Optimum Solutions of Fractional Order Zakharov–Kuznetsov Equations · 2019. 12. 10. · Research Article Optimum Solutions of Fractional Order Zakharov–Kuznetsov Equations RashidNawaz

Research ArticleOptimum Solutions of Fractional OrderZakharovndashKuznetsov Equations

Rashid Nawaz 1 Laiq Zada1 Abraiz Khattak2 Muhammad Jibran3 and AdamKhan 45

1Department of Mathematics Abdul Wali Khan University Mardan Mardan Khaber Pakhtunkhwa Pakistan2Department of Electrical Power Engineering US Pakistan Center for Advanced Studies in EnergyNational University of Sciences and Technology (NUST) Islamabad Pakistan3Department of Computer Science and IT Sarhad University of Science and Information Technology PeshawarKhaber Pakhtunkhwa Pakistan4Department for Management of Science and Technology Development Ton Duc ang University Ho Chi Minh City Vietnam5Faculty of Electrical amp Electronics Engineering Ton Duc ang University Ho Chi Minh City Vietnam

Correspondence should be addressed to Adam Khan adamkhantdtueduvn

Received 11 June 2019 Revised 24 August 2019 Accepted 12 November 2019 Published 10 December 2019

Academic Editor Marcio Eisencraft

Copyright copy 2019 Rashid Nawaz et alis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper the Optimal Homotopy Asymptotic Method is extended to derive the approximate solutions of fractional order two-dimensional partial dierential equations e fractional order ZakharovndashKuznetsov equation is solved as a test example whilethe time fractional derivatives are described in the Caputo sensee solutions of the problem are computed in the form of rapidlyconvergent series with easily calculable components using Mathematica Reliability of the proposed method is given bycomparison with other methods in the literature e obtained results showed that the method is powerful and ecient fordetermination of solution of higher-dimensional fractional order partial dierential equations

1 Introduction

Fractional calculus is simply an extension of integer ordercalculus For many years it was assumed that fractionalcalculus is a pure subject of mathematics and having no suchapplications in real-world phenomena but this concept isnow wrong because of the recent applications of fractionalcalculus inmodeling of the sound waves propagation in rigidporous materials [1] ultrasonic wave propagation in humancancellous bone [2] viscoelastic properties of soft biologicaltissues [3] the path tracking problem in an autonomouselectric vehicles [4] etc Dierential equations of fractionalorder are the center of attention of many studies due to theirfrequent applications in the areas of electromagnetic elec-trochemistry acoustics material science physics visco-elasticity and engineering [5ndash9] ese kinds of problemsare more complex as compared to integer order dierentialequations Due to the complexities of fractional calculusmost of the fractional order dierential equations do not

have the exact solutions and as an alternative the ap-proximate methods are extensively used for solution of thesetypes of equations [10ndash14] Some of the recent methods forapproximate solutions of fractional order dierentialequations are the Adomian Decomposition Method (ADM)the Homotopy PerturbationMethod (HPM) the VariationalIteration Method (VIM) Homotopy Analysis Method(HAM) etc [15ndash26]

Marinca and Herisanu introduced the OptimalHomotopy Asymptotic Method (OHAM) for solving non-linear dierential equations which made the perturbationmethods independent of the assumption of small parametersand huge computational work [27ndash31] e method wasrecently extended by Sarwar et al for solution of fractionalorder dierential equations [32ndash35]

In this paper OHAM formulation is extended to two-dimensional fractional order partial dierential equationsParticularly the extended formulation is demonstrated byillustrative examples of the following fractional version of

HindawiComplexityVolume 2019 Article ID 1741958 9 pageshttpsdoiorg10115520191741958

the ZakharovndashKuznetsov equations shortly called FZK(p q r)

Dαt ζ + a ζp

( 1113857x + b ζq( 1113857xxx + c ζr

( 1113857yyx 0 (1)

In above equation α is a parameter describing theory ofthe fractional derivative (0lt αle 1) a b and c are arbitraryconstants and p q rne 0 are integers which govern thebehavior of weakly nonlinear ion-acoustic waves in plasmacomprising cold ions and hot isothermal electrons in thepresence of a uniform magnetic field e FZK equation hasbeen solved by many researchers using different techniquesSome recent well-known techniques are [36ndash40]

e present paper is divided into six sections In Section2 some basic definitions and properties from fractionalcalculus are given Section 3 is devoted to analysis of theOHAM for two-dimensional partial differential equations offractional order In Section 4 the 1st order approximatesolutions of FZK (2 2 2) and FZK (3 3 3) equations aregiven in which the time fractional derivatives are describedin the Caputo sense In Section 5 comparisons of the resultsof 1st order approximate solution by the proposed methodare made with 3rd order variational iteration method (VIM)Perturbation-Iteration Algorithm (PIA) and residual powerseries method (RPS) solutions [36 37] In all cases theproposed method yields better results

2 Basic Definitions

In this section some definitions and results from the lit-erature are stated which are relevant to the current workRiemannndashLiouville Welyl Reize Compos and Caputoproposed many definitions

Definition 1 A real function f(x) xgt 0 is said to be inspace Cη η isin R if there is a real number pgt η such thatf(x) xpf1(x) where f1(x) isin C(0infin) and it is said to bein the space Cη

m if only if fm isin Cη m isin N

Definition 2 e RiemannndashLiouville fractional integraloperator of order αge 0 of a function f isin Cη ηge minus 1 isdefined as

Iαaf(x)

1Γ(α)

1113946x

a(x minus η)

αminus 1f(η)dη αgt 0 xgt 0

I0af(x) f(x)

(2)

When we formulate the model of real-world problemswith fractional calculus the RiemannndashLiouville operatorhave certain disadvantages Caputo proposed a modifiedfractional differential operator Dα

lowast in his work on the theoryof viscoelasticity

Definition 3 e fractional derivative of f(x) in Caputosense is defined as

Dαaf(x) I

mminus αa D

mf(x)

1Γ(m minus a)

1113946x

a(x minus ζ)

mminus αminus 1f

m(ζ)dζ

form minus 1lt αlem m isin N xgt 0 f isin Cmminus 1

(3)

Definition 4 If m minus 1lt αlem m isin N and f isin Cmη η ge minus 1

then

DαaI

αaf(x) f(x)

DαaI

αaf(x) f(x) minus 1113944

mminus 1

k0f

(k)(x minus a)

k xgt 0

(4)

One can found the properties of the operator Iα in theliterature We mention the following

For f isin Cmη α βgt 0 η ge minus 1 and c ge minus 1

Iαaf(x) exists for almost every x isin [a b]IαaIβaf(x) Iα+β

a f(x)IαaIβaf(x) IβaIαaf(x)Iαa(x minus a)c Γ(c + 1)Γ(α + c + 1)(x minus a)α+c

3 OHAM Analysis for Fractional Order PDEs

In this section the OHAM for fractional order partial dif-ferential equation is introduced e proposed method ispresented in the following steps

Step 1 write the general fractional order partial dif-ferential equation as

zαζ(r t)

ztα A(ζ(r t)) + f(r t) αgt 0 (5)

Subject to the initial conditions

Dαminus k0 ζ(r 0) hk(r) k 0 1 2 n minus 1

Dαminus n0 ζ(r 0) 0 n [α]

Dk0ζ(r 0) gk(r) k 0 1 2 n minus 1

Dn0ζ(r 0) 0 n [α]

(6)

In above equations zαztα is the Caputo orRiemannndashLiouville fraction derivative operator A isthe differential operator ζ(r t) is the unknown func-tion and f(r t) is known analytic function r isin Ω is ann-tuple which denotes spatial independent variablesand t represents the temporal independent variablerespectivelyStep 2 construct an optimal homotopy for fractionalorder partial differential equation ϕ(r t p) Ωtimes

[0 1]⟶ R which is

2 Complexity

(1 minus p)zαϕ(r t)

ztαminus f(r t)1113888 1113889 minus H(r p)

zαϕ(r t)

ztα1113888

minus (A( ϕ(r t)) + f(r t)1113889 0

(7)

In equation (7) p isin [0 1] is the embedding parameterand H(r p) is auxiliary function which satisfies thefollowing relationH(r p)ne 0 for pne 0 and H(r 0) 0e solution ϕ(r t) converges rapidly to the exactsolution as the value of p increases in the interval [0 1]e efficiency of OHAM depends upon the construc-tion and determination of the auxiliary function whichcontrols the convergence of the solutionAn auxiliary function H(r p) can be written in the form

H(r p) pk1 r Ci( 1113857 + p2k2 r Ci( 1113857 + p

3k3 r Ci( 1113857 +

+ pm

km r Ci( 1113857

(8)

In the above equation Ci i 1 2 3 are the con-vergence control parameters and ki(r) i 1 2 3 isa function of rStep 3 expanding ϕ(r t p Ci) in Taylorrsquos series aboutp we have

ϕ r t Ci( 1113857 ζ0(r t) + 1113944

m

k1ζk r t Ci( 1113857( 1113857p

k i 1 2 3

(9)

Remarks it is clear from equation (9) the convergenceof the series depends upon the auxiliary convergencecontrol parameter Ci i 1 2 3 4 m

If it converges at p 1 one has

ζ r t Ci( 1113857 ζ0(r t) + 1113944infin

k1ζk r t Ci( 1113857 i 1 2 3

(10)

Step 4 equating the coefficients of like powers of p aftersubstituting equation (10) in equation (7) we get zeroorder 1st order 2nd order and high-order problems

p0

zαζ0(r t)

ztαminus f 0

p1

zαζ1 r t C1( 1113857

ztαminus 1 + C1( 1113857

zαζ0(r t)

ztα+ 1 + C1( 1113857f

+ C1A ζ0(r t)( 1113857 0

p2

zαζ2 r t C1 C2( 1113857

ztαminus 1 + C1( 1113857

zαζ1 r t C1( 1113857

ztαminus C2

zαζ0(r t)

ztα

+ C1A ζ1 r t C1( 1113857( 1113857 + C2 f + A ζ0(r t)( 1113857( 1113857 0

(11)

Step 5 these problems contain the time fractionalderivatives erefore we apply the Iα operator on theabove problems and obtain a series of solutions asfollows

ζ0(r t) Iα[f]

ζ1 r t C1( 1113857 Iα 1 + C1( 1113857

zαζ0(r t)

ztαminus 1 + C1( 1113857f1113890

minus C1A ζ0(r t)( 11138571113891

ζ2 r t C1 C2( 1113857 Iα 1 + C1( 1113857

zαζ1 r t C1( 1113857

ztα+ C2

zαζ0(r t)

ztα1113890

minus C1A ζ1 r t C1( 1113857( 1113857 minus C2 f + A ζ0(r t)( 1113857( 11138571113891

(12)

By putting the above solutions in equation (12) one canget the approximate solution ζ(r t Ci) e residualR(r t Ci) is obtained by substituting approximatesolution ζ(r t Ci) in equation (5)Step 6 the convergence control parameters C1 C2

can be found either by the Ritz method Collocationmethod Galerkinrsquos method or least square method Inthis presentation least square method is used to cal-culate the convergence control parameters in which wefirst construct the functional

χ Ci( 1113857 1113946t

01113946

x

0R2

r t Ci( 1113857dr dt (13)

And then the convergence control parameters arecalculated by solving the following system

zχzC1

zχ

zC2 middot middot middot

zχzCm

0 (14)

e approximate solution is obtained by putting theoptimum values of the convergence control parametersin equation (10) e method of least squares is apowerful technique and has been used in many othermethods such as Optimal Homotopy PerturbationMethod (OHPM) and Optimal Auxiliary FunctionsMethod (OAFM) for calculating the optimum values ofarbitrary constants [41 42]

4 OHAM Convergence

If the series (10) converges to ζ(x t) where ζk(x t) isin L(R+)

is produced by zero order problem and the K-order de-formation then ζ(x t) is the exact solution of (5)

Proof since the series

1113944

infin

k1ζ ik x t C1 C2 Ck( 1113857 (15)

Complexity 3

converges it can be written as

ψi(x t) 1113944

infin

k1ζ ik x t C1 C2 Ck( 1113857 (16)

and its holds that

limk⟶infin

ζ ik x t C1 C2 Ck( 1113857 0 (17)

In fact the following equation is satisfied

ζ i1 x t C1( 1113857 + 1113944

n

k2ζ ik x t C

⟶k1113874 1113875

minus 1113944n

k2ζ ikminus 1 x t C

⟶kminus 11113874 1113875

ζ i2 x t C⟶

21113874 1113875 minus ζ i1 x t C1( 1113857 + middot middot middot + ζ in x t C⟶

n1113874 1113875

minus ζ inminus 1 x t C⟶

nminus 11113874 1113875

ζ in x t C⟶

n1113874 1113875

(18)

Now we have

Li1 ζ i1 x t C1( 11138571113872 1113873 + 1113944infin

k2L1 ζ ik x t C

⟶k1113874 11138751113874 1113875

minus 1113944infin

k2Li ζ ikminus 1 x t C

⟶kminus 11113874 11138751113874 1113875

Li ζ i1 x t C1( 11138571113872 1113873 + 1113944infin

k2Li ζ ik x t C

⟶k1113874 11138751113874 1113875

minus 1113944infin


⟶kminus 11113874 11138751113874 1113875 0

(19)

which satisfies

Li1 ζ i1 x t C1( 11138571113872 1113873 + Li 1113944

infin

k2ζ ik x t C

⟶k1113874 11138751113874 1113875

minus Li 1113944

infin


⟶kminus 11113874 11138751113874 1113875

1113944infin

k2Cm Li ζ ikminus m x t C

⟶kminus m1113874 11138751113874 1113875 + Nikminus m1113876

middot ζ ikminus 1 x t Ckminus 1( 111385711138751113874 1113877 + gi(x t) 0

(20)

Now if Cm m 1 2 3 is properly chosen then theequation leads to

Li ζ i(x t) + A( 1113857 0 (21)

which is the exact solution

5 Application of OHAM

51 Time Fractional FZK (2 2 2) Consider the followingTime Fractional FZK (2 2 2) equation with initial conditionas

zαw

ztα+

zw2

zx2 +18

z3w2

zx3 +18

z3w2

zx zy2 0 0lt αle 1

w(x y 0) 43λ sinh2(x + y)

(22)

e exact solution of equation (22) for α 1

w(x y t) 43λ sinh2(x + y minus λt) (23)

where λ is an arbitrary constantUsing the OHAM formulation discussed in Section 3 we

have

Zero-order problem

zαw

ztα 0

w0(x y 0) 43λ sinh2(x + y)

(24)

First-order problem

zαw1(x y t)

ztα

zαw0(x y t)

ztα+ C1

zαw0(x y t)

ztα

+ 2C1zαw0(x y t)

ztαw0(x y t) +

14C1

z2w0(x y t)

zy2

zw0(x y t)

zx+12C1

zw0(x y t)

zy

z2w0(x y t)

zx zy

+14C1w0(x y t)

z3w0(x y t)

zx zy2 +34C1

zw0(x y t)

zx

z2w0(x y t)

zx2 +14C1w0(x y t)

z3w0(x y t)

zx3 w1(x y 0) 0

(25)

e solutions of above problems are as follows

w0(x y t) 43λ sinh2(x + y)

w1 x y t C1( 1113857 minus 8C1t

αλ2(4 sinh(2(x + y)) minus 5 sinh(4(x + y)))

9Γ(1 + α)

(26)

e 1st order approximate solution by the OHAM isgiven by the following expression

4 Complexity

1113957w x y t Ci( 1113857 w0(x y t) + w1 x y t C1( 1113857

1113957w x y t C1( 1113857 43λ sinh2(x + y) +

19Γ(1 + a)

middot minus 8C1tαλ2(4 sinh(2(x + y))1113872

minus 5 sinh(4(x + y)))1113857

(27)


zαw

ztα+

zw3

zx+ 2

z3w3

zx3 + 2z3w3

zx y2 0 0lt αle 1

w(x y 0) 32λ sinh

16

(x + y)1113874 1113875

(28)


w(x y t) 32λ sinh

16

(x + y minus λt)1113874 1113875 (29)


have

Zero-order problemzαw(x y t)

ztα 0

w0(x y 0) 32λ sinh

16

(x + y)1113874 1113875

(30)

First-order problemzαw1(x y t)

ztα

zαw0(x y t)

ztα+ C1

zαw0(x y t)

ztα

+ 3C1zw0(x y t)

zxw

20(x y t)

+ 12C1z2w2

0(x y t)

zy2

middotzw0(x y t)

zx+ 12C1w0(x y t)

z2w0(x y t)

zy2zw0(x y t)

zx

+ 12C1zw3

0(x y t)

zx3 + 24C1w0(x y t)

zw0(x y t)

zx

z2w0(x y t)

zx zy+ 6C1w

20(x y t)

z3w0(x y t)

zx zy2

+ 36C1w20(x y t)

zw0(x y t)

zx

z2w0(x y t)

zx2

+ 6C1w20(x y t)

z3w0(x y t)

zy3 w1(x y 0) 0

(31)


w0(x y t) 32λ sinh

16

(x + y)1113874 1113875

w1 x y t C1( 1113857 1

32Γ(1 + α)3C1t

αλ3 minus 5 cosh16

(x + y)1113874 111387511138741113874

+ 9 cosh12

(x + y)1113874 111387511138751113875

(32)


1113957w x y t Ci( 1113857 w0(x y t) + w1 x y t C1( 1113857 (33)

6 Results and Discussion

OHAM formulation is tested upon the FZKequationMathematica 7 is used for most of computational work

Table 1 shows the optimum values of the convergencecontrol parameters for FZK (2 2 2) and FZK (3 3 3)equations at different values of α In Tables 2 and 3 theresults obtained by 1st order approximation of proposedmethod for the FZK (2 2 2) equation are compared with 3rdorder approximation of Perturbation-Iteration Algorithm(PIA) and Residual power Series (RPS) method at differentvalues of α In Tables 4 and 5 the results obtained by 1storder approximation of the proposed method are comparedwith 3rd order approximation of VIM for FZK (3 3 3)equation Figures 1ndash4 show the 3D plots of exact versusapproximate solution by the proposed method for FZK (2 22) equation Figures 1 and 2 show the 3D plots of exactversus approximate solution by the proposed method forFZK (3 3 3) equation Figure 5 shows the 2D plots ofapproximate solution by the proposed method for FZK (2 22) equation at different values of α Figure 6 shows the 2Dplots of approximate solution by the proposed method forFZK (3 3 3) equation at different values of α

It is clear from 2D figures that as value of α increases to 1the approximate solutions tend close to the exact solutions

Table 1 Convergence-control parameters for FZK (2 2 2) and FZK (3 3 3)

FZK (2 2 2) FZK (3 3 3)α C1 α C1

10 minus 01654570202126229 10 minus 10008903783207066075 minus 011770797863128038 075 minus 0999990856107855067 minus 011303202695535328 067 minus 09999963529361133

Complexity 5

Table 3 1st order approximate solution obtained by the OHAM in comparison with 3 terms approximate solution obtained by PIA and RPSfor FZK (2 2 2) at λ 0001

α 067 α 075x y t PIA [36] solution RPS [36] solution OHAM solution PIA [36] solution RPS [36] solution OHAM solution

01 0102 531854times10minus 5 531244times10minus 5 539424times10minus 5 532747times10minus 5 532479times10minus 5 53953times10minus 5

03 528631times 10minus 5 528410times10minus 5 539094times10minus 5 529757times10minus 5 529675times10minus 5 539191times 10minus 5

04 525777times10minus 5 525897times10minus 5 538798times10minus 5 527039times10minus 5 527119times10minus 5 538881times 10minus 5

06 0602 295493times10minus 3 295185times10minus 3 30273times10minus 3 296356times10minus 3 296251times 10minus 3 302837times10minus 3

03 292662times10minus 3 292709times10minus 3 302397times10minus 3 293717times10minus 3 293780times10minus 3 302496times10minus 3

04 290307times10minus 3 290522times10minus 3 302099times10minus 3 291448times10minus 3 291561times 10minus 3 302182times10minus 3




Table 4 1st order solution compared with three terms of VIM at λ 0001 for FZK (3 3 3)

Solution α 1 Absolute errorsx y t VIM [37] solution OHAM solution Exact solution OHAM error

01 0102 500091times 10minus 5 500092times10minus 5 499592times10minus 5 499519times10ndash8

03 500091times 10minus 5 500091times 10minus 5 499342times10minus 5 749278times10ndash8


06 0602 302003times10minus 4 302004times10minus 4 301953times10minus 4 508987times10ndash8

03 302003times10minus 4 302004times10minus 4 301927times10minus 4 763479times10ndash8





Table 5 1st order solution compared with 3 terms of VIM for λ 0001 for FZK (3 3 3)

α 067 α 075x y t VIM [37] solution OHAM solution VIM [37] solution OHAM solution

01 0102 500091times 10minus 5 500091times 10minus 5 500091times 10minus 5 500091times 10minus 5

03 500090times10minus 5 500091times 10minus 5 500090times10minus 5 500091times 10minus 5

04 500090times10minus 5 50009times10minus 5 500090times10minus 5 500091times 10minus 5

06 0602 302003times10minus 4 302004times10minus 4 302003times10minus 4 302004times10minus 4

03 302003times10minus 4 302004times10minus 4 302003times10minus 4 302004times10minus 4






Solution for α 10 Absolute errors for α 10x y t OHAM solution Exact solution PIA [34] error RPS [34] error OHAM error

01 0102 53966times10minus 5 539388times10minus 5 385217times10minus 7 385217times10minus 7 271884times10minus 8

03 539248times10minus 5 538841times 10minus 5 575911times 10minus 7 575912times10minus 7 407394times10minus 8

04 538837times10minus 5 538294times10minus 5 765359times10minus 7 765352times10minus 7 542615times10minus 8

06 0602 302967times10minus 3 303651times 10minus 3 466337times10minus 5 466389times10minus 5 683433times10minus 6



09 0902 114455times10minus 2 11537times10minus 2 512131times 10minus 4 514241times 10minus 4 914704times10minus 5


04 113492times10minus 2 115321times 10minus 2 957942times10minus 4 989139times10minus 4 182943times10minus 4

6 Complexity

7 Conclusion

e 1st order OHAM solution gives more encouraging re-sults in comparison to 3rd order approximations of PIARPS and VIM From obtained results it is concluded that

the proposed method is very effective and convenient forsolving higher-dimensional partial differential equations offractional order e accuracy of the method can be furtherimproved by taking higher-order approximations

0010

0005

0002

0406

08 00

01

02

03

04

Figure 1 Approximate solution for equation (22) for α 1(y 09)

0010

0005

0002

0406

08 00

01

02

03

04

Figure 2 Exact solution for equation (22) for α 1 (y 09)

000045000040000035000030000025

0002

0406

08 00

01

02

03

04


000045000040000035000030000025

0002

0406

08 00

01

02

03

04


0 1 2 3 4ndash06

ndash04

ndash02

00

02

04

06

08

x

α = 067α = 075

α = 10Exact

Figure 5 Convergence at different value of alpha for equation (22)at t 01 y 02

0 1 2 3 400000

00002

00004

00006

00008

00010

00012

x

α = 067α = 075

α = 10Exact


Complexity 7

Data Availability

No data were generated or analyzed during the study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

All authors contributed equally and significantly in writingthis article All authors read and approved the finalmanuscript

Acknowledgments

e authors acknowledge the help and support from theDepartment of Mathematics of AWKUM for completion ofthis work

References

[1] Z E A Fellah C Depollier and M Fellah ldquoApplication offractional calculus to sound wave propagation in rigid porousmaterials validation via ultrasonic measurementsrdquo ActaAcustica United with Acustica vol 88 no 1 pp 34ndash39 2002

[2] N Sebaa Z E A Fellah W Lauriks and C DepollierldquoApplication of fractional calculus to ultrasonic wave prop-agation in human cancellous bonerdquo Signal Processing-Frac-tional Calculus Applications in Signals and Systems vol 86no 10 pp 2668ndash2677 2006

[3] F C Meral T J Royston and R Magin ldquoFractional calculusin viscoelasticity an experimental studyrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 4pp 939ndash945 2010

[4] J I Suaacuterez B M Vinagre A J Calderacuteon C A Monje andY Q Chen ldquoUsing fractional calculus for lateral and longi-tudinal control of autonomous vehiclesrdquo in Lecture Notes inComputer Science Springer Berlin Germany 2004

[5] K B Oldham and J Spanier -e Fractional Calculus Aca-demic Press New York NY USA 1974

[6] M M Meerschaert H-P Scheffler and C Tadjeran ldquoFinitedifference methods for two-dimensional fractional dispersionequationrdquo Journal of Computational Physics vol 211 no 1pp 249ndash261 2006

[7] R Metzler and J Klafter ldquoe restaurant at the end of therandom walk recent developments in the description ofanomalous transport by fractional dynamicsrdquo Journal ofPhysic vol 37 pp 161ndash208 2004

[8] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

[9] W R Schneider and W Wyess ldquoFractional diffusion andwave equationsrdquo Journal of Mathematic and Physic vol 30pp 134ndash144 1989

[10] S Momani ldquoNon-perturbative analytical solutions of thespace- and time-fractional Burgers equationsrdquo Chaos Solitonsamp Fractals vol 28 no 4 pp 930ndash937 2006

[11] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 15ndash27 2006

[12] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional NavierndashStokes equation by Adomian de-composition methodrdquo Applied Mathematics and Computa-tional vol 177 no 2 pp 488ndash494 2006

[13] S Momani and Z Odibat ldquoNumerical comparison ofmethods for solving linear differential equations of fractionalorderrdquo Chaos Solitons amp Fractals vol 31 no 5pp 1248ndash1255 2007

[14] Z M Odibat and S Momani ldquoApproximate solutions forboundary value problems of time-fractional wave equationrdquoApplied Mathematics and Computation vol 181 no 1pp 767ndash774 2006

[15] S S Ray ldquoAnalytical solution for the space fractional diffusionequation by two-step Adomian decomposition methodrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 14 no 4 pp 1295ndash1306 2009

[16] O Abdulaziz I Hashim and E S Ismail ldquoApproximateanalytical solution to fractional modified KdV equationsrdquoMathematical and Computer Modelling vol 49 no 1-2pp 136ndash145 2009

[17] S H Hosseinnia A Ranjbar and S Momani ldquoUsing anenhanced homotopy perturbation method in fractionalequations via deforming the linear partrdquo Computers ampMathematics with Applications vol 56 no 12 pp 3138ndash31492008

[18] O Abdulaziz I Hashim and S Momani ldquoSolving systems offractional differential equations by homotopy-perturbationmethodrdquo Physics Letters A vol 372 no 4 pp 451ndash459 2008

[19] O Abdulaziz I Hashim and S Momani ldquoApplication ofhomotopy-perturbation method to fractional IVPsrdquo Journalof Computational and Applied Mathematics vol 216 no 2pp 574ndash584 2008

[20] L Song and H Zhang ldquoApplication of homotopy analysismethod to fractional KdV-Burgers-Kuramoto equationrdquoPhysics Letters A vol 367 no 1-2 pp 88ndash94 2007

[21] I Hashim O Abdulaziz and S Momani ldquoHomotopy analysismethod for fractional IVPsrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 3 pp 674ndash6842008

[22] O Abdulaziz I Hashim and A Saif ldquoSeries solutions of time-fractional PDEs by homotopy analysis methodrdquo DifferentialEquations and Nonlinear Mechanics vol 2008 Article ID686512 16 pages 2008

[23] O Abdulaziz I Hashim M S H Chowdhury andA K Zulkie ldquoAssessment of decomposition method forlinearand nonlinear fractional di_erential equationsrdquo Far EastJournal of Applied Mathematics vol 28 pp 95ndash112 2007

[24] A S Bataineh A K Alomari M S M Noorani I Hashimand R Nazar ldquoSeries solutions of systems of nonlinearfractional differential equationsrdquo Acta Applicandae Mathe-maticae vol 105 no 2 pp 189ndash198 2009

[25] J H He ldquoApproximate analytical solutions for seepage owwith fractional derivatives in porous mediardquo ComputerMethods in Applied Mechanics and Engineering vol 167no 1-2 pp 57ndash68 1998

[26] A Neamaty B Agheli and R Darzi ldquoVariational iterationmethod and Hersquos polynomials for time-fractional partialdifferential equationsrdquo Progress in Fractional Differentiationand Applications vol 1 no 1 pp 47ndash55 2015

[27] V Marinca N Herisanu and I Nemes ldquoOptimal homotopyasymptotic method with application to thin film flowrdquo OpenPhysics vol 6 no 3 pp 648ndash653 2008

8 Complexity

[28] V Marinca and N Herisanu ldquoe optimal homotopy as-ymptotic method for solving Blasius equationrdquo AppliedMathematics and Computation vol 231 pp pp134ndash139 2014

[29] V Marinca N Herisanu and Gh Madescu ldquoAn analyticalapproach to non-linear dynamical model of a permanentmagnet synchronous generatorrdquo Wind Energy vol 18 no 9pp 1657ndash1670 2015

[30] V Marinca and N Herisanu ldquoAn optimal homotopy as-ymptotic method for solving nonlinear equations arising inheat transferrdquo International Communications in Heat andMass Transfer vol 35 no 6 pp 710ndash715 2008

[31] N Herisanu V Marinca and B Marinca ldquoAn optimalhomotopy asymptotic method applied to steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

[32] S Sarwar S Alkhalaf S Iqbal and M A Zahid ldquoA note onoptimal homotopy asymptotic method for the solutions offractional order heat- and wave-like partial differentialequationsrdquo Computers amp Mathematics with Applicationsvol 70 no 5 pp 942ndash953 2015

[33] S Sarwar M A Zahid and S Iqbal ldquoMathematical study offractional-order biological population model using optimalhomotopy asymptotic methodrdquo International Journal ofBiomathematics vol 9 no 6 Article ID 1650081 2016

[34] S Sarwar and M M Rashidi ldquoApproximate solution of twoterm fractional order diffusion wave-diffision and telegraphmodels arising in mathematical physics using optimalhomotopy asymptotic methodrdquo Waves in Random andComplex Media vol 26 no 3 pp pp365ndash382 2016

[35] S Iqbal F Sarwar M R Mufti and I Siddique ldquoUse ofoptimal homotopy asymptotic method for fractional ordernonlinear fredholm integro-differential equationsrdquo ScienceInternational vol 27 no 4 pp 3033ndash3040 2015

[36] S Mehmet M Alquran and H D Kasmaei ldquoOn the com-parison of perturbation-iteration algorithm and residualpower series method to solve fractional Zakharov-Kuznetsovequationrdquo Results in Physics vol 9 pp 321ndash327 2018

[37] R Y Molliq M S M Noorani I Hashim and R R AhmadldquoApproximate solutions of fractional ZakharovndashKuznetsovequations by VIMrdquo Journal of Computational and AppliedMathematics vol 233 no 2 pp 103ndash108 2009

[38] L Fu and H Yang ldquoAn application of (3 + 1)-dimensionaltime-space fractional ZK model to analyze the complex dustacoustic wavesrdquo Complexity vol 2019 Article ID 280672415 pages 2019

[39] Q Jin T Xia and J Wang ldquoe exact solution of the space-time fractional modified Kdv-Zakharov-Kuznetsov equationrdquoJournal of Applied Mathematics and Physics vol 5 no 4pp 844ndash852 2017

[40] C Li and J Zhang ldquoLie symmetry analysis and exact solutionsof generalized fractional Zakharov-Kuznetsov equationsrdquoSymmetry vol 11 no 5 601 pages 2019

[41] H abet and S Kendre ldquoModified least squares homotopyperturbation method for solving fractional partial differentialequationsrdquo Malaya Journal of Matematik vol 6 no 2pp 420ndash427 2018

[42] N Herisanu V Marinca G Madescu and F Dragan ldquoDy-namic response of a permanent magnet synchronous gen-erator to a wind gustrdquo Energies vol 12 no 5 p 915 2019

Complexity 9

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

the ZakharovndashKuznetsov equations shortly called FZK(p q r)

Dαt ζ + a ζp

( 1113857x + b ζq( 1113857xxx + c ζr

( 1113857yyx 0 (1)

In above equation α is a parameter describing theory ofthe fractional derivative (0lt αle 1) a b and c are arbitraryconstants and p q rne 0 are integers which govern thebehavior of weakly nonlinear ion-acoustic waves in plasmacomprising cold ions and hot isothermal electrons in thepresence of a uniform magnetic field e FZK equation hasbeen solved by many researchers using different techniquesSome recent well-known techniques are [36ndash40]

e present paper is divided into six sections In Section2 some basic definitions and properties from fractionalcalculus are given Section 3 is devoted to analysis of theOHAM for two-dimensional partial differential equations offractional order In Section 4 the 1st order approximatesolutions of FZK (2 2 2) and FZK (3 3 3) equations aregiven in which the time fractional derivatives are describedin the Caputo sense In Section 5 comparisons of the resultsof 1st order approximate solution by the proposed methodare made with 3rd order variational iteration method (VIM)Perturbation-Iteration Algorithm (PIA) and residual powerseries method (RPS) solutions [36 37] In all cases theproposed method yields better results

2 Basic Definitions

In this section some definitions and results from the lit-erature are stated which are relevant to the current workRiemannndashLiouville Welyl Reize Compos and Caputoproposed many definitions

Definition 1 A real function f(x) xgt 0 is said to be inspace Cη η isin R if there is a real number pgt η such thatf(x) xpf1(x) where f1(x) isin C(0infin) and it is said to bein the space Cη

m if only if fm isin Cη m isin N

Definition 2 e RiemannndashLiouville fractional integraloperator of order αge 0 of a function f isin Cη ηge minus 1 isdefined as

Iαaf(x)

1Γ(α)

1113946x

a(x minus η)

αminus 1f(η)dη αgt 0 xgt 0

I0af(x) f(x)

(2)

When we formulate the model of real-world problemswith fractional calculus the RiemannndashLiouville operatorhave certain disadvantages Caputo proposed a modifiedfractional differential operator Dα

lowast in his work on the theoryof viscoelasticity

Definition 3 e fractional derivative of f(x) in Caputosense is defined as

Dαaf(x) I

mminus αa D

mf(x)

1Γ(m minus a)

1113946x

a(x minus ζ)

mminus αminus 1f

m(ζ)dζ

form minus 1lt αlem m isin N xgt 0 f isin Cmminus 1

(3)

Definition 4 If m minus 1lt αlem m isin N and f isin Cmη η ge minus 1

then

DαaI

αaf(x) f(x)

DαaI

αaf(x) f(x) minus 1113944

mminus 1

k0f

(k)(x minus a)

k xgt 0

(4)

One can found the properties of the operator Iα in theliterature We mention the following

For f isin Cmη α βgt 0 η ge minus 1 and c ge minus 1

Iαaf(x) exists for almost every x isin [a b]IαaIβaf(x) Iα+β

a f(x)IαaIβaf(x) IβaIαaf(x)Iαa(x minus a)c Γ(c + 1)Γ(α + c + 1)(x minus a)α+c

3 OHAM Analysis for Fractional Order PDEs

In this section the OHAM for fractional order partial dif-ferential equation is introduced e proposed method ispresented in the following steps

Step 1 write the general fractional order partial dif-ferential equation as

zαζ(r t)

ztα A(ζ(r t)) + f(r t) αgt 0 (5)

Subject to the initial conditions

Dαminus k0 ζ(r 0) hk(r) k 0 1 2 n minus 1

Dαminus n0 ζ(r 0) 0 n [α]

Dk0ζ(r 0) gk(r) k 0 1 2 n minus 1

Dn0ζ(r 0) 0 n [α]

(6)

In above equations zαztα is the Caputo orRiemannndashLiouville fraction derivative operator A isthe differential operator ζ(r t) is the unknown func-tion and f(r t) is known analytic function r isin Ω is ann-tuple which denotes spatial independent variablesand t represents the temporal independent variablerespectivelyStep 2 construct an optimal homotopy for fractionalorder partial differential equation ϕ(r t p) Ωtimes

[0 1]⟶ R which is

2 Complexity



zαϕ(r t)

ztα1113888

minus (A( ϕ(r t)) + f(r t)1113889 0

(7)


H(r p) pk1 r Ci( 1113857 + p2k2 r Ci( 1113857 + p

3k3 r Ci( 1113857 +

+ pm

km r Ci( 1113857

(8)


ϕ r t Ci( 1113857 ζ0(r t) + 1113944

m

k1ζk r t Ci( 1113857( 1113857p

k i 1 2 3

(9)



ζ r t Ci( 1113857 ζ0(r t) + 1113944infin

k1ζk r t Ci( 1113857 i 1 2 3

(10)


p0

zαζ0(r t)

ztαminus f 0

p1

zαζ1 r t C1( 1113857

ztαminus 1 + C1( 1113857

zαζ0(r t)

ztα+ 1 + C1( 1113857f

+ C1A ζ0(r t)( 1113857 0

p2

zαζ2 r t C1 C2( 1113857

ztαminus 1 + C1( 1113857

zαζ1 r t C1( 1113857

ztαminus C2

zαζ0(r t)

ztα

+ C1A ζ1 r t C1( 1113857( 1113857 + C2 f + A ζ0(r t)( 1113857( 1113857 0

(11)


ζ0(r t) Iα[f]

ζ1 r t C1( 1113857 Iα 1 + C1( 1113857

zαζ0(r t)

ztαminus 1 + C1( 1113857f1113890

minus C1A ζ0(r t)( 11138571113891

ζ2 r t C1 C2( 1113857 Iα 1 + C1( 1113857

zαζ1 r t C1( 1113857

ztα+ C2

zαζ0(r t)

ztα1113890


(12)



χ Ci( 1113857 1113946t

01113946

x

0R2

r t Ci( 1113857dr dt (13)


zχzC1

zχ


zχzCm

0 (14)


4 OHAM Convergence




1113944

infin

k1ζ ik x t C1 C2 Ck( 1113857 (15)

Complexity 3


ψi(x t) 1113944

infin

k1ζ ik x t C1 C2 Ck( 1113857 (16)

and its holds that

limk⟶infin

ζ ik x t C1 C2 Ck( 1113857 0 (17)


ζ i1 x t C1( 1113857 + 1113944

n

k2ζ ik x t C

⟶k1113874 1113875

minus 1113944n


⟶kminus 11113874 1113875

ζ i2 x t C⟶


n1113874 1113875


nminus 11113874 1113875

ζ in x t C⟶

n1113874 1113875

(18)

Now we have

Li1 ζ i1 x t C1( 11138571113872 1113873 + 1113944infin

k2L1 ζ ik x t C

⟶k1113874 11138751113874 1113875

minus 1113944infin


⟶kminus 11113874 11138751113874 1113875

Li ζ i1 x t C1( 11138571113872 1113873 + 1113944infin

k2Li ζ ik x t C

⟶k1113874 11138751113874 1113875

minus 1113944infin


⟶kminus 11113874 11138751113874 1113875 0

(19)

which satisfies

Li1 ζ i1 x t C1( 11138571113872 1113873 + Li 1113944

infin

k2ζ ik x t C

⟶k1113874 11138751113874 1113875

minus Li 1113944

infin


⟶kminus 11113874 11138751113874 1113875

1113944infin




(20)


Li ζ i(x t) + A( 1113857 0 (21)




zαw

ztα+

zw2

zx2 +18

z3w2

zx3 +18

z3w2

zx zy2 0 0lt αle 1


(22)




have

Zero-order problem

zαw

ztα 0

w0(x y 0) 43λ sinh2(x + y)

(24)

First-order problem

zαw1(x y t)

ztα

zαw0(x y t)

ztα+ C1

zαw0(x y t)

ztα

+ 2C1zαw0(x y t)

ztαw0(x y t) +

14C1

z2w0(x y t)

zy2

zw0(x y t)

zx+12C1

zw0(x y t)

zy

z2w0(x y t)

zx zy

+14C1w0(x y t)

z3w0(x y t)

zx zy2 +34C1

zw0(x y t)

zx

z2w0(x y t)

zx2 +14C1w0(x y t)

z3w0(x y t)

zx3 w1(x y 0) 0

(25)



w1 x y t C1( 1113857 minus 8C1t


9Γ(1 + α)

(26)


4 Complexity


1113957w x y t C1( 1113857 43λ sinh2(x + y) +

19Γ(1 + a)


minus 5 sinh(4(x + y)))1113857

(27)


zαw

ztα+

zw3

zx+ 2

z3w3

zx3 + 2z3w3

zx y2 0 0lt αle 1

w(x y 0) 32λ sinh

16

(x + y)1113874 1113875

(28)


w(x y t) 32λ sinh

16

(x + y minus λt)1113874 1113875 (29)


have


ztα 0

w0(x y 0) 32λ sinh

16

(x + y)1113874 1113875

(30)


ztα

zαw0(x y t)

ztα+ C1

zαw0(x y t)

ztα

+ 3C1zw0(x y t)

zxw

20(x y t)

+ 12C1z2w2

0(x y t)

zy2

middotzw0(x y t)

zx+ 12C1w0(x y t)

z2w0(x y t)

zy2zw0(x y t)

zx

+ 12C1zw3

0(x y t)

zx3 + 24C1w0(x y t)

zw0(x y t)

zx

z2w0(x y t)

zx zy+ 6C1w

20(x y t)

z3w0(x y t)

zx zy2

+ 36C1w20(x y t)

zw0(x y t)

zx

z2w0(x y t)

zx2

+ 6C1w20(x y t)

z3w0(x y t)

zy3 w1(x y 0) 0

(31)


w0(x y t) 32λ sinh

16

(x + y)1113874 1113875

w1 x y t C1( 1113857 1

32Γ(1 + α)3C1t


(x + y)1113874 111387511138741113874

+ 9 cosh12

(x + y)1113874 111387511138751113875

(32)








FZK (2 2 2) FZK (3 3 3)α C1 α C1


Complexity 5













































6 Complexity

7 Conclusion



0010

0005

0002

0406

08 00

01

02

03

04


0010

0005

0002

0406

08 00

01

02

03

04


000045000040000035000030000025

0002

0406

08 00

01

02

03

04


000045000040000035000030000025

0002

0406

08 00

01

02

03

04


0 1 2 3 4ndash06

ndash04

ndash02

00

02

04

06

08

x

α = 067α = 075

α = 10Exact


0 1 2 3 400000

00002

00004

00006

00008

00010

00012

x

α = 067α = 075

α = 10Exact


Complexity 7

Data Availability






Acknowledgments


References




























8 Complexity
















Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018










zαϕ(r t)

ztα1113888

minus (A( ϕ(r t)) + f(r t)1113889 0

(7)


H(r p) pk1 r Ci( 1113857 + p2k2 r Ci( 1113857 + p

3k3 r Ci( 1113857 +

+ pm

km r Ci( 1113857

(8)


ϕ r t Ci( 1113857 ζ0(r t) + 1113944

m

k1ζk r t Ci( 1113857( 1113857p

k i 1 2 3

(9)



ζ r t Ci( 1113857 ζ0(r t) + 1113944infin

k1ζk r t Ci( 1113857 i 1 2 3

(10)


p0

zαζ0(r t)

ztαminus f 0

p1

zαζ1 r t C1( 1113857

ztαminus 1 + C1( 1113857

zαζ0(r t)

ztα+ 1 + C1( 1113857f

+ C1A ζ0(r t)( 1113857 0

p2

zαζ2 r t C1 C2( 1113857

ztαminus 1 + C1( 1113857

zαζ1 r t C1( 1113857

ztαminus C2

zαζ0(r t)

ztα

+ C1A ζ1 r t C1( 1113857( 1113857 + C2 f + A ζ0(r t)( 1113857( 1113857 0

(11)


ζ0(r t) Iα[f]

ζ1 r t C1( 1113857 Iα 1 + C1( 1113857

zαζ0(r t)

ztαminus 1 + C1( 1113857f1113890

minus C1A ζ0(r t)( 11138571113891

ζ2 r t C1 C2( 1113857 Iα 1 + C1( 1113857

zαζ1 r t C1( 1113857

ztα+ C2

zαζ0(r t)

ztα1113890


(12)



χ Ci( 1113857 1113946t

01113946

x

0R2

r t Ci( 1113857dr dt (13)


zχzC1

zχ


zχzCm

0 (14)


4 OHAM Convergence




1113944

infin

k1ζ ik x t C1 C2 Ck( 1113857 (15)

Complexity 3


ψi(x t) 1113944

infin

k1ζ ik x t C1 C2 Ck( 1113857 (16)

and its holds that

limk⟶infin

ζ ik x t C1 C2 Ck( 1113857 0 (17)


ζ i1 x t C1( 1113857 + 1113944

n

k2ζ ik x t C

⟶k1113874 1113875

minus 1113944n


⟶kminus 11113874 1113875

ζ i2 x t C⟶


n1113874 1113875


nminus 11113874 1113875

ζ in x t C⟶

n1113874 1113875

(18)

Now we have

Li1 ζ i1 x t C1( 11138571113872 1113873 + 1113944infin

k2L1 ζ ik x t C

⟶k1113874 11138751113874 1113875

minus 1113944infin


⟶kminus 11113874 11138751113874 1113875

Li ζ i1 x t C1( 11138571113872 1113873 + 1113944infin

k2Li ζ ik x t C

⟶k1113874 11138751113874 1113875

minus 1113944infin


⟶kminus 11113874 11138751113874 1113875 0

(19)

which satisfies

Li1 ζ i1 x t C1( 11138571113872 1113873 + Li 1113944

infin

k2ζ ik x t C

⟶k1113874 11138751113874 1113875

minus Li 1113944

infin


⟶kminus 11113874 11138751113874 1113875

1113944infin




(20)


Li ζ i(x t) + A( 1113857 0 (21)




zαw

ztα+

zw2

zx2 +18

z3w2

zx3 +18

z3w2

zx zy2 0 0lt αle 1


(22)




have

Zero-order problem

zαw

ztα 0

w0(x y 0) 43λ sinh2(x + y)

(24)

First-order problem

zαw1(x y t)

ztα

zαw0(x y t)

ztα+ C1

zαw0(x y t)

ztα

+ 2C1zαw0(x y t)

ztαw0(x y t) +

14C1

z2w0(x y t)

zy2

zw0(x y t)

zx+12C1

zw0(x y t)

zy

z2w0(x y t)

zx zy

+14C1w0(x y t)

z3w0(x y t)

zx zy2 +34C1

zw0(x y t)

zx

z2w0(x y t)

zx2 +14C1w0(x y t)

z3w0(x y t)

zx3 w1(x y 0) 0

(25)



w1 x y t C1( 1113857 minus 8C1t


9Γ(1 + α)

(26)


4 Complexity


1113957w x y t C1( 1113857 43λ sinh2(x + y) +

19Γ(1 + a)


minus 5 sinh(4(x + y)))1113857

(27)


zαw

ztα+

zw3

zx+ 2

z3w3

zx3 + 2z3w3

zx y2 0 0lt αle 1

w(x y 0) 32λ sinh

16

(x + y)1113874 1113875

(28)


w(x y t) 32λ sinh

16

(x + y minus λt)1113874 1113875 (29)


have


ztα 0

w0(x y 0) 32λ sinh

16

(x + y)1113874 1113875

(30)


ztα

zαw0(x y t)

ztα+ C1

zαw0(x y t)

ztα

+ 3C1zw0(x y t)

zxw

20(x y t)

+ 12C1z2w2

0(x y t)

zy2

middotzw0(x y t)

zx+ 12C1w0(x y t)

z2w0(x y t)

zy2zw0(x y t)

zx

+ 12C1zw3

0(x y t)

zx3 + 24C1w0(x y t)

zw0(x y t)

zx

z2w0(x y t)

zx zy+ 6C1w

20(x y t)

z3w0(x y t)

zx zy2

+ 36C1w20(x y t)

zw0(x y t)

zx

z2w0(x y t)

zx2

+ 6C1w20(x y t)

z3w0(x y t)

zy3 w1(x y 0) 0

(31)


w0(x y t) 32λ sinh

16

(x + y)1113874 1113875

w1 x y t C1( 1113857 1

32Γ(1 + α)3C1t


(x + y)1113874 111387511138741113874

+ 9 cosh12

(x + y)1113874 111387511138751113875

(32)








FZK (2 2 2) FZK (3 3 3)α C1 α C1


Complexity 5













































6 Complexity

7 Conclusion



0010

0005

0002

0406

08 00

01

02

03

04


0010

0005

0002

0406

08 00

01

02

03

04


000045000040000035000030000025

0002

0406

08 00

01

02

03

04


000045000040000035000030000025

0002

0406

08 00

01

02

03

04


0 1 2 3 4ndash06

ndash04

ndash02

00

02

04

06

08

x

α = 067α = 075

α = 10Exact


0 1 2 3 400000

00002

00004

00006

00008

00010

00012

x

α = 067α = 075

α = 10Exact


Complexity 7

Data Availability






Acknowledgments


References




























8 Complexity
















Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018









ψi(x t) 1113944

infin

k1ζ ik x t C1 C2 Ck( 1113857 (16)

and its holds that

limk⟶infin

ζ ik x t C1 C2 Ck( 1113857 0 (17)


ζ i1 x t C1( 1113857 + 1113944

n

k2ζ ik x t C

⟶k1113874 1113875

minus 1113944n


⟶kminus 11113874 1113875

ζ i2 x t C⟶


n1113874 1113875


nminus 11113874 1113875

ζ in x t C⟶

n1113874 1113875

(18)

Now we have

Li1 ζ i1 x t C1( 11138571113872 1113873 + 1113944infin

k2L1 ζ ik x t C

⟶k1113874 11138751113874 1113875

minus 1113944infin


⟶kminus 11113874 11138751113874 1113875

Li ζ i1 x t C1( 11138571113872 1113873 + 1113944infin

k2Li ζ ik x t C

⟶k1113874 11138751113874 1113875

minus 1113944infin


⟶kminus 11113874 11138751113874 1113875 0

(19)

which satisfies

Li1 ζ i1 x t C1( 11138571113872 1113873 + Li 1113944

infin

k2ζ ik x t C

⟶k1113874 11138751113874 1113875

minus Li 1113944

infin


⟶kminus 11113874 11138751113874 1113875

1113944infin




(20)


Li ζ i(x t) + A( 1113857 0 (21)




zαw

ztα+

zw2

zx2 +18

z3w2

zx3 +18

z3w2

zx zy2 0 0lt αle 1


(22)




have

Zero-order problem

zαw

ztα 0

w0(x y 0) 43λ sinh2(x + y)

(24)

First-order problem

zαw1(x y t)

ztα

zαw0(x y t)

ztα+ C1

zαw0(x y t)

ztα

+ 2C1zαw0(x y t)

ztαw0(x y t) +

14C1

z2w0(x y t)

zy2

zw0(x y t)

zx+12C1

zw0(x y t)

zy

z2w0(x y t)

zx zy

+14C1w0(x y t)

z3w0(x y t)

zx zy2 +34C1

zw0(x y t)

zx

z2w0(x y t)

zx2 +14C1w0(x y t)

z3w0(x y t)

zx3 w1(x y 0) 0

(25)



w1 x y t C1( 1113857 minus 8C1t


9Γ(1 + α)

(26)


4 Complexity


1113957w x y t C1( 1113857 43λ sinh2(x + y) +

19Γ(1 + a)


minus 5 sinh(4(x + y)))1113857

(27)


zαw

ztα+

zw3

zx+ 2

z3w3

zx3 + 2z3w3

zx y2 0 0lt αle 1

w(x y 0) 32λ sinh

16

(x + y)1113874 1113875

(28)


w(x y t) 32λ sinh

16

(x + y minus λt)1113874 1113875 (29)


have


ztα 0

w0(x y 0) 32λ sinh

16

(x + y)1113874 1113875

(30)


ztα

zαw0(x y t)

ztα+ C1

zαw0(x y t)

ztα

+ 3C1zw0(x y t)

zxw

20(x y t)

+ 12C1z2w2

0(x y t)

zy2

middotzw0(x y t)

zx+ 12C1w0(x y t)

z2w0(x y t)

zy2zw0(x y t)

zx

+ 12C1zw3

0(x y t)

zx3 + 24C1w0(x y t)

zw0(x y t)

zx

z2w0(x y t)

zx zy+ 6C1w

20(x y t)

z3w0(x y t)

zx zy2

+ 36C1w20(x y t)

zw0(x y t)

zx

z2w0(x y t)

zx2

+ 6C1w20(x y t)

z3w0(x y t)

zy3 w1(x y 0) 0

(31)


w0(x y t) 32λ sinh

16

(x + y)1113874 1113875

w1 x y t C1( 1113857 1

32Γ(1 + α)3C1t


(x + y)1113874 111387511138741113874

+ 9 cosh12

(x + y)1113874 111387511138751113875

(32)








FZK (2 2 2) FZK (3 3 3)α C1 α C1


Complexity 5













































6 Complexity

7 Conclusion



0010

0005

0002

0406

08 00

01

02

03

04


0010

0005

0002

0406

08 00

01

02

03

04


000045000040000035000030000025

0002

0406

08 00

01

02

03

04


000045000040000035000030000025

0002

0406

08 00

01

02

03

04


0 1 2 3 4ndash06

ndash04

ndash02

00

02

04

06

08

x

α = 067α = 075

α = 10Exact


0 1 2 3 400000

00002

00004

00006

00008

00010

00012

x

α = 067α = 075

α = 10Exact


Complexity 7

Data Availability






Acknowledgments


References




























8 Complexity
















Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018









1113957w x y t C1( 1113857 43λ sinh2(x + y) +

19Γ(1 + a)


minus 5 sinh(4(x + y)))1113857

(27)


zαw

ztα+

zw3

zx+ 2

z3w3

zx3 + 2z3w3

zx y2 0 0lt αle 1

w(x y 0) 32λ sinh

16

(x + y)1113874 1113875

(28)


w(x y t) 32λ sinh

16

(x + y minus λt)1113874 1113875 (29)


have


ztα 0

w0(x y 0) 32λ sinh

16

(x + y)1113874 1113875

(30)


ztα

zαw0(x y t)

ztα+ C1

zαw0(x y t)

ztα

+ 3C1zw0(x y t)

zxw

20(x y t)

+ 12C1z2w2

0(x y t)

zy2

middotzw0(x y t)

zx+ 12C1w0(x y t)

z2w0(x y t)

zy2zw0(x y t)

zx

+ 12C1zw3

0(x y t)

zx3 + 24C1w0(x y t)

zw0(x y t)

zx

z2w0(x y t)

zx zy+ 6C1w

20(x y t)

z3w0(x y t)

zx zy2

+ 36C1w20(x y t)

zw0(x y t)

zx

z2w0(x y t)

zx2

+ 6C1w20(x y t)

z3w0(x y t)

zy3 w1(x y 0) 0

(31)


w0(x y t) 32λ sinh

16

(x + y)1113874 1113875

w1 x y t C1( 1113857 1

32Γ(1 + α)3C1t


(x + y)1113874 111387511138741113874

+ 9 cosh12

(x + y)1113874 111387511138751113875

(32)








FZK (2 2 2) FZK (3 3 3)α C1 α C1


Complexity 5













































6 Complexity

7 Conclusion



0010

0005

0002

0406

08 00

01

02

03

04


0010

0005

0002

0406

08 00

01

02

03

04


000045000040000035000030000025

0002

0406

08 00

01

02

03

04


000045000040000035000030000025

0002

0406

08 00

01

02

03

04


0 1 2 3 4ndash06

ndash04

ndash02

00

02

04

06

08

x

α = 067α = 075

α = 10Exact


0 1 2 3 400000

00002

00004

00006

00008

00010

00012

x

α = 067α = 075

α = 10Exact


Complexity 7

Data Availability






Acknowledgments


References




























8 Complexity
















Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018




















































6 Complexity

7 Conclusion



0010

0005

0002

0406

08 00

01

02

03

04


0010

0005

0002

0406

08 00

01

02

03

04


000045000040000035000030000025

0002

0406

08 00

01

02

03

04


000045000040000035000030000025

0002

0406

08 00

01

02

03

04


0 1 2 3 4ndash06

ndash04

ndash02

00

02

04

06

08

x

α = 067α = 075

α = 10Exact


0 1 2 3 400000

00002

00004

00006

00008

00010

00012

x

α = 067α = 075

α = 10Exact


Complexity 7

Data Availability






Acknowledgments


References




























8 Complexity
















Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018








7 Conclusion



0010

0005

0002

0406

08 00

01

02

03

04


0010

0005

0002

0406

08 00

01

02

03

04


000045000040000035000030000025

0002

0406

08 00

01

02

03

04


000045000040000035000030000025

0002

0406

08 00

01

02

03

04


0 1 2 3 4ndash06

ndash04

ndash02

00

02

04

06

08

x

α = 067α = 075

α = 10Exact


0 1 2 3 400000

00002

00004

00006

00008

00010

00012

x

α = 067α = 075

α = 10Exact


Complexity 7

Data Availability






Acknowledgments


References




























8 Complexity
















Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018








Data Availability






Acknowledgments


References




























8 Complexity
















Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018























Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

Optimum Solutions of Fractional Order Zakharov–Kuznetsov Equations · 2019. 12. 10. · Research Article Optimum Solutions of Fractional Order Zakharov–Kuznetsov Equations RashidNawaz