90)}

Solving(90),wehave M=

α

9

σ

^−kX2−

C0

9

σ

^−kX+

C1

9

σ

^{−k, (91)}

and

M= 2

α

9

σ

^−kX−

C0

9

σ

^{−k. (92)}

Substituting(92)into(89),wehave

₂

α

9

σ

^−k−2

X+9

σ

^k−k2−₉

σ

^{C0−k−1=0. (93)}

Sowehave 2

α

9

σ

−k−2=0, (94)

σ

^k−k2−₉

σ

^{C0−k−1=0. (95)}

Solving(94)and(95)comesto

k=9

σ

−

α

, C0=

α (

⁹

σα

−

α

²−1

)

. (96)

SubstitutingvaluesofkandC₀ into(91),wehave M=X²+

(

¹+

α

²−9

σα )

^X+C1

α

^{, (97)}

andthusweobtain

A=X²+

(

¹+

α

²−9

σα )

^X+C1

α

⁺

⁽

⁹

σ

−

α )

Y. (98)

Andinthiscase,(86)turnstobethefollowingODE

G+

α

^G=0. (99)

Solving(99)gives

G

(

^z

)

=C2exp

(

−

α

^z

)

. (100)

Asaresult,weobtainthefollowingfirstintegral Z+

(

⁹

σ

−

α )

^Y+X²+

(

¹+

α

²−9

ασ )

^X+C1

α

^+C2exp

⁽

⁻

α

^z

)

=0, (101)

i.e.,

yzz+

(

⁹

σ

−

α )

^yz+y²+

(

¹+

α

²−9

ασ )

^y+C1

α

^+C2exp

⁽

⁻

α

^z

)

=0. (102)

Remark5. Thefirstintegral(102)isinaccordwith(78)whichhasbeennewlypresentedbyKudryashov[38]. 3.5. TheKdV–Burgers–Fisherequation

Recently, Kocak proposed a third-order dispersion-dissipation-reaction model called the KdV–Burgers–Fisher equa- tion[39],namely

ut+

ε

^uux−

ν

^uxx+

μ

^uxxx=ru

(

^1−u

)

^{, (103)}

where

ε

,

ν

,

μ

,andrarerealparametersforconvection,diffusion,dispersion,andreactionterms,respectively.

TheKdV–Burgers–FisherequationcanbereducedtotheKdVequationwhen

ν

=r=0,Burgersequationwhen

μ

=r=0, Fisherequationwhen

ε

=

μ

=0,Burgers–Fisherequationwhen

μ

=0,anddispersive-Fisherequationwhen

ε

=

ν

=0.

Applyingthefollowingtravelingwavetransformation

u=u

( ξ )

,

ξ

=kx+

ω

^t+

ξ

0 (104)

to(103),wehave

ω

^u+

ε

^kuu−

ν

^k2u+

μ

^k3u=ru

(

¹−u

)

. (105)

Kocakusedthe tanhmethodtoinvestigateexactsolutionsof(105).Inwhatfollows,we employourmethodtoobtainits firstintegral.

Wesupposethat(105)hasasolutionintheform

u

( ξ )

=X

( ξ )

. (106)

Then(105)canbechangedtoasystemofnonlinearODEsas X=Y,

Y=Z,

Z=rX

(

¹−X

)

−

ω

^Y−

ε

^kXY+

ν

^k2Z

μ

^{k3 , (107)}

Wesupposethat(107)admitsafirstintegralintheform

p=0, where p^{de f}=Z+f

(

X,Y

)

. (108)

Sowehave dp

d

ξ

^{=f1Y+f2Z+Z=f1Y+f2Z+}

rX

(

¹−X

)

−

ω

^Y−

ε

^kXY+

ν

^k2Z

μ

^{k3 =f1Y+}

rX

(

¹−X

)

−

ω

^Y−

ε

^kXY

μ

^{k3 +}

f₂+

ν μ

^k

Z.

(109) Accordingtoourmodification,wetake

f₁Y+rX

(

¹−X

)

−

ω

^Y−

ε

^kXY

μ

^{k3 +}

⁽

^f2+

ν

μ

^k

⁾

^Z=

⁽

^f2+

ν

μ

^k

⁾

^·p=

⁽

^f2+

ν

μ

^k

⁾

^·

⁽

^Z+f

⁽

^X,Y

⁾⁾

^{, (110)}

whichcanbesimplifiedas

f₁Y+rX

(

¹−X

)

−

ω

^Y−

ε

^kXY

μ

^{k3 =}

⁽

^f2+

ν

μ

^k

⁾

^{·f. (111)}

Byobservation,weconsiderthefollowingtwocases:

• CaseI: f= f(^X)

Inthiscase,(111)becomes rX

(

¹−X

)

μ

^{k3 +}

⁽

^f−

ω

+

ε

^kX

μ

^k3

⁾

^Y=

ν

μ

^{k·f. (112)}

Therefore,wetake f−

ω

+

ε

^kX

μ

^{k3 =0. (113)}

Solving(113),weobtain f=2

ω

^X+

ε

^kX2

2

μ

^{k3 (114)}

Substituting(114)to(112),weobtain rX

(

¹−X

)

=

ν

μ ⁽ ω

kX+

ε

2X²

)

. (115)

Thus,weobtainthefollowingconstraintconditions

ω

k =−

ε

2 =r

μ

ν

^{. (116)}

Inoneword,underconstraintconditions(116),wehaveobtainedthefollowingfirstintegralof(105). u+

ω

μ

^k3

⁽

^u−u2

⁾

^{=0. (117)}

Multiplying2uandintegrating(117)once,wehave

(

^u

)

²= 2

ω

3

μ

^k3u3−

ω

μ

^{k3u2+C, (118)}

whereCisintegrationconstant.Thenwehave u=6k³

μ

ω

^{℘+1, (119)}

where℘=℘(

ξ

;g₂,g₃)^isWeierstrassellipticfunction,whichadmitsthefollowingequation:

℘˙²=4℘³−g2℘−g3. (120)

Theinvariantsg₂ andg₃aredeterminedby g₂=

ω

²

12

μ

^2k6,g3=

ω

³−6C

μω

^2k3

216

μ

^{3k9 . (121)}

• CaseII: f=a(^X)+b(^X)^Y Inthiscase,(111)becomes

bY²+

a−

b²+

ν

^b

μ

^k+

ω

+

ε

^kX

μ

^k3

Y+rX

(

¹−X

)

μ

^{k3 −}

⁽

^b+

ν

μ

^k

⁾

^{a=0. (122)}

Sowehave

b=0 (123)

a=b²+

ν

^b

μ

^k+

ω

+

ε

^kX

μ

^{k3 (124)}

rX

(

¹−X

)

μ

^{k3 =}

⁽

^b+

ν

μ

^k

⁾

^{a. (125)}

From(123),weobtainthatbisaconstanttobedetermined.From(124),wetake a

(

^X

)

=

(

^b2+

ν

^b

μ

^k+

ω

μ

^k3

⁾

^X+

ε

2

μ

^{k2X2. (126)}

Substituting(126)into(125),weobtain rX

(

^1−X

)

μ

^{k3 =}

⁽

^b+

ν μ

^k

⁾

^·

b²+

ν

^b

μ

^k+

ω

μ

^k3

X+

ε

2

μ

^k2X2

(127)

From(127),weobtain r

μ

^{k3 =−}

⁽

^b+

ν μ

^k

⁾

^·

ε

2

μ

^{k2 (128)}

b²+

ν

^b

μ

^k+

ω

μ

^{k3 =−}

ε

2

μ

^{k2. (129)}

From(128),weobtain b=−2r

ε

^k−

ν

μ

^{k. (130)}

Substituting(130)into(129),weobtainthefollowingconstraintcondition

8k

μ

^r2+4kr

εν

⁺²

ωε

^2+k

ε

^{3=0. (131)}

Inoneword,undertheconstraintcondition(131),wehaveobtainedthefollowingfirstintegralof(105). u−

_2r

ε

^k+

ν μ

^k

u−

ε

2

μ

^k2

⁽

^u−u2

⁾

^{=0. (132)}

Remark 6. Weshouldpointout thatexcepttheveryfirst,therelationsofparameters inthe restsixsolutionsof(105)in Kocak[39]alsomeetourconstraintcondition(131).

Remark7. Tothebestofourknowledge,bothoffirstintegrals(117)and(132)arenew.