Research ArticleNew Exponential and Complex Traveling Wave Solutions tothe Konopelchenko-Dubrovsky Model

Faruk Dusunceli

Faculty of Architecture and Engineering Mardin Artuklu University Mardin Turkey

Correspondence should be addressed to Faruk Dusunceli farukdusunceliartukluedutr

Received 23 November 2018 Accepted 17 January 2019 Published 11 February 2019

Academic Editor Antonio Scarfone

Copyright copy 2019 Faruk Dusunceli This 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

The Konopelchenko-Dubrovsky (KD) system is presented by the application of the improved Bernoulli subequation functionmethod (IBSEFM) First The KD system being Nonlinear partial differential equations system is transformed into nonlinearordinary differential equation by using a wave transformation Last the resulting equation is successfully explored for new explicitexact solutions including singular soliton kink and periodic wave solutions All the obtained solutions in this study satisfy theKonopelchenko-Dubrovsky model Under suitable choice of the parameter values interesting two- and three-dimensional graphsof all the obtained solutions are plotted

1 Introduction

Various complex nonlinear phenomena in different fields ofnonlinear sciences such as fluid mechanic plasma physicsand optical fibers can be expressed in the form of nonlin-ear partial differential equations (NPDEs) Many analyticalmethods for solving such type of equations have beendeveloped and used by different researchers from all overthe world Zayed et al [1] used the generalized Kudryashovin addressing some NPDEs arising in mathematical physicsWang [2] investigated the Sharma-Tosso-Olver by usingthe extended hamoclinic test function method Nofal [3]solved some nonlinear partial differential equation by usingthe simple equation method Shang [4] obtained the exactsolutions of the long-short wave resonance equation by usingthe extended hyperbolic function method Wazzan [5] usedthe modified tanh-coth method in solving the generalizedBurgers-fisher and Kuramoto-Sivashinsky equations Renet al [6] applied the generalized algebra method to the(2+1)-dimensional Boiti-Leon-Pempinelli system In generalvarious studies in this context have been submitted to theliterature [7ndash23]

In this paper the Konopelchenko-Dubrovsky (KD) equa-tions [24] are investigated by using the improved Bernoullisubequation function method [25 26]

The Konopelchenko-Dubrovsky (KD) equations aregiven by [24]

119906119905 minus 119906119909119909119909 minus 6119887119906119906119909 + 3211988621199062119906119909 minus 3V119910 + 3119886V119909119906 = 0119906119910 = V119909 (1)

where 119906 = 119906(119909 119910 119905) V = V(119909 119910 119905) and 119886 119887 are constantsVarious analytical approaches have been used in obtain-

ing the exact solutions to the Konopelchenko-Dubrovskyequations Sheng [27] used the improved F-expansionmethod in addressing (1) Wazwaz [28] employed the tanh-sech method the cosh-sinh method and the exponentialfunctions method for obtaining the analytical solutions to (1)and Kumar et al [29] solved the KD equations by travelingwave hypothesis and lie symmetry approach Song and Zhang[30] utilized the extended Riccati equation rational expan-sion method and Feng and Lin [31] applied the improvedRiccati mapping approach Bekir [32] used the extended tanhmethod Xia et al [33] employed the new modified extendedtanh function method

2 The IBSEFM

In this section we present Improved Bernoulli subequa-tion function method (IBSEFM) formed by modifying the

HindawiAdvances in Mathematical PhysicsVolume 2019 Article ID 7801247 9 pageshttpsdoiorg10115520197801247

2 Advances in Mathematical Physics

Bernoulli subequation function method Therefore we con-sider the following four steps

Step 1 Let us consider the nonlinear partial differentialequation given as 119875 (119906 119906119909 119906119905 119906119909119905 sdot sdot sdot) = 0 (2)

and the wave transformation119906 (119909 119905) = 119880 (120578) 120578 = 119896119909 minus 119888119905 (3)

Substituting (3) into (2) gives the following nonlinear ordi-nary differential equation119873(1198801198801015840 11988010158401015840 119880101584010158401015840 sdot sdot sdot) = 0 (4)

Step 2 The trial solution of (4) is assumed to be

119880 (120578) = sum119899119894=0 119886119894119865119894 (120578)sum119898119895=0 119887119895119865119895 (120578)= 1198860 + 1198861119865 (120578) + 11988621198652 (120578) + sdot sdot sdot + 119886119899119865119899 (120578)1198870 + 1198871119865 (120578) + 11988721198652 (120578) + sdot sdot sdot + 119887119898119865119898 (120578)

(5)

According to the Bernoulli theory we have the general formof the Bernoulli differential equation as1198651015840 = 119908119865 + 119889119865119872 119908 = 0 119889 = 0 119872 isin 119877 minus 0 1 2 (6)

where 119865 = 119865(120578) is Bernoulli differential polynomialSubstituting (6) into (4) gives equations in degrees of 119865 asΩ (119865) = 120588119904119865119904 + sdot sdot sdot + 1205881119865 + 1205880 = 0 (7)

The values of 119899 119898 and 119872 are to be determined by using thehomogeneous balance principle

Step 3 Setting the summation of the coefficients ofΩ(119865) andequating each summation to zero yields an algebraic systemof equation 120588119894 = 0 119894 = 0 sdot sdot sdot 119904 (8)

Solving this system of equation we reach the values of1198860 sdot sdot sdot 119886119899 and 1198870 sdot sdot sdot 119887119898Step 4 Equation (6) has the following solutions dependingon the values of 119887 and 119889119865 (120578) = [minus119889119908 + 119864119890119908(119872minus1)120578 ]1(1minus119872) 119908 = 119889 (9)

119865 (120578)= [(119864 minus 1) + (119864 + 1) tanh (119908 (1 minus 119872) 1205782)1 minus tanh (119908 (1 minus 119872) 1205782) ]1(1minus119872)

119908 = 119889(10)

Using a complete discrimination system for the polynomialof 119865 we obtain the analytical solutions to (4) with the help ofWolfram Mathematica For a better interpretation of resultsobtained in this way we can plot two- and three-dimensionalfigures of analytical solutions by considering suitable valuesof parameters

3 Application

In this section we present the application of IBSEFMmethodto the Konopelchenko-Dubrovsky (KD) equations Using thewave transformation on (1)

119906 (119909 119910 119905) = 119880 (120574) 120574 = 119909 + 119898119910 minus 119888119905V (119909 119910 119905) = 119881 (120574) 120574 = 119909 + 119898119910 minus 119888119905 (11)

we get the following system of nonlinear ordinary differentialequations

minus1198881198801015840 minus 119880101584010158401015840 minus 61198871198801198801015840 + 32119886211988021198801015840 minus 31198811015840 + 31198861198811015840119880 = 01198981198801015840 = 1198811015840 (12)

Integrating the second equation in the system (12) we get

119898119880 = 119881 (13)

Inserting (13) into the first equation of (12) we get thefollowing single nonlinear ordinary differential equation

(6119898119886 minus 12119887)1198801198801015840 minus (2119888 + 61198982)1198801015840 minus 2119880101584010158401015840 + 3119886211988021198801015840= 0 (14)

Finally integrating (14) we have

(3119898119886 minus 6119887)1198802 minus (2119888 + 61198982)119880 minus 211988010158401015840 + 11988621198803 = 0 (15)

Balancing (15) by considering the highest derivative and thehighest power we obtain 119898 + 119872 = 119899 + 1

Choosing 119872 = 3 and 119898 = 1 gives 119899 = 3 Thus the trialsolution to (1) takes the following form

119880(120574) = 1198860 + 1198861119865 (120574) + 11988621198652 (120574) + 11988631198653 (120574)1198870 + 1198871119865 (120574) (16)

where 1198651015840 = 119908119865 + 1198891198653 119908 = 0 119889 = 0 Substituting (16) itssecond derivative along with 1198651015840 = 119908119865 + 1198891198653 119908 = 0 119889 = 0into (15) yields a polynomial in 119865 We collect a system ofalgebraic equations from the polynomial by equating eachsummation of the coefficients of 119865 which have the samepower Solving the systemof the algebraic equations yields thevalues of the parameter involved Substituting the obtainedvalues of the parameters into (16) yields the solutions to (1)

Advances in Mathematical Physics 3

Case 1

1198860 = minus2 (minus2119882 + 119898120572) 11988701205722 1198863 = 12057221198861119886241198821198870 minus 21198981205721198870 1198871 = 120572211988614119882 minus 2119898120572119888 = minus4 (1198822 minus 119898119882120572 + 11989821205722)1205722 120590 = minus1198982 + 119882120572 119889 = 120572119886241198870

(17)

Case 2

1198860 = 119886111988701198871 1198862 = 119889119886111988701205901198871 1198863 = 1198891198861120590 119888 = minus31198982 minus 41205902120572 = minus412059011988711198861

119882 = minus2 (119898 minus 2120590) 12059011988711198861

(18)

Case 3

1198860 = 119886111988701198871 1198862 = minus41198891198870120572 1198863 = minus41198891198871120572 119888 = minus31198982 + 120572211988621211988721

119882 = 1198981205722 120590 = minus120572119886121198871

(19)

Case 4

1198860 = 119886111988701198871 1198862 = minus 2radic211988911988611198870radic(119888 + 31198982) 11988721 1198863 = minus 2radic211988911988611198871radic(119888 + 31198982) 11988721 120572 = minusradic2radic(119888 + 31198982) 119887211198861

119882 = minus119898radic(119888 + 31198982) 11988721radic21198861 120590 = minusradic(119888 + 31198982) 11988721radic21198871

(20)

Case 5

1198860 = 119886111988701198871 1198862 = minus 211989411988911988611198870radic119888 + 311989821198871 1198863 = minus 21198941198891198861radic119888 + 31198982 120572 = 2119894radic119888 + 3119898211988711198861

119882 = minus(119888 + 119898(3119898 minus 119894radic119888 + 31198982)) 11988711198861 120590 = 12 119894radic119888 + 31198982

(21)

Substituting (17) into (16) gives

1199061 (119909 119910 119905) = 4119882 minus 21198981205721205722 + 1198862(119890minus2(minus119888119905+119909+119898119910)120590120598 minus 119889120590) 1198870 (22)

Substituting (18) into (16) gives

1199062 (119909 119910 119905) = 1205981205901198861(minus1198891198902(minus119888119905+119909+119898119910)120590 + 120598120590) 1198871 (23)

Substituting (19) into (16) gives

1199063 (119909 119910 119905) = 4120590 (1 + 120598120590 (1198891198902(minus119888119905+119909+119898119910)120590 minus 120598120590))120572 + 11988611198871 (24)

4 Advances in Mathematical Physics

Substituting (20) into (16) gives

1199064 (119909 119910 119905)= (minus21198892 + 1198902radic2(minus119888119905+119909+119898119910)radic(119888+31198982)11988721 1198871 (119888 + 31198982) 1205982) 1198861radic(119888 + 31198982) 11988721

(119890radic2(minus119888119905+119909+119898119910)radic(119888+31198982)11988721 1198871 (119888 + 31198982) 1205981198871 + radic2119889radic(119888 + 31198982) 11988721)(radic21198891198871 + 119890radic2(minus119888119905+119909+119898119910)radic(119888+31198982)11988721 1198871120598radic(119888 + 31198982) 11988721) (25)

Substituting (21) into (16) gives

1199065 (119909 119910 119905) = radic119888 + 311989821205981198861(2119894119889119890119894radic119888+31198982(minus119888119905+119909+119898119910) + radic119888 + 31198982120598) 1198871 (26)

4 Results and Discussion

This study uses the IBSEFM to obtain some travelling wavesolutions to the Konopelchenko-Dubrovsky equations suchas the singular soliton kink-type soliton and the periodicwave solutions The results are presented in exponentialfunction structure in Section 3 When the basic relation119890119909 = sinh(119909) + cosh(119909) is used solutions (22)-(26)become

1199061 (119909 119910 119905) = 41198821205722 minus 2119898120572 + 11988701198862 (minus119889120590 + 120598 cosh [2119888119905120590 minus 2119909120590 minus 2119898119910120590] + 120598 sinh [2119888119905120590 minus 2119909120590 minus 2119898119910120590])minus1 (27)

1199062 (119909 119910 119905) = 12059812059011988611198871 (120598120590 minus 119889 cosh [2119888119905120590 minus 2119909120590 minus 2119898119910120590] + 119889 sinh [2119888119905120590 minus 2119909120590 minus 2119898119910120590])minus1 (28)

1199063 (119909 119910 119905) = 4120590120572 + 41205981205902120572 (minus120598120590 + 119889 cosh [2119888119905120590 minus 2119909120590 minus 2119898119910120590] minus 119889 sinh [2119888119905120590 minus 2119909120590 minus 2119898119910120590]) + 11988611198871 (29)

1199064 (119909 119910 119905) = ((minus21198892

+ (119888 + 31198982) 1205982(cosh[[[2radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]] + sinh[[[

2radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]]))sdot 1198861radic11988811988721 + 3119898211988721)times (((119888 + 31198982) 120598(cosh[[[

radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]] + sinh[[[radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]])1198871

+ radic2119889radic11988811988721 + 3119898211988721)(radic21198891198871 + 120598 cosh[[[radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]]radic11988811988721 + 3119898211988721

+ 120598 sinh[[[radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]]radic11988811988721 + 3119898211988721) ))

minus1

(30)

Advances in Mathematical Physics 5

2 4 6 8

minus150

minus100

minus50

50

100

150u

2 4 6 8

minus150

minus100

minus50

50

100

150v

50

0

minus50

50

0

minus50

0

1

2

3

minus10

0

10

minus10

0

10

u

x

xx

x

tt

minus10

minus05

00

05

10 v

Figure 1The 3D and 2D surfaces of the solution (22) by considering the values 1198870 = 05 1198862 = 02 119896 = 16119882 = 03 119888 = 02 119889 = 001prop= 08120590 = 03 120598 = 09 119910 = 15 and 119905 = 0007 for 2119863

where 11988811988721 + 3119898211988721 gt 0 for valid solution

1199065 (119909 119910 119905) = radic119888 + 3119898212059811988611198871(radic119888 + 31198982120598+ 2119894119889 cos [radic119888 + 31198982 (minus119888119905 + 119909 + 119898119910)]minus 2119889 sin [radic119888 + 31198982 (minus119888119905 + 119909 + 119898119910)]

(31)

where 119888 + 31198982 gt 0 for valid solutionIt has been reported in the literature that some computa-

tional approaches have been utilized to obtain the solutionsof the Konopelchenko-Dubrovsky equations Sheng [27]reported that some Jacobi elliptic function solutions soliton-like solutions trigonometric function solutions improvedF-expansion method Wazwaz [28] obtained some solitarywave periodic wave and kink solutions to this equation byusing the tanhndashsech method and the coshndashsinh scheme Thetraveling wave hypothesis and the lie group approach wereused on the Konopelchenko-Dubrovsky equations by Kumaret al [29] and some solitary wave periodic singular wavesand cnoidal and snoidal solutions were reported Song andZhang [30] employed the extended Riccati equation rationalexpansion method to (1) and obtained some rational formalhyperbolic function solutions rational formal triangularperiodic solutions and rational solutions Via the improvedmapping approach and variable separation method Feng andLin [31] constructed some solitary wave solutions periodic

wave solutions and rational function solutions Bekir [32]constructed some kink and singular solitons by using theextended tanh method Xia et al [33] employed the modifiedextended tanh function method to the Konopelchenko-Dubrovsky equation and soliton-like solutions periodicformal solution and rational function solutionswere success-fully constructed We observed that the analytical methoduse in this study revealed the wave solutions in exponentialfunction structure that can be converted to solutions withhyperbolic and trigonometric function structure

Remark 1 Solutions (22) (23) and (24) are singular solitonsolutions Solution (25) is a kink-type soliton and solution(26) is a periodic wave solution

5 Conclusions

In this paper the IBSEFM is applied to the KD equations Wesuccessfully obtained some new traveling wave solutions tothe studied model such as complex and exponential functionsolutions We presented the 2D and 3D graphs to each of theobtained solutions with the help of some powerful softwareprogram

It has been observed that these solutions have verifiednonlinear KD equations These traveling wave solutionsFigures 1 2 3 4 5 and 6 have been formed Moreover if wechose 119872 = 3 119898 = 2 and 119899 = 4 we could obtain some other

6 Advances in Mathematical Physics

minus2 2 4 6 8x

minus15

minus10

minus5

5

10

15u

minus2 2 4 6 8x

minus15

minus10

minus5

5

10

15 v

u v

x

t t

x

5

0

minus5

5

0

0

10

minus5

0

1

2

3

0

2

4

6

8

minus10

minus10

minus05

00

05

10

Figure 2The 3D and 2D surfaces of the solution (23) by considering the values 1198871 = 05 1198861 = 02 119896 = 16 119888 = 02 119889 = 001 120590 = 03 120598 = 09119910 = 15 and 119905 = 0007 for 2119863

minus1 1 2 3 4 5 6x

minus40

minus20

20

40

u

minus1 1 2 3 4 5 6x

minus40

minus20

20

40

v

u vt t

0

1

2

3

minus10

minus0500

0510

xx0

10

minus10

15

10

5

0

minus5

15

10

5

0

minus5

15

10

5

minus50

Figure 3The 3D and 2D surfaces of the solution (24) by considering the values 1198871 = 05 1198861 = 02 119896 = 16 119888 = 02 119889 = 001prop= 08 120590 = 03120598 = 09 119910 = 15 and 119905 = 0007 for 2119863

Advances in Mathematical Physics 7

minus6 minus4 minus2 2x

minus10

minus05

05

10u

minus6 minus4 minus2 2x

minus15

minus10

minus05

05

10

v

u

t

0

1

2

3

0

1

2

3

5

0

minus5x x0

10

minus10

minus1

v

t

minus10

minus05

minus0500

00

05

05

10

Figure 4The 3D and 2D surfaces of the solution (25) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08 119889 = 001 119910 = 15 120598 = 09and 119905 = 0007 for 2119863

minus4 minus4minus2 0 2 4x

0655

0660

0665

0670

0675

0680

minus2 0 2 4x

1055

1060

1065

1070

1075

Re[u]

Re[u]

Re[v]

Re[v]

t

0

1

2

3

t

xx

0

1

2

3

5

minus50

5

minus50

1060

1065

1070

1075

0665

0670

Figure 5 The 3D and 2D surfaces of the analytical solution (26) (real part) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08119889 = 001 119910 = 15 120598 = 09 and 119905 = 0007 for 2119863

8 Advances in Mathematical Physics

minus4 minus2 2 4x

minus0010

minus0005

0005

0010

minus4 minus2 2 4x

minus0010

minus0005

0005

0010Im[v]Im[u]

x5

minus50

x

5

minus50

Im[v]Im[u]

t

0

1

2

3

t

0

1

2

3

minus0005

0000

0005

minus0005

0000

0005

Figure 6The 3D and 2D surfaces of the analytical solution (26) (imaginer part) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08119889 = 001 119910 = 15 120598 = 09 and 119905 = 0007 for 2119863

new traveling wave solutions to the nonlinear KD equationsWhen we compare our results with some of the existingresults in the literature we observe that the results reportedin this study especially (25) and (26) are new complex andexponential solutions All the reported solutions in this studyverify the Konopelchenko-Dubrovsky equation Our resultsmight be useful in explaining the physical meaning of variousnonlinear models arising in the field of nonlinear sciencesIBSEFM is powerful and efficient mathematical tool that canbe used to handle various nonlinear mathematical models

The performance of this method is effective and reliableIn our research these traveling wave solutions have notsubmitted to the literature in advance

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

Part of this article is presented in 3rd International Con-ference on Computational Mathematics and EngineeringSciences (CMES 2018)

Conflicts of Interest

The author declares that he has no conflicts of interest

Authorsrsquo Contributions

The author read and approved the final manuscript

Acknowledgments

This work is supported by MAUBAP18MMF020

References

[1] E M E Zayed G M Moatimid and A G Al-Nowehy ldquoThegeneralized kudryashovmethod and its applications for solvingnonlinear PDEs in mathematical physicsrdquo Scientific Journal ofMathematics Research vol 5 no 2 pp 19ndash39 2015

[2] CWang ldquoDynamic behavior of travelingwaves for the Sharma-Tasso-Olver equationrdquo Nonlinear Dynamics vol 85 no 2 pp1119ndash1126 2016

[3] T A Nofal ldquoSimple equation method for nonlinear partial dif-ferential equations and its applicationsrdquo Journal of the EgyptianMathematical Society vol 24 no 2 pp 204ndash209 2016

[4] Y Shang ldquoThe extended hyperbolic functionmethod and exactsolutions of the long-short wave resonance equationsrdquo ChaosSolitons and Fractals vol 36 no 3 pp 762ndash771 2008

Advances in Mathematical Physics 9

[5] L Wazzan ldquoA modified tanh-coth method for solving thegeneral Burgers-Fisher and the Kuramoto-Sivashinsky equa-tionsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 6 pp 2642ndash2652 2009

[6] Y-J Ren S-T Liu andH-Q Zhang ldquoA new generalized algebramethod and its application in the (2+1)-dimensional boiti-leon-pempinelli equationrdquoChaos Solitons and Fractals vol 32 no 5pp 1655ndash1665 2007

[7] C Cattani ldquoHarmonic wavelet solutions of the Schrodingerequationrdquo International Journal of Fluid Mechanics Researchvol 30 no 5 pp 463ndash472 2003

[8] C Cattani and Y Y Rushchitskii ldquoCubically nonlinear elasticwaves wave equations and methods of analysisrdquo InternationalApplied Mechanics vol 39 no 10 pp 1115ndash1145 2003

[9] A-MWazwaz andM SMehanna ldquoA variety of exact travellingwave solutions for the (2+1)-dimensional boiti-leon-pempinelliequationrdquo Applied Mathematics and Computation vol 217 no4 pp 1484ndash1490 2010

[10] W-G Feng K-M Li Y-Z Li and C Lin ldquoExplicit exactsolutions for (2+1)-dimensional boiti-leon-pempinelli equa-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2013ndash2017 2009

[11] M Eslami A Neyrame and M Ebrahimi ldquoExplicit solutionsof nonlinear (2+1)-dimensional dispersive long wave equationrdquoJournal of King SaudUniversity - Science vol 24 no 1 pp 69ndash712012

[12] M Eslami andH Rezazadeh ldquoThefirst integralmethod forWu-Zhang system with conformable time-fractional derivativerdquoCalcolo vol 53 no 3 pp 475ndash485 2016

[13] S T R Rizvi and K Ali ldquoJacobian elliptic periodic travelingwave solutions in the negative-index materialsrdquo NonlinearDynamics vol 87 no 3 pp 1967ndash1972 2017

[14] H M Baskonus T A Sulaiman and H Bulut ldquoOn the novelwave behaviors to the coupled nonlinear Maccarirsquos system withcomplex structurerdquoOptik vol 131 pp 1036ndash1043 2017

[15] H M Baskonus T A Sulaiman and H Bulut ldquoNew solitarywave solutions to the (2+1)-dimensional CalogerondashBogoyav-lenskiindashSchiff and the KadomtsevndashPetviashvili hierarchy equa-tionsrdquo Indian Journal of Physics vol 91 no 10 pp 1237ndash12432017

[16] C Cattani and A Ciancio ldquoHybrid two scales mathematicaltools for active particles modelling complex systems withlearning hiding dynamicsrdquo Mathematical Models and Methodsin Applied Sciences vol 17 no 2 pp 171ndash187 2007

[17] M H Heydari M R Hooshmandasl G B Loghmani and CCattani ldquoWavelets Galerkin method for solving stochastic heatequationrdquo International Journal of Computer Mathematics vol93 no 9 pp 1579ndash1596 2016

[18] A R Seadawy ldquoFractional solitary wave solutions of the non-linear higher-order extended KdV equation in a stratified shearflow Part IrdquoComputers andMathematics with Applications vol70 no 4 pp 345ndash352 2015

[19] A R Seadawy ldquoIon acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equa-tion in quantum plasmardquoMathematical Methods in the AppliedSciences vol 40 no 5 pp 1598ndash1607 2017

[20] A Yokus H M Baskonus T A Sulaiman and H BulutldquoNumerical simulation and solutions of the two-componentsecond order KdV evolutionary systemrdquoNumericalMethods forPartial Differential Equations vol 34 no 1 pp 211ndash227 2018

[21] J Liu M Eslami H Rezazadeh and M Mirzazadeh ldquoRationalsolutions and lump solutions to a non-isospectral and gen-eralized variable-coefficient KadomtsevndashPetviashvili equationrdquoNonlinear Dynamics vol 95 no 2 pp 1027ndash1033 2019

[22] N Zhang and T-C Xia ldquoA new negative discrete hierarchyand its N-fold darboux transformationrdquo Communications ineoretical Physics vol 68 no 6 pp 687ndash692 2017

[23] H M Baskonus T A Sulaiman and H Bulut ldquoNovel complexandhyperbolic forms to the strainwave equation inmicrostruc-tured solidsrdquo Optical and Quantum Electronics vol 50 no 142018

[24] B G Konopelchenko and VG Dubrovsky ldquoSome new inte-grable nonlinear evolution equations in (2+1) dimensionsrdquoPhysics Letters A vol 102 no 1-2 pp 15ndash17 1984

[25] H M Baskonus and H Bulut ldquoOn the complex structures ofKundu-Eckhaus equation via improved Bernoulli sub-equationfunction methodrdquo Waves in Random and Complex Media vol25 no 4 pp 720ndash728 2015

[26] FDusunceli ldquoSolutions for the drinfeld-sokolov equation usingan IBSEFM methodrdquo MSU Journal of Science vol 6 no 1 pp505ndash510 2018

[27] Z Sheng ldquoFurther Improved F-expansion method and newexact solutions of KD equationrdquo Chaos Solitons and Fractalsvol 32 no 4 pp 1375ndash1383 2007

[28] A-M Wazwaz ldquoNew kinks and solitons solutions to the(2+1) dimensional KD equationrdquo Mathematical and ComputerModelling vol 45 no 3-4 pp 473ndash479 2007

[29] SKumarAHama andABiswas ldquoSolutions ofKDequation byTravelingwave hypothesis and lie symmetry approachrdquoAppliedMathematics and Information Sciences vol 8 no 4 pp 1533ndash1539 2014

[30] L Song and H Zhang ldquoNew exact solutions for theKonopelchenko-Dubrovsky equation using an extendedRiccati equation rational expansion method and symboliccomputationrdquo Applied Mathematics and Computation vol 187no 2 pp 1373ndash1388 2007

[31] W-G Feng and C Lin ldquoExplicit exact solutions for the (2+1)dimensional KD equationrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 298ndash302 2009

[32] A Bekir ldquoApplications of the extended tanhmethod for couplednonlinear evolution equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1748ndash17572008

[33] T Xia Z Lu and H Zhang ldquoSymbolic computation andnew families of exact soliton-like solutions of Konopelchenko-Dubrovsky equationsrdquo Chaos Solitons and Fractals vol 20 no3 pp 561ndash566 2004

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

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

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Advances in Mathematical Physics 3

Case 1

1198860 = minus2 (minus2119882 + 119898120572) 11988701205722 1198863 = 12057221198861119886241198821198870 minus 21198981205721198870 1198871 = 120572211988614119882 minus 2119898120572119888 = minus4 (1198822 minus 119898119882120572 + 11989821205722)1205722 120590 = minus1198982 + 119882120572 119889 = 120572119886241198870

(17)

Case 2

1198860 = 119886111988701198871 1198862 = 119889119886111988701205901198871 1198863 = 1198891198861120590 119888 = minus31198982 minus 41205902120572 = minus412059011988711198861

119882 = minus2 (119898 minus 2120590) 12059011988711198861

(18)

Case 3

1198860 = 119886111988701198871 1198862 = minus41198891198870120572 1198863 = minus41198891198871120572 119888 = minus31198982 + 120572211988621211988721

119882 = 1198981205722 120590 = minus120572119886121198871

(19)

Case 4

1198860 = 119886111988701198871 1198862 = minus 2radic211988911988611198870radic(119888 + 31198982) 11988721 1198863 = minus 2radic211988911988611198871radic(119888 + 31198982) 11988721 120572 = minusradic2radic(119888 + 31198982) 119887211198861

119882 = minus119898radic(119888 + 31198982) 11988721radic21198861 120590 = minusradic(119888 + 31198982) 11988721radic21198871

(20)

Case 5

1198860 = 119886111988701198871 1198862 = minus 211989411988911988611198870radic119888 + 311989821198871 1198863 = minus 21198941198891198861radic119888 + 31198982 120572 = 2119894radic119888 + 3119898211988711198861

119882 = minus(119888 + 119898(3119898 minus 119894radic119888 + 31198982)) 11988711198861 120590 = 12 119894radic119888 + 31198982

(21)

Substituting (17) into (16) gives

1199061 (119909 119910 119905) = 4119882 minus 21198981205721205722 + 1198862(119890minus2(minus119888119905+119909+119898119910)120590120598 minus 119889120590) 1198870 (22)

Substituting (18) into (16) gives

1199062 (119909 119910 119905) = 1205981205901198861(minus1198891198902(minus119888119905+119909+119898119910)120590 + 120598120590) 1198871 (23)

Substituting (19) into (16) gives

1199063 (119909 119910 119905) = 4120590 (1 + 120598120590 (1198891198902(minus119888119905+119909+119898119910)120590 minus 120598120590))120572 + 11988611198871 (24)

4 Advances in Mathematical Physics

Substituting (20) into (16) gives

1199064 (119909 119910 119905)= (minus21198892 + 1198902radic2(minus119888119905+119909+119898119910)radic(119888+31198982)11988721 1198871 (119888 + 31198982) 1205982) 1198861radic(119888 + 31198982) 11988721

(119890radic2(minus119888119905+119909+119898119910)radic(119888+31198982)11988721 1198871 (119888 + 31198982) 1205981198871 + radic2119889radic(119888 + 31198982) 11988721)(radic21198891198871 + 119890radic2(minus119888119905+119909+119898119910)radic(119888+31198982)11988721 1198871120598radic(119888 + 31198982) 11988721) (25)

Substituting (21) into (16) gives

1199065 (119909 119910 119905) = radic119888 + 311989821205981198861(2119894119889119890119894radic119888+31198982(minus119888119905+119909+119898119910) + radic119888 + 31198982120598) 1198871 (26)

4 Results and Discussion

This study uses the IBSEFM to obtain some travelling wavesolutions to the Konopelchenko-Dubrovsky equations suchas the singular soliton kink-type soliton and the periodicwave solutions The results are presented in exponentialfunction structure in Section 3 When the basic relation119890119909 = sinh(119909) + cosh(119909) is used solutions (22)-(26)become

1199061 (119909 119910 119905) = 41198821205722 minus 2119898120572 + 11988701198862 (minus119889120590 + 120598 cosh [2119888119905120590 minus 2119909120590 minus 2119898119910120590] + 120598 sinh [2119888119905120590 minus 2119909120590 minus 2119898119910120590])minus1 (27)

1199062 (119909 119910 119905) = 12059812059011988611198871 (120598120590 minus 119889 cosh [2119888119905120590 minus 2119909120590 minus 2119898119910120590] + 119889 sinh [2119888119905120590 minus 2119909120590 minus 2119898119910120590])minus1 (28)

1199063 (119909 119910 119905) = 4120590120572 + 41205981205902120572 (minus120598120590 + 119889 cosh [2119888119905120590 minus 2119909120590 minus 2119898119910120590] minus 119889 sinh [2119888119905120590 minus 2119909120590 minus 2119898119910120590]) + 11988611198871 (29)

1199064 (119909 119910 119905) = ((minus21198892

+ (119888 + 31198982) 1205982(cosh[[[2radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]] + sinh[[[

2radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]]))sdot 1198861radic11988811988721 + 3119898211988721)times (((119888 + 31198982) 120598(cosh[[[

radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]] + sinh[[[radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]])1198871

+ radic2119889radic11988811988721 + 3119898211988721)(radic21198891198871 + 120598 cosh[[[radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]]radic11988811988721 + 3119898211988721

+ 120598 sinh[[[radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]]radic11988811988721 + 3119898211988721) ))

minus1

(30)

Advances in Mathematical Physics 5

2 4 6 8

minus150

minus100

minus50

50

100

150u

2 4 6 8

minus150

minus100

minus50

50

100

150v

50

0

minus50

50

0

minus50

0

1

2

3

minus10

0

10

minus10

0

10

u

x

xx

x

tt

minus10

minus05

00

05

10 v

Figure 1The 3D and 2D surfaces of the solution (22) by considering the values 1198870 = 05 1198862 = 02 119896 = 16119882 = 03 119888 = 02 119889 = 001prop= 08120590 = 03 120598 = 09 119910 = 15 and 119905 = 0007 for 2119863

where 11988811988721 + 3119898211988721 gt 0 for valid solution

1199065 (119909 119910 119905) = radic119888 + 3119898212059811988611198871(radic119888 + 31198982120598+ 2119894119889 cos [radic119888 + 31198982 (minus119888119905 + 119909 + 119898119910)]minus 2119889 sin [radic119888 + 31198982 (minus119888119905 + 119909 + 119898119910)]

(31)

where 119888 + 31198982 gt 0 for valid solutionIt has been reported in the literature that some computa-

tional approaches have been utilized to obtain the solutionsof the Konopelchenko-Dubrovsky equations Sheng [27]reported that some Jacobi elliptic function solutions soliton-like solutions trigonometric function solutions improvedF-expansion method Wazwaz [28] obtained some solitarywave periodic wave and kink solutions to this equation byusing the tanhndashsech method and the coshndashsinh scheme Thetraveling wave hypothesis and the lie group approach wereused on the Konopelchenko-Dubrovsky equations by Kumaret al [29] and some solitary wave periodic singular wavesand cnoidal and snoidal solutions were reported Song andZhang [30] employed the extended Riccati equation rationalexpansion method to (1) and obtained some rational formalhyperbolic function solutions rational formal triangularperiodic solutions and rational solutions Via the improvedmapping approach and variable separation method Feng andLin [31] constructed some solitary wave solutions periodic

wave solutions and rational function solutions Bekir [32]constructed some kink and singular solitons by using theextended tanh method Xia et al [33] employed the modifiedextended tanh function method to the Konopelchenko-Dubrovsky equation and soliton-like solutions periodicformal solution and rational function solutionswere success-fully constructed We observed that the analytical methoduse in this study revealed the wave solutions in exponentialfunction structure that can be converted to solutions withhyperbolic and trigonometric function structure

Remark 1 Solutions (22) (23) and (24) are singular solitonsolutions Solution (25) is a kink-type soliton and solution(26) is a periodic wave solution

5 Conclusions

In this paper the IBSEFM is applied to the KD equations Wesuccessfully obtained some new traveling wave solutions tothe studied model such as complex and exponential functionsolutions We presented the 2D and 3D graphs to each of theobtained solutions with the help of some powerful softwareprogram

It has been observed that these solutions have verifiednonlinear KD equations These traveling wave solutionsFigures 1 2 3 4 5 and 6 have been formed Moreover if wechose 119872 = 3 119898 = 2 and 119899 = 4 we could obtain some other

6 Advances in Mathematical Physics

minus2 2 4 6 8x

minus15

minus10

minus5

5

10

15u

minus2 2 4 6 8x

minus15

minus10

minus5

5

10

15 v

u v

x

t t

x

5

0

minus5

5

0

0

10

minus5

0

1

2

3

0

2

4

6

8

minus10

minus10

minus05

00

05

10

Figure 2The 3D and 2D surfaces of the solution (23) by considering the values 1198871 = 05 1198861 = 02 119896 = 16 119888 = 02 119889 = 001 120590 = 03 120598 = 09119910 = 15 and 119905 = 0007 for 2119863

minus1 1 2 3 4 5 6x

minus40

minus20

20

40

u

minus1 1 2 3 4 5 6x

minus40

minus20

20

40

v

u vt t

0

1

2

3

minus10

minus0500

0510

xx0

10

minus10

15

10

5

0

minus5

15

10

5

0

minus5

15

10

5

minus50

Figure 3The 3D and 2D surfaces of the solution (24) by considering the values 1198871 = 05 1198861 = 02 119896 = 16 119888 = 02 119889 = 001prop= 08 120590 = 03120598 = 09 119910 = 15 and 119905 = 0007 for 2119863

Advances in Mathematical Physics 7

minus6 minus4 minus2 2x

minus10

minus05

05

10u

minus6 minus4 minus2 2x

minus15

minus10

minus05

05

10

v

u

t

0

1

2

3

0

1

2

3

5

0

minus5x x0

10

minus10

minus1

v

t

minus10

minus05

minus0500

00

05

05

10

Figure 4The 3D and 2D surfaces of the solution (25) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08 119889 = 001 119910 = 15 120598 = 09and 119905 = 0007 for 2119863

minus4 minus4minus2 0 2 4x

0655

0660

0665

0670

0675

0680

minus2 0 2 4x

1055

1060

1065

1070

1075

Re[u]

Re[u]

Re[v]

Re[v]

t

0

1

2

3

t

xx

0

1

2

3

5

minus50

5

minus50

1060

1065

1070

1075

0665

0670

Figure 5 The 3D and 2D surfaces of the analytical solution (26) (real part) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08119889 = 001 119910 = 15 120598 = 09 and 119905 = 0007 for 2119863

8 Advances in Mathematical Physics

minus4 minus2 2 4x

minus0010

minus0005

0005

0010

minus4 minus2 2 4x

minus0010

minus0005

0005

0010Im[v]Im[u]

x5

minus50

x

5

minus50

Im[v]Im[u]

t

0

1

2

3

t

0

1

2

3

minus0005

0000

0005

minus0005

0000

0005

Figure 6The 3D and 2D surfaces of the analytical solution (26) (imaginer part) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08119889 = 001 119910 = 15 120598 = 09 and 119905 = 0007 for 2119863

new traveling wave solutions to the nonlinear KD equationsWhen we compare our results with some of the existingresults in the literature we observe that the results reportedin this study especially (25) and (26) are new complex andexponential solutions All the reported solutions in this studyverify the Konopelchenko-Dubrovsky equation Our resultsmight be useful in explaining the physical meaning of variousnonlinear models arising in the field of nonlinear sciencesIBSEFM is powerful and efficient mathematical tool that canbe used to handle various nonlinear mathematical models

The performance of this method is effective and reliableIn our research these traveling wave solutions have notsubmitted to the literature in advance

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

Part of this article is presented in 3rd International Con-ference on Computational Mathematics and EngineeringSciences (CMES 2018)

Conflicts of Interest

The author declares that he has no conflicts of interest

Authorsrsquo Contributions

The author read and approved the final manuscript

Acknowledgments

This work is supported by MAUBAP18MMF020

References

[1] E M E Zayed G M Moatimid and A G Al-Nowehy ldquoThegeneralized kudryashovmethod and its applications for solvingnonlinear PDEs in mathematical physicsrdquo Scientific Journal ofMathematics Research vol 5 no 2 pp 19ndash39 2015

[2] CWang ldquoDynamic behavior of travelingwaves for the Sharma-Tasso-Olver equationrdquo Nonlinear Dynamics vol 85 no 2 pp1119ndash1126 2016

[3] T A Nofal ldquoSimple equation method for nonlinear partial dif-ferential equations and its applicationsrdquo Journal of the EgyptianMathematical Society vol 24 no 2 pp 204ndash209 2016

[4] Y Shang ldquoThe extended hyperbolic functionmethod and exactsolutions of the long-short wave resonance equationsrdquo ChaosSolitons and Fractals vol 36 no 3 pp 762ndash771 2008

Advances in Mathematical Physics 9

[5] L Wazzan ldquoA modified tanh-coth method for solving thegeneral Burgers-Fisher and the Kuramoto-Sivashinsky equa-tionsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 6 pp 2642ndash2652 2009

[6] Y-J Ren S-T Liu andH-Q Zhang ldquoA new generalized algebramethod and its application in the (2+1)-dimensional boiti-leon-pempinelli equationrdquoChaos Solitons and Fractals vol 32 no 5pp 1655ndash1665 2007

[7] C Cattani ldquoHarmonic wavelet solutions of the Schrodingerequationrdquo International Journal of Fluid Mechanics Researchvol 30 no 5 pp 463ndash472 2003

[8] C Cattani and Y Y Rushchitskii ldquoCubically nonlinear elasticwaves wave equations and methods of analysisrdquo InternationalApplied Mechanics vol 39 no 10 pp 1115ndash1145 2003

[9] A-MWazwaz andM SMehanna ldquoA variety of exact travellingwave solutions for the (2+1)-dimensional boiti-leon-pempinelliequationrdquo Applied Mathematics and Computation vol 217 no4 pp 1484ndash1490 2010

[10] W-G Feng K-M Li Y-Z Li and C Lin ldquoExplicit exactsolutions for (2+1)-dimensional boiti-leon-pempinelli equa-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2013ndash2017 2009

[11] M Eslami A Neyrame and M Ebrahimi ldquoExplicit solutionsof nonlinear (2+1)-dimensional dispersive long wave equationrdquoJournal of King SaudUniversity - Science vol 24 no 1 pp 69ndash712012

[12] M Eslami andH Rezazadeh ldquoThefirst integralmethod forWu-Zhang system with conformable time-fractional derivativerdquoCalcolo vol 53 no 3 pp 475ndash485 2016

[13] S T R Rizvi and K Ali ldquoJacobian elliptic periodic travelingwave solutions in the negative-index materialsrdquo NonlinearDynamics vol 87 no 3 pp 1967ndash1972 2017

[14] H M Baskonus T A Sulaiman and H Bulut ldquoOn the novelwave behaviors to the coupled nonlinear Maccarirsquos system withcomplex structurerdquoOptik vol 131 pp 1036ndash1043 2017

[15] H M Baskonus T A Sulaiman and H Bulut ldquoNew solitarywave solutions to the (2+1)-dimensional CalogerondashBogoyav-lenskiindashSchiff and the KadomtsevndashPetviashvili hierarchy equa-tionsrdquo Indian Journal of Physics vol 91 no 10 pp 1237ndash12432017

[16] C Cattani and A Ciancio ldquoHybrid two scales mathematicaltools for active particles modelling complex systems withlearning hiding dynamicsrdquo Mathematical Models and Methodsin Applied Sciences vol 17 no 2 pp 171ndash187 2007

[17] M H Heydari M R Hooshmandasl G B Loghmani and CCattani ldquoWavelets Galerkin method for solving stochastic heatequationrdquo International Journal of Computer Mathematics vol93 no 9 pp 1579ndash1596 2016

[18] A R Seadawy ldquoFractional solitary wave solutions of the non-linear higher-order extended KdV equation in a stratified shearflow Part IrdquoComputers andMathematics with Applications vol70 no 4 pp 345ndash352 2015

[19] A R Seadawy ldquoIon acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equa-tion in quantum plasmardquoMathematical Methods in the AppliedSciences vol 40 no 5 pp 1598ndash1607 2017

[20] A Yokus H M Baskonus T A Sulaiman and H BulutldquoNumerical simulation and solutions of the two-componentsecond order KdV evolutionary systemrdquoNumericalMethods forPartial Differential Equations vol 34 no 1 pp 211ndash227 2018

[21] J Liu M Eslami H Rezazadeh and M Mirzazadeh ldquoRationalsolutions and lump solutions to a non-isospectral and gen-eralized variable-coefficient KadomtsevndashPetviashvili equationrdquoNonlinear Dynamics vol 95 no 2 pp 1027ndash1033 2019

[22] N Zhang and T-C Xia ldquoA new negative discrete hierarchyand its N-fold darboux transformationrdquo Communications ineoretical Physics vol 68 no 6 pp 687ndash692 2017

[23] H M Baskonus T A Sulaiman and H Bulut ldquoNovel complexandhyperbolic forms to the strainwave equation inmicrostruc-tured solidsrdquo Optical and Quantum Electronics vol 50 no 142018

[24] B G Konopelchenko and VG Dubrovsky ldquoSome new inte-grable nonlinear evolution equations in (2+1) dimensionsrdquoPhysics Letters A vol 102 no 1-2 pp 15ndash17 1984

[25] H M Baskonus and H Bulut ldquoOn the complex structures ofKundu-Eckhaus equation via improved Bernoulli sub-equationfunction methodrdquo Waves in Random and Complex Media vol25 no 4 pp 720ndash728 2015

[26] FDusunceli ldquoSolutions for the drinfeld-sokolov equation usingan IBSEFM methodrdquo MSU Journal of Science vol 6 no 1 pp505ndash510 2018

[27] Z Sheng ldquoFurther Improved F-expansion method and newexact solutions of KD equationrdquo Chaos Solitons and Fractalsvol 32 no 4 pp 1375ndash1383 2007

[28] A-M Wazwaz ldquoNew kinks and solitons solutions to the(2+1) dimensional KD equationrdquo Mathematical and ComputerModelling vol 45 no 3-4 pp 473ndash479 2007

[29] SKumarAHama andABiswas ldquoSolutions ofKDequation byTravelingwave hypothesis and lie symmetry approachrdquoAppliedMathematics and Information Sciences vol 8 no 4 pp 1533ndash1539 2014

[30] L Song and H Zhang ldquoNew exact solutions for theKonopelchenko-Dubrovsky equation using an extendedRiccati equation rational expansion method and symboliccomputationrdquo Applied Mathematics and Computation vol 187no 2 pp 1373ndash1388 2007

[31] W-G Feng and C Lin ldquoExplicit exact solutions for the (2+1)dimensional KD equationrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 298ndash302 2009

[32] A Bekir ldquoApplications of the extended tanhmethod for couplednonlinear evolution equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1748ndash17572008

[33] T Xia Z Lu and H Zhang ldquoSymbolic computation andnew families of exact soliton-like solutions of Konopelchenko-Dubrovsky equationsrdquo Chaos Solitons and Fractals vol 20 no3 pp 561ndash566 2004

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

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

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 Advances in Mathematical Physics

Substituting (20) into (16) gives

1199064 (119909 119910 119905)= (minus21198892 + 1198902radic2(minus119888119905+119909+119898119910)radic(119888+31198982)11988721 1198871 (119888 + 31198982) 1205982) 1198861radic(119888 + 31198982) 11988721

(119890radic2(minus119888119905+119909+119898119910)radic(119888+31198982)11988721 1198871 (119888 + 31198982) 1205981198871 + radic2119889radic(119888 + 31198982) 11988721)(radic21198891198871 + 119890radic2(minus119888119905+119909+119898119910)radic(119888+31198982)11988721 1198871120598radic(119888 + 31198982) 11988721) (25)

Substituting (21) into (16) gives

1199065 (119909 119910 119905) = radic119888 + 311989821205981198861(2119894119889119890119894radic119888+31198982(minus119888119905+119909+119898119910) + radic119888 + 31198982120598) 1198871 (26)

4 Results and Discussion

This study uses the IBSEFM to obtain some travelling wavesolutions to the Konopelchenko-Dubrovsky equations suchas the singular soliton kink-type soliton and the periodicwave solutions The results are presented in exponentialfunction structure in Section 3 When the basic relation119890119909 = sinh(119909) + cosh(119909) is used solutions (22)-(26)become

1199061 (119909 119910 119905) = 41198821205722 minus 2119898120572 + 11988701198862 (minus119889120590 + 120598 cosh [2119888119905120590 minus 2119909120590 minus 2119898119910120590] + 120598 sinh [2119888119905120590 minus 2119909120590 minus 2119898119910120590])minus1 (27)

1199062 (119909 119910 119905) = 12059812059011988611198871 (120598120590 minus 119889 cosh [2119888119905120590 minus 2119909120590 minus 2119898119910120590] + 119889 sinh [2119888119905120590 minus 2119909120590 minus 2119898119910120590])minus1 (28)

1199063 (119909 119910 119905) = 4120590120572 + 41205981205902120572 (minus120598120590 + 119889 cosh [2119888119905120590 minus 2119909120590 minus 2119898119910120590] minus 119889 sinh [2119888119905120590 minus 2119909120590 minus 2119898119910120590]) + 11988611198871 (29)

1199064 (119909 119910 119905) = ((minus21198892

+ (119888 + 31198982) 1205982(cosh[[[2radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]] + sinh[[[

2radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]]))sdot 1198861radic11988811988721 + 3119898211988721)times (((119888 + 31198982) 120598(cosh[[[

radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]] + sinh[[[radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]])1198871

+ radic2119889radic11988811988721 + 3119898211988721)(radic21198891198871 + 120598 cosh[[[radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]]radic11988811988721 + 3119898211988721

+ 120598 sinh[[[radic2 (minus119888119905 + 119909 + 119898119910)radic11988811988721 + 31198982119887211198871 ]]]radic11988811988721 + 3119898211988721) ))

minus1

(30)

Advances in Mathematical Physics 5

2 4 6 8

minus150

minus100

minus50

50

100

150u

2 4 6 8

minus150

minus100

minus50

50

100

150v

50

0

minus50

50

0

minus50

0

1

2

3

minus10

0

10

minus10

0

10

u

x

xx

x

tt

minus10

minus05

00

05

10 v

Figure 1The 3D and 2D surfaces of the solution (22) by considering the values 1198870 = 05 1198862 = 02 119896 = 16119882 = 03 119888 = 02 119889 = 001prop= 08120590 = 03 120598 = 09 119910 = 15 and 119905 = 0007 for 2119863

where 11988811988721 + 3119898211988721 gt 0 for valid solution

1199065 (119909 119910 119905) = radic119888 + 3119898212059811988611198871(radic119888 + 31198982120598+ 2119894119889 cos [radic119888 + 31198982 (minus119888119905 + 119909 + 119898119910)]minus 2119889 sin [radic119888 + 31198982 (minus119888119905 + 119909 + 119898119910)]

(31)

where 119888 + 31198982 gt 0 for valid solutionIt has been reported in the literature that some computa-

tional approaches have been utilized to obtain the solutionsof the Konopelchenko-Dubrovsky equations Sheng [27]reported that some Jacobi elliptic function solutions soliton-like solutions trigonometric function solutions improvedF-expansion method Wazwaz [28] obtained some solitarywave periodic wave and kink solutions to this equation byusing the tanhndashsech method and the coshndashsinh scheme Thetraveling wave hypothesis and the lie group approach wereused on the Konopelchenko-Dubrovsky equations by Kumaret al [29] and some solitary wave periodic singular wavesand cnoidal and snoidal solutions were reported Song andZhang [30] employed the extended Riccati equation rationalexpansion method to (1) and obtained some rational formalhyperbolic function solutions rational formal triangularperiodic solutions and rational solutions Via the improvedmapping approach and variable separation method Feng andLin [31] constructed some solitary wave solutions periodic

wave solutions and rational function solutions Bekir [32]constructed some kink and singular solitons by using theextended tanh method Xia et al [33] employed the modifiedextended tanh function method to the Konopelchenko-Dubrovsky equation and soliton-like solutions periodicformal solution and rational function solutionswere success-fully constructed We observed that the analytical methoduse in this study revealed the wave solutions in exponentialfunction structure that can be converted to solutions withhyperbolic and trigonometric function structure

Remark 1 Solutions (22) (23) and (24) are singular solitonsolutions Solution (25) is a kink-type soliton and solution(26) is a periodic wave solution

5 Conclusions

In this paper the IBSEFM is applied to the KD equations Wesuccessfully obtained some new traveling wave solutions tothe studied model such as complex and exponential functionsolutions We presented the 2D and 3D graphs to each of theobtained solutions with the help of some powerful softwareprogram

It has been observed that these solutions have verifiednonlinear KD equations These traveling wave solutionsFigures 1 2 3 4 5 and 6 have been formed Moreover if wechose 119872 = 3 119898 = 2 and 119899 = 4 we could obtain some other

6 Advances in Mathematical Physics

minus2 2 4 6 8x

minus15

minus10

minus5

5

10

15u

minus2 2 4 6 8x

minus15

minus10

minus5

5

10

15 v

u v

x

t t

x

5

0

minus5

5

0

0

10

minus5

0

1

2

3

0

2

4

6

8

minus10

minus10

minus05

00

05

10

Figure 2The 3D and 2D surfaces of the solution (23) by considering the values 1198871 = 05 1198861 = 02 119896 = 16 119888 = 02 119889 = 001 120590 = 03 120598 = 09119910 = 15 and 119905 = 0007 for 2119863

minus1 1 2 3 4 5 6x

minus40

minus20

20

40

u

minus1 1 2 3 4 5 6x

minus40

minus20

20

40

v

u vt t

0

1

2

3

minus10

minus0500

0510

xx0

10

minus10

15

10

5

0

minus5

15

10

5

0

minus5

15

10

5

minus50

Figure 3The 3D and 2D surfaces of the solution (24) by considering the values 1198871 = 05 1198861 = 02 119896 = 16 119888 = 02 119889 = 001prop= 08 120590 = 03120598 = 09 119910 = 15 and 119905 = 0007 for 2119863

Advances in Mathematical Physics 7

minus6 minus4 minus2 2x

minus10

minus05

05

10u

minus6 minus4 minus2 2x

minus15

minus10

minus05

05

10

v

u

t

0

1

2

3

0

1

2

3

5

0

minus5x x0

10

minus10

minus1

v

t

minus10

minus05

minus0500

00

05

05

10

Figure 4The 3D and 2D surfaces of the solution (25) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08 119889 = 001 119910 = 15 120598 = 09and 119905 = 0007 for 2119863

minus4 minus4minus2 0 2 4x

0655

0660

0665

0670

0675

0680

minus2 0 2 4x

1055

1060

1065

1070

1075

Re[u]

Re[u]

Re[v]

Re[v]

t

0

1

2

3

t

xx

0

1

2

3

5

minus50

5

minus50

1060

1065

1070

1075

0665

0670

Figure 5 The 3D and 2D surfaces of the analytical solution (26) (real part) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08119889 = 001 119910 = 15 120598 = 09 and 119905 = 0007 for 2119863

8 Advances in Mathematical Physics

minus4 minus2 2 4x

minus0010

minus0005

0005

0010

minus4 minus2 2 4x

minus0010

minus0005

0005

0010Im[v]Im[u]

x5

minus50

x

5

minus50

Im[v]Im[u]

t

0

1

2

3

t

0

1

2

3

minus0005

0000

0005

minus0005

0000

0005

Figure 6The 3D and 2D surfaces of the analytical solution (26) (imaginer part) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08119889 = 001 119910 = 15 120598 = 09 and 119905 = 0007 for 2119863

new traveling wave solutions to the nonlinear KD equationsWhen we compare our results with some of the existingresults in the literature we observe that the results reportedin this study especially (25) and (26) are new complex andexponential solutions All the reported solutions in this studyverify the Konopelchenko-Dubrovsky equation Our resultsmight be useful in explaining the physical meaning of variousnonlinear models arising in the field of nonlinear sciencesIBSEFM is powerful and efficient mathematical tool that canbe used to handle various nonlinear mathematical models

The performance of this method is effective and reliableIn our research these traveling wave solutions have notsubmitted to the literature in advance

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

Part of this article is presented in 3rd International Con-ference on Computational Mathematics and EngineeringSciences (CMES 2018)

Conflicts of Interest

The author declares that he has no conflicts of interest

Authorsrsquo Contributions

The author read and approved the final manuscript

Acknowledgments

This work is supported by MAUBAP18MMF020

References

[1] E M E Zayed G M Moatimid and A G Al-Nowehy ldquoThegeneralized kudryashovmethod and its applications for solvingnonlinear PDEs in mathematical physicsrdquo Scientific Journal ofMathematics Research vol 5 no 2 pp 19ndash39 2015

[2] CWang ldquoDynamic behavior of travelingwaves for the Sharma-Tasso-Olver equationrdquo Nonlinear Dynamics vol 85 no 2 pp1119ndash1126 2016

[3] T A Nofal ldquoSimple equation method for nonlinear partial dif-ferential equations and its applicationsrdquo Journal of the EgyptianMathematical Society vol 24 no 2 pp 204ndash209 2016

[4] Y Shang ldquoThe extended hyperbolic functionmethod and exactsolutions of the long-short wave resonance equationsrdquo ChaosSolitons and Fractals vol 36 no 3 pp 762ndash771 2008

Advances in Mathematical Physics 9

[5] L Wazzan ldquoA modified tanh-coth method for solving thegeneral Burgers-Fisher and the Kuramoto-Sivashinsky equa-tionsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 6 pp 2642ndash2652 2009

[6] Y-J Ren S-T Liu andH-Q Zhang ldquoA new generalized algebramethod and its application in the (2+1)-dimensional boiti-leon-pempinelli equationrdquoChaos Solitons and Fractals vol 32 no 5pp 1655ndash1665 2007

[7] C Cattani ldquoHarmonic wavelet solutions of the Schrodingerequationrdquo International Journal of Fluid Mechanics Researchvol 30 no 5 pp 463ndash472 2003

[8] C Cattani and Y Y Rushchitskii ldquoCubically nonlinear elasticwaves wave equations and methods of analysisrdquo InternationalApplied Mechanics vol 39 no 10 pp 1115ndash1145 2003

[9] A-MWazwaz andM SMehanna ldquoA variety of exact travellingwave solutions for the (2+1)-dimensional boiti-leon-pempinelliequationrdquo Applied Mathematics and Computation vol 217 no4 pp 1484ndash1490 2010

[10] W-G Feng K-M Li Y-Z Li and C Lin ldquoExplicit exactsolutions for (2+1)-dimensional boiti-leon-pempinelli equa-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2013ndash2017 2009

[11] M Eslami A Neyrame and M Ebrahimi ldquoExplicit solutionsof nonlinear (2+1)-dimensional dispersive long wave equationrdquoJournal of King SaudUniversity - Science vol 24 no 1 pp 69ndash712012

[12] M Eslami andH Rezazadeh ldquoThefirst integralmethod forWu-Zhang system with conformable time-fractional derivativerdquoCalcolo vol 53 no 3 pp 475ndash485 2016

[13] S T R Rizvi and K Ali ldquoJacobian elliptic periodic travelingwave solutions in the negative-index materialsrdquo NonlinearDynamics vol 87 no 3 pp 1967ndash1972 2017

[14] H M Baskonus T A Sulaiman and H Bulut ldquoOn the novelwave behaviors to the coupled nonlinear Maccarirsquos system withcomplex structurerdquoOptik vol 131 pp 1036ndash1043 2017

[15] H M Baskonus T A Sulaiman and H Bulut ldquoNew solitarywave solutions to the (2+1)-dimensional CalogerondashBogoyav-lenskiindashSchiff and the KadomtsevndashPetviashvili hierarchy equa-tionsrdquo Indian Journal of Physics vol 91 no 10 pp 1237ndash12432017

[16] C Cattani and A Ciancio ldquoHybrid two scales mathematicaltools for active particles modelling complex systems withlearning hiding dynamicsrdquo Mathematical Models and Methodsin Applied Sciences vol 17 no 2 pp 171ndash187 2007

[17] M H Heydari M R Hooshmandasl G B Loghmani and CCattani ldquoWavelets Galerkin method for solving stochastic heatequationrdquo International Journal of Computer Mathematics vol93 no 9 pp 1579ndash1596 2016

[18] A R Seadawy ldquoFractional solitary wave solutions of the non-linear higher-order extended KdV equation in a stratified shearflow Part IrdquoComputers andMathematics with Applications vol70 no 4 pp 345ndash352 2015

[19] A R Seadawy ldquoIon acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equa-tion in quantum plasmardquoMathematical Methods in the AppliedSciences vol 40 no 5 pp 1598ndash1607 2017

[20] A Yokus H M Baskonus T A Sulaiman and H BulutldquoNumerical simulation and solutions of the two-componentsecond order KdV evolutionary systemrdquoNumericalMethods forPartial Differential Equations vol 34 no 1 pp 211ndash227 2018

[21] J Liu M Eslami H Rezazadeh and M Mirzazadeh ldquoRationalsolutions and lump solutions to a non-isospectral and gen-eralized variable-coefficient KadomtsevndashPetviashvili equationrdquoNonlinear Dynamics vol 95 no 2 pp 1027ndash1033 2019

[22] N Zhang and T-C Xia ldquoA new negative discrete hierarchyand its N-fold darboux transformationrdquo Communications ineoretical Physics vol 68 no 6 pp 687ndash692 2017

[23] H M Baskonus T A Sulaiman and H Bulut ldquoNovel complexandhyperbolic forms to the strainwave equation inmicrostruc-tured solidsrdquo Optical and Quantum Electronics vol 50 no 142018

[24] B G Konopelchenko and VG Dubrovsky ldquoSome new inte-grable nonlinear evolution equations in (2+1) dimensionsrdquoPhysics Letters A vol 102 no 1-2 pp 15ndash17 1984

[25] H M Baskonus and H Bulut ldquoOn the complex structures ofKundu-Eckhaus equation via improved Bernoulli sub-equationfunction methodrdquo Waves in Random and Complex Media vol25 no 4 pp 720ndash728 2015

[26] FDusunceli ldquoSolutions for the drinfeld-sokolov equation usingan IBSEFM methodrdquo MSU Journal of Science vol 6 no 1 pp505ndash510 2018

[27] Z Sheng ldquoFurther Improved F-expansion method and newexact solutions of KD equationrdquo Chaos Solitons and Fractalsvol 32 no 4 pp 1375ndash1383 2007

[28] A-M Wazwaz ldquoNew kinks and solitons solutions to the(2+1) dimensional KD equationrdquo Mathematical and ComputerModelling vol 45 no 3-4 pp 473ndash479 2007

[29] SKumarAHama andABiswas ldquoSolutions ofKDequation byTravelingwave hypothesis and lie symmetry approachrdquoAppliedMathematics and Information Sciences vol 8 no 4 pp 1533ndash1539 2014

[30] L Song and H Zhang ldquoNew exact solutions for theKonopelchenko-Dubrovsky equation using an extendedRiccati equation rational expansion method and symboliccomputationrdquo Applied Mathematics and Computation vol 187no 2 pp 1373ndash1388 2007

[31] W-G Feng and C Lin ldquoExplicit exact solutions for the (2+1)dimensional KD equationrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 298ndash302 2009

[32] A Bekir ldquoApplications of the extended tanhmethod for couplednonlinear evolution equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1748ndash17572008

[33] T Xia Z Lu and H Zhang ldquoSymbolic computation andnew families of exact soliton-like solutions of Konopelchenko-Dubrovsky equationsrdquo Chaos Solitons and Fractals vol 20 no3 pp 561ndash566 2004

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

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

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Advances in Mathematical Physics 5

2 4 6 8

minus150

minus100

minus50

50

100

150u

2 4 6 8

minus150

minus100

minus50

50

100

150v

50

0

minus50

50

0

minus50

0

1

2

3

minus10

0

10

minus10

0

10

u

x

xx

x

tt

minus10

minus05

00

05

10 v

Figure 1The 3D and 2D surfaces of the solution (22) by considering the values 1198870 = 05 1198862 = 02 119896 = 16119882 = 03 119888 = 02 119889 = 001prop= 08120590 = 03 120598 = 09 119910 = 15 and 119905 = 0007 for 2119863

where 11988811988721 + 3119898211988721 gt 0 for valid solution

1199065 (119909 119910 119905) = radic119888 + 3119898212059811988611198871(radic119888 + 31198982120598+ 2119894119889 cos [radic119888 + 31198982 (minus119888119905 + 119909 + 119898119910)]minus 2119889 sin [radic119888 + 31198982 (minus119888119905 + 119909 + 119898119910)]

(31)

where 119888 + 31198982 gt 0 for valid solutionIt has been reported in the literature that some computa-

tional approaches have been utilized to obtain the solutionsof the Konopelchenko-Dubrovsky equations Sheng [27]reported that some Jacobi elliptic function solutions soliton-like solutions trigonometric function solutions improvedF-expansion method Wazwaz [28] obtained some solitarywave periodic wave and kink solutions to this equation byusing the tanhndashsech method and the coshndashsinh scheme Thetraveling wave hypothesis and the lie group approach wereused on the Konopelchenko-Dubrovsky equations by Kumaret al [29] and some solitary wave periodic singular wavesand cnoidal and snoidal solutions were reported Song andZhang [30] employed the extended Riccati equation rationalexpansion method to (1) and obtained some rational formalhyperbolic function solutions rational formal triangularperiodic solutions and rational solutions Via the improvedmapping approach and variable separation method Feng andLin [31] constructed some solitary wave solutions periodic

wave solutions and rational function solutions Bekir [32]constructed some kink and singular solitons by using theextended tanh method Xia et al [33] employed the modifiedextended tanh function method to the Konopelchenko-Dubrovsky equation and soliton-like solutions periodicformal solution and rational function solutionswere success-fully constructed We observed that the analytical methoduse in this study revealed the wave solutions in exponentialfunction structure that can be converted to solutions withhyperbolic and trigonometric function structure

Remark 1 Solutions (22) (23) and (24) are singular solitonsolutions Solution (25) is a kink-type soliton and solution(26) is a periodic wave solution

5 Conclusions

In this paper the IBSEFM is applied to the KD equations Wesuccessfully obtained some new traveling wave solutions tothe studied model such as complex and exponential functionsolutions We presented the 2D and 3D graphs to each of theobtained solutions with the help of some powerful softwareprogram

It has been observed that these solutions have verifiednonlinear KD equations These traveling wave solutionsFigures 1 2 3 4 5 and 6 have been formed Moreover if wechose 119872 = 3 119898 = 2 and 119899 = 4 we could obtain some other

6 Advances in Mathematical Physics

minus2 2 4 6 8x

minus15

minus10

minus5

5

10

15u

minus2 2 4 6 8x

minus15

minus10

minus5

5

10

15 v

u v

x

t t

x

5

0

minus5

5

0

0

10

minus5

0

1

2

3

0

2

4

6

8

minus10

minus10

minus05

00

05

10

Figure 2The 3D and 2D surfaces of the solution (23) by considering the values 1198871 = 05 1198861 = 02 119896 = 16 119888 = 02 119889 = 001 120590 = 03 120598 = 09119910 = 15 and 119905 = 0007 for 2119863

minus1 1 2 3 4 5 6x

minus40

minus20

20

40

u

minus1 1 2 3 4 5 6x

minus40

minus20

20

40

v

u vt t

0

1

2

3

minus10

minus0500

0510

xx0

10

minus10

15

10

5

0

minus5

15

10

5

0

minus5

15

10

5

minus50

Figure 3The 3D and 2D surfaces of the solution (24) by considering the values 1198871 = 05 1198861 = 02 119896 = 16 119888 = 02 119889 = 001prop= 08 120590 = 03120598 = 09 119910 = 15 and 119905 = 0007 for 2119863

Advances in Mathematical Physics 7

minus6 minus4 minus2 2x

minus10

minus05

05

10u

minus6 minus4 minus2 2x

minus15

minus10

minus05

05

10

v

u

t

0

1

2

3

0

1

2

3

5

0

minus5x x0

10

minus10

minus1

v

t

minus10

minus05

minus0500

00

05

05

10

Figure 4The 3D and 2D surfaces of the solution (25) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08 119889 = 001 119910 = 15 120598 = 09and 119905 = 0007 for 2119863

minus4 minus4minus2 0 2 4x

0655

0660

0665

0670

0675

0680

minus2 0 2 4x

1055

1060

1065

1070

1075

Re[u]

Re[u]

Re[v]

Re[v]

t

0

1

2

3

t

xx

0

1

2

3

5

minus50

5

minus50

1060

1065

1070

1075

0665

0670

Figure 5 The 3D and 2D surfaces of the analytical solution (26) (real part) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08119889 = 001 119910 = 15 120598 = 09 and 119905 = 0007 for 2119863

8 Advances in Mathematical Physics

minus4 minus2 2 4x

minus0010

minus0005

0005

0010

minus4 minus2 2 4x

minus0010

minus0005

0005

0010Im[v]Im[u]

x5

minus50

x

5

minus50

Im[v]Im[u]

t

0

1

2

3

t

0

1

2

3

minus0005

0000

0005

minus0005

0000

0005

Figure 6The 3D and 2D surfaces of the analytical solution (26) (imaginer part) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08119889 = 001 119910 = 15 120598 = 09 and 119905 = 0007 for 2119863

new traveling wave solutions to the nonlinear KD equationsWhen we compare our results with some of the existingresults in the literature we observe that the results reportedin this study especially (25) and (26) are new complex andexponential solutions All the reported solutions in this studyverify the Konopelchenko-Dubrovsky equation Our resultsmight be useful in explaining the physical meaning of variousnonlinear models arising in the field of nonlinear sciencesIBSEFM is powerful and efficient mathematical tool that canbe used to handle various nonlinear mathematical models

The performance of this method is effective and reliableIn our research these traveling wave solutions have notsubmitted to the literature in advance

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

Part of this article is presented in 3rd International Con-ference on Computational Mathematics and EngineeringSciences (CMES 2018)

Conflicts of Interest

The author declares that he has no conflicts of interest

Authorsrsquo Contributions

The author read and approved the final manuscript

Acknowledgments

This work is supported by MAUBAP18MMF020

References

[1] E M E Zayed G M Moatimid and A G Al-Nowehy ldquoThegeneralized kudryashovmethod and its applications for solvingnonlinear PDEs in mathematical physicsrdquo Scientific Journal ofMathematics Research vol 5 no 2 pp 19ndash39 2015

[2] CWang ldquoDynamic behavior of travelingwaves for the Sharma-Tasso-Olver equationrdquo Nonlinear Dynamics vol 85 no 2 pp1119ndash1126 2016

[3] T A Nofal ldquoSimple equation method for nonlinear partial dif-ferential equations and its applicationsrdquo Journal of the EgyptianMathematical Society vol 24 no 2 pp 204ndash209 2016

[4] Y Shang ldquoThe extended hyperbolic functionmethod and exactsolutions of the long-short wave resonance equationsrdquo ChaosSolitons and Fractals vol 36 no 3 pp 762ndash771 2008

Advances in Mathematical Physics 9

[5] L Wazzan ldquoA modified tanh-coth method for solving thegeneral Burgers-Fisher and the Kuramoto-Sivashinsky equa-tionsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 6 pp 2642ndash2652 2009

[6] Y-J Ren S-T Liu andH-Q Zhang ldquoA new generalized algebramethod and its application in the (2+1)-dimensional boiti-leon-pempinelli equationrdquoChaos Solitons and Fractals vol 32 no 5pp 1655ndash1665 2007

[7] C Cattani ldquoHarmonic wavelet solutions of the Schrodingerequationrdquo International Journal of Fluid Mechanics Researchvol 30 no 5 pp 463ndash472 2003

[8] C Cattani and Y Y Rushchitskii ldquoCubically nonlinear elasticwaves wave equations and methods of analysisrdquo InternationalApplied Mechanics vol 39 no 10 pp 1115ndash1145 2003

[9] A-MWazwaz andM SMehanna ldquoA variety of exact travellingwave solutions for the (2+1)-dimensional boiti-leon-pempinelliequationrdquo Applied Mathematics and Computation vol 217 no4 pp 1484ndash1490 2010

[10] W-G Feng K-M Li Y-Z Li and C Lin ldquoExplicit exactsolutions for (2+1)-dimensional boiti-leon-pempinelli equa-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2013ndash2017 2009

[11] M Eslami A Neyrame and M Ebrahimi ldquoExplicit solutionsof nonlinear (2+1)-dimensional dispersive long wave equationrdquoJournal of King SaudUniversity - Science vol 24 no 1 pp 69ndash712012

[12] M Eslami andH Rezazadeh ldquoThefirst integralmethod forWu-Zhang system with conformable time-fractional derivativerdquoCalcolo vol 53 no 3 pp 475ndash485 2016

[13] S T R Rizvi and K Ali ldquoJacobian elliptic periodic travelingwave solutions in the negative-index materialsrdquo NonlinearDynamics vol 87 no 3 pp 1967ndash1972 2017

[14] H M Baskonus T A Sulaiman and H Bulut ldquoOn the novelwave behaviors to the coupled nonlinear Maccarirsquos system withcomplex structurerdquoOptik vol 131 pp 1036ndash1043 2017

[15] H M Baskonus T A Sulaiman and H Bulut ldquoNew solitarywave solutions to the (2+1)-dimensional CalogerondashBogoyav-lenskiindashSchiff and the KadomtsevndashPetviashvili hierarchy equa-tionsrdquo Indian Journal of Physics vol 91 no 10 pp 1237ndash12432017

[16] C Cattani and A Ciancio ldquoHybrid two scales mathematicaltools for active particles modelling complex systems withlearning hiding dynamicsrdquo Mathematical Models and Methodsin Applied Sciences vol 17 no 2 pp 171ndash187 2007

[17] M H Heydari M R Hooshmandasl G B Loghmani and CCattani ldquoWavelets Galerkin method for solving stochastic heatequationrdquo International Journal of Computer Mathematics vol93 no 9 pp 1579ndash1596 2016

[18] A R Seadawy ldquoFractional solitary wave solutions of the non-linear higher-order extended KdV equation in a stratified shearflow Part IrdquoComputers andMathematics with Applications vol70 no 4 pp 345ndash352 2015

[19] A R Seadawy ldquoIon acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equa-tion in quantum plasmardquoMathematical Methods in the AppliedSciences vol 40 no 5 pp 1598ndash1607 2017

[20] A Yokus H M Baskonus T A Sulaiman and H BulutldquoNumerical simulation and solutions of the two-componentsecond order KdV evolutionary systemrdquoNumericalMethods forPartial Differential Equations vol 34 no 1 pp 211ndash227 2018

[21] J Liu M Eslami H Rezazadeh and M Mirzazadeh ldquoRationalsolutions and lump solutions to a non-isospectral and gen-eralized variable-coefficient KadomtsevndashPetviashvili equationrdquoNonlinear Dynamics vol 95 no 2 pp 1027ndash1033 2019

[22] N Zhang and T-C Xia ldquoA new negative discrete hierarchyand its N-fold darboux transformationrdquo Communications ineoretical Physics vol 68 no 6 pp 687ndash692 2017

[23] H M Baskonus T A Sulaiman and H Bulut ldquoNovel complexandhyperbolic forms to the strainwave equation inmicrostruc-tured solidsrdquo Optical and Quantum Electronics vol 50 no 142018

[24] B G Konopelchenko and VG Dubrovsky ldquoSome new inte-grable nonlinear evolution equations in (2+1) dimensionsrdquoPhysics Letters A vol 102 no 1-2 pp 15ndash17 1984

[25] H M Baskonus and H Bulut ldquoOn the complex structures ofKundu-Eckhaus equation via improved Bernoulli sub-equationfunction methodrdquo Waves in Random and Complex Media vol25 no 4 pp 720ndash728 2015

[26] FDusunceli ldquoSolutions for the drinfeld-sokolov equation usingan IBSEFM methodrdquo MSU Journal of Science vol 6 no 1 pp505ndash510 2018

[27] Z Sheng ldquoFurther Improved F-expansion method and newexact solutions of KD equationrdquo Chaos Solitons and Fractalsvol 32 no 4 pp 1375ndash1383 2007

[28] A-M Wazwaz ldquoNew kinks and solitons solutions to the(2+1) dimensional KD equationrdquo Mathematical and ComputerModelling vol 45 no 3-4 pp 473ndash479 2007

[29] SKumarAHama andABiswas ldquoSolutions ofKDequation byTravelingwave hypothesis and lie symmetry approachrdquoAppliedMathematics and Information Sciences vol 8 no 4 pp 1533ndash1539 2014

[30] L Song and H Zhang ldquoNew exact solutions for theKonopelchenko-Dubrovsky equation using an extendedRiccati equation rational expansion method and symboliccomputationrdquo Applied Mathematics and Computation vol 187no 2 pp 1373ndash1388 2007

[31] W-G Feng and C Lin ldquoExplicit exact solutions for the (2+1)dimensional KD equationrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 298ndash302 2009

[32] A Bekir ldquoApplications of the extended tanhmethod for couplednonlinear evolution equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1748ndash17572008

[33] T Xia Z Lu and H Zhang ldquoSymbolic computation andnew families of exact soliton-like solutions of Konopelchenko-Dubrovsky equationsrdquo Chaos Solitons and Fractals vol 20 no3 pp 561ndash566 2004

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

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

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Advances in Mathematical Physics

minus2 2 4 6 8x

minus15

minus10

minus5

5

10

15u

minus2 2 4 6 8x

minus15

minus10

minus5

5

10

15 v

u v

x

t t

x

5

0

minus5

5

0

0

10

minus5

0

1

2

3

0

2

4

6

8

minus10

minus10

minus05

00

05

10

Figure 2The 3D and 2D surfaces of the solution (23) by considering the values 1198871 = 05 1198861 = 02 119896 = 16 119888 = 02 119889 = 001 120590 = 03 120598 = 09119910 = 15 and 119905 = 0007 for 2119863

minus1 1 2 3 4 5 6x

minus40

minus20

20

40

u

minus1 1 2 3 4 5 6x

minus40

minus20

20

40

v

u vt t

0

1

2

3

minus10

minus0500

0510

xx0

10

minus10

15

10

5

0

minus5

15

10

5

0

minus5

15

10

5

minus50

Figure 3The 3D and 2D surfaces of the solution (24) by considering the values 1198871 = 05 1198861 = 02 119896 = 16 119888 = 02 119889 = 001prop= 08 120590 = 03120598 = 09 119910 = 15 and 119905 = 0007 for 2119863

Advances in Mathematical Physics 7

minus6 minus4 minus2 2x

minus10

minus05

05

10u

minus6 minus4 minus2 2x

minus15

minus10

minus05

05

10

v

u

t

0

1

2

3

0

1

2

3

5

0

minus5x x0

10

minus10

minus1

v

t

minus10

minus05

minus0500

00

05

05

10

Figure 4The 3D and 2D surfaces of the solution (25) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08 119889 = 001 119910 = 15 120598 = 09and 119905 = 0007 for 2119863

minus4 minus4minus2 0 2 4x

0655

0660

0665

0670

0675

0680

minus2 0 2 4x

1055

1060

1065

1070

1075

Re[u]

Re[u]

Re[v]

Re[v]

t

0

1

2

3

t

xx

0

1

2

3

5

minus50

5

minus50

1060

1065

1070

1075

0665

0670

Figure 5 The 3D and 2D surfaces of the analytical solution (26) (real part) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08119889 = 001 119910 = 15 120598 = 09 and 119905 = 0007 for 2119863

8 Advances in Mathematical Physics

minus4 minus2 2 4x

minus0010

minus0005

0005

0010

minus4 minus2 2 4x

minus0010

minus0005

0005

0010Im[v]Im[u]

x5

minus50

x

5

minus50

Im[v]Im[u]

t

0

1

2

3

t

0

1

2

3

minus0005

0000

0005

minus0005

0000

0005

Figure 6The 3D and 2D surfaces of the analytical solution (26) (imaginer part) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08119889 = 001 119910 = 15 120598 = 09 and 119905 = 0007 for 2119863

new traveling wave solutions to the nonlinear KD equationsWhen we compare our results with some of the existingresults in the literature we observe that the results reportedin this study especially (25) and (26) are new complex andexponential solutions All the reported solutions in this studyverify the Konopelchenko-Dubrovsky equation Our resultsmight be useful in explaining the physical meaning of variousnonlinear models arising in the field of nonlinear sciencesIBSEFM is powerful and efficient mathematical tool that canbe used to handle various nonlinear mathematical models

The performance of this method is effective and reliableIn our research these traveling wave solutions have notsubmitted to the literature in advance

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

Part of this article is presented in 3rd International Con-ference on Computational Mathematics and EngineeringSciences (CMES 2018)

Conflicts of Interest

The author declares that he has no conflicts of interest

Authorsrsquo Contributions

The author read and approved the final manuscript

Acknowledgments

This work is supported by MAUBAP18MMF020

References

[1] E M E Zayed G M Moatimid and A G Al-Nowehy ldquoThegeneralized kudryashovmethod and its applications for solvingnonlinear PDEs in mathematical physicsrdquo Scientific Journal ofMathematics Research vol 5 no 2 pp 19ndash39 2015

[2] CWang ldquoDynamic behavior of travelingwaves for the Sharma-Tasso-Olver equationrdquo Nonlinear Dynamics vol 85 no 2 pp1119ndash1126 2016

[3] T A Nofal ldquoSimple equation method for nonlinear partial dif-ferential equations and its applicationsrdquo Journal of the EgyptianMathematical Society vol 24 no 2 pp 204ndash209 2016

[4] Y Shang ldquoThe extended hyperbolic functionmethod and exactsolutions of the long-short wave resonance equationsrdquo ChaosSolitons and Fractals vol 36 no 3 pp 762ndash771 2008

Advances in Mathematical Physics 9

[5] L Wazzan ldquoA modified tanh-coth method for solving thegeneral Burgers-Fisher and the Kuramoto-Sivashinsky equa-tionsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 6 pp 2642ndash2652 2009

[6] Y-J Ren S-T Liu andH-Q Zhang ldquoA new generalized algebramethod and its application in the (2+1)-dimensional boiti-leon-pempinelli equationrdquoChaos Solitons and Fractals vol 32 no 5pp 1655ndash1665 2007

[7] C Cattani ldquoHarmonic wavelet solutions of the Schrodingerequationrdquo International Journal of Fluid Mechanics Researchvol 30 no 5 pp 463ndash472 2003

[8] C Cattani and Y Y Rushchitskii ldquoCubically nonlinear elasticwaves wave equations and methods of analysisrdquo InternationalApplied Mechanics vol 39 no 10 pp 1115ndash1145 2003

[9] A-MWazwaz andM SMehanna ldquoA variety of exact travellingwave solutions for the (2+1)-dimensional boiti-leon-pempinelliequationrdquo Applied Mathematics and Computation vol 217 no4 pp 1484ndash1490 2010

[10] W-G Feng K-M Li Y-Z Li and C Lin ldquoExplicit exactsolutions for (2+1)-dimensional boiti-leon-pempinelli equa-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2013ndash2017 2009

[11] M Eslami A Neyrame and M Ebrahimi ldquoExplicit solutionsof nonlinear (2+1)-dimensional dispersive long wave equationrdquoJournal of King SaudUniversity - Science vol 24 no 1 pp 69ndash712012

[12] M Eslami andH Rezazadeh ldquoThefirst integralmethod forWu-Zhang system with conformable time-fractional derivativerdquoCalcolo vol 53 no 3 pp 475ndash485 2016

[13] S T R Rizvi and K Ali ldquoJacobian elliptic periodic travelingwave solutions in the negative-index materialsrdquo NonlinearDynamics vol 87 no 3 pp 1967ndash1972 2017

[14] H M Baskonus T A Sulaiman and H Bulut ldquoOn the novelwave behaviors to the coupled nonlinear Maccarirsquos system withcomplex structurerdquoOptik vol 131 pp 1036ndash1043 2017

[15] H M Baskonus T A Sulaiman and H Bulut ldquoNew solitarywave solutions to the (2+1)-dimensional CalogerondashBogoyav-lenskiindashSchiff and the KadomtsevndashPetviashvili hierarchy equa-tionsrdquo Indian Journal of Physics vol 91 no 10 pp 1237ndash12432017

[16] C Cattani and A Ciancio ldquoHybrid two scales mathematicaltools for active particles modelling complex systems withlearning hiding dynamicsrdquo Mathematical Models and Methodsin Applied Sciences vol 17 no 2 pp 171ndash187 2007

[17] M H Heydari M R Hooshmandasl G B Loghmani and CCattani ldquoWavelets Galerkin method for solving stochastic heatequationrdquo International Journal of Computer Mathematics vol93 no 9 pp 1579ndash1596 2016

[18] A R Seadawy ldquoFractional solitary wave solutions of the non-linear higher-order extended KdV equation in a stratified shearflow Part IrdquoComputers andMathematics with Applications vol70 no 4 pp 345ndash352 2015

[19] A R Seadawy ldquoIon acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equa-tion in quantum plasmardquoMathematical Methods in the AppliedSciences vol 40 no 5 pp 1598ndash1607 2017

[20] A Yokus H M Baskonus T A Sulaiman and H BulutldquoNumerical simulation and solutions of the two-componentsecond order KdV evolutionary systemrdquoNumericalMethods forPartial Differential Equations vol 34 no 1 pp 211ndash227 2018

[21] J Liu M Eslami H Rezazadeh and M Mirzazadeh ldquoRationalsolutions and lump solutions to a non-isospectral and gen-eralized variable-coefficient KadomtsevndashPetviashvili equationrdquoNonlinear Dynamics vol 95 no 2 pp 1027ndash1033 2019

[22] N Zhang and T-C Xia ldquoA new negative discrete hierarchyand its N-fold darboux transformationrdquo Communications ineoretical Physics vol 68 no 6 pp 687ndash692 2017

[23] H M Baskonus T A Sulaiman and H Bulut ldquoNovel complexandhyperbolic forms to the strainwave equation inmicrostruc-tured solidsrdquo Optical and Quantum Electronics vol 50 no 142018

[24] B G Konopelchenko and VG Dubrovsky ldquoSome new inte-grable nonlinear evolution equations in (2+1) dimensionsrdquoPhysics Letters A vol 102 no 1-2 pp 15ndash17 1984

[25] H M Baskonus and H Bulut ldquoOn the complex structures ofKundu-Eckhaus equation via improved Bernoulli sub-equationfunction methodrdquo Waves in Random and Complex Media vol25 no 4 pp 720ndash728 2015

[26] FDusunceli ldquoSolutions for the drinfeld-sokolov equation usingan IBSEFM methodrdquo MSU Journal of Science vol 6 no 1 pp505ndash510 2018

[27] Z Sheng ldquoFurther Improved F-expansion method and newexact solutions of KD equationrdquo Chaos Solitons and Fractalsvol 32 no 4 pp 1375ndash1383 2007

[28] A-M Wazwaz ldquoNew kinks and solitons solutions to the(2+1) dimensional KD equationrdquo Mathematical and ComputerModelling vol 45 no 3-4 pp 473ndash479 2007

[29] SKumarAHama andABiswas ldquoSolutions ofKDequation byTravelingwave hypothesis and lie symmetry approachrdquoAppliedMathematics and Information Sciences vol 8 no 4 pp 1533ndash1539 2014

[30] L Song and H Zhang ldquoNew exact solutions for theKonopelchenko-Dubrovsky equation using an extendedRiccati equation rational expansion method and symboliccomputationrdquo Applied Mathematics and Computation vol 187no 2 pp 1373ndash1388 2007

[31] W-G Feng and C Lin ldquoExplicit exact solutions for the (2+1)dimensional KD equationrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 298ndash302 2009

[32] A Bekir ldquoApplications of the extended tanhmethod for couplednonlinear evolution equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1748ndash17572008

[33] T Xia Z Lu and H Zhang ldquoSymbolic computation andnew families of exact soliton-like solutions of Konopelchenko-Dubrovsky equationsrdquo Chaos Solitons and Fractals vol 20 no3 pp 561ndash566 2004

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

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

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Advances in Mathematical Physics 7

minus6 minus4 minus2 2x

minus10

minus05

05

10u

minus6 minus4 minus2 2x

minus15

minus10

minus05

05

10

v

u

t

0

1

2

3

0

1

2

3

5

0

minus5x x0

10

minus10

minus1

v

t

minus10

minus05

minus0500

00

05

05

10

Figure 4The 3D and 2D surfaces of the solution (25) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08 119889 = 001 119910 = 15 120598 = 09and 119905 = 0007 for 2119863

minus4 minus4minus2 0 2 4x

0655

0660

0665

0670

0675

0680

minus2 0 2 4x

1055

1060

1065

1070

1075

Re[u]

Re[u]

Re[v]

Re[v]

t

0

1

2

3

t

xx

0

1

2

3

5

minus50

5

minus50

1060

1065

1070

1075

0665

0670

Figure 5 The 3D and 2D surfaces of the analytical solution (26) (real part) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08119889 = 001 119910 = 15 120598 = 09 and 119905 = 0007 for 2119863

8 Advances in Mathematical Physics

minus4 minus2 2 4x

minus0010

minus0005

0005

0010

minus4 minus2 2 4x

minus0010

minus0005

0005

0010Im[v]Im[u]

x5

minus50

x

5

minus50

Im[v]Im[u]

t

0

1

2

3

t

0

1

2

3

minus0005

0000

0005

minus0005

0000

0005

Figure 6The 3D and 2D surfaces of the analytical solution (26) (imaginer part) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08119889 = 001 119910 = 15 120598 = 09 and 119905 = 0007 for 2119863

new traveling wave solutions to the nonlinear KD equationsWhen we compare our results with some of the existingresults in the literature we observe that the results reportedin this study especially (25) and (26) are new complex andexponential solutions All the reported solutions in this studyverify the Konopelchenko-Dubrovsky equation Our resultsmight be useful in explaining the physical meaning of variousnonlinear models arising in the field of nonlinear sciencesIBSEFM is powerful and efficient mathematical tool that canbe used to handle various nonlinear mathematical models

The performance of this method is effective and reliableIn our research these traveling wave solutions have notsubmitted to the literature in advance

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

Part of this article is presented in 3rd International Con-ference on Computational Mathematics and EngineeringSciences (CMES 2018)

Conflicts of Interest

The author declares that he has no conflicts of interest

Authorsrsquo Contributions

The author read and approved the final manuscript

Acknowledgments

This work is supported by MAUBAP18MMF020

References

[1] E M E Zayed G M Moatimid and A G Al-Nowehy ldquoThegeneralized kudryashovmethod and its applications for solvingnonlinear PDEs in mathematical physicsrdquo Scientific Journal ofMathematics Research vol 5 no 2 pp 19ndash39 2015

[2] CWang ldquoDynamic behavior of travelingwaves for the Sharma-Tasso-Olver equationrdquo Nonlinear Dynamics vol 85 no 2 pp1119ndash1126 2016

[3] T A Nofal ldquoSimple equation method for nonlinear partial dif-ferential equations and its applicationsrdquo Journal of the EgyptianMathematical Society vol 24 no 2 pp 204ndash209 2016

[4] Y Shang ldquoThe extended hyperbolic functionmethod and exactsolutions of the long-short wave resonance equationsrdquo ChaosSolitons and Fractals vol 36 no 3 pp 762ndash771 2008

Advances in Mathematical Physics 9

[5] L Wazzan ldquoA modified tanh-coth method for solving thegeneral Burgers-Fisher and the Kuramoto-Sivashinsky equa-tionsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 6 pp 2642ndash2652 2009

[6] Y-J Ren S-T Liu andH-Q Zhang ldquoA new generalized algebramethod and its application in the (2+1)-dimensional boiti-leon-pempinelli equationrdquoChaos Solitons and Fractals vol 32 no 5pp 1655ndash1665 2007

[7] C Cattani ldquoHarmonic wavelet solutions of the Schrodingerequationrdquo International Journal of Fluid Mechanics Researchvol 30 no 5 pp 463ndash472 2003

[8] C Cattani and Y Y Rushchitskii ldquoCubically nonlinear elasticwaves wave equations and methods of analysisrdquo InternationalApplied Mechanics vol 39 no 10 pp 1115ndash1145 2003

[9] A-MWazwaz andM SMehanna ldquoA variety of exact travellingwave solutions for the (2+1)-dimensional boiti-leon-pempinelliequationrdquo Applied Mathematics and Computation vol 217 no4 pp 1484ndash1490 2010

[10] W-G Feng K-M Li Y-Z Li and C Lin ldquoExplicit exactsolutions for (2+1)-dimensional boiti-leon-pempinelli equa-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2013ndash2017 2009

[11] M Eslami A Neyrame and M Ebrahimi ldquoExplicit solutionsof nonlinear (2+1)-dimensional dispersive long wave equationrdquoJournal of King SaudUniversity - Science vol 24 no 1 pp 69ndash712012

[12] M Eslami andH Rezazadeh ldquoThefirst integralmethod forWu-Zhang system with conformable time-fractional derivativerdquoCalcolo vol 53 no 3 pp 475ndash485 2016

[13] S T R Rizvi and K Ali ldquoJacobian elliptic periodic travelingwave solutions in the negative-index materialsrdquo NonlinearDynamics vol 87 no 3 pp 1967ndash1972 2017

[14] H M Baskonus T A Sulaiman and H Bulut ldquoOn the novelwave behaviors to the coupled nonlinear Maccarirsquos system withcomplex structurerdquoOptik vol 131 pp 1036ndash1043 2017

[15] H M Baskonus T A Sulaiman and H Bulut ldquoNew solitarywave solutions to the (2+1)-dimensional CalogerondashBogoyav-lenskiindashSchiff and the KadomtsevndashPetviashvili hierarchy equa-tionsrdquo Indian Journal of Physics vol 91 no 10 pp 1237ndash12432017

[16] C Cattani and A Ciancio ldquoHybrid two scales mathematicaltools for active particles modelling complex systems withlearning hiding dynamicsrdquo Mathematical Models and Methodsin Applied Sciences vol 17 no 2 pp 171ndash187 2007

[17] M H Heydari M R Hooshmandasl G B Loghmani and CCattani ldquoWavelets Galerkin method for solving stochastic heatequationrdquo International Journal of Computer Mathematics vol93 no 9 pp 1579ndash1596 2016

[18] A R Seadawy ldquoFractional solitary wave solutions of the non-linear higher-order extended KdV equation in a stratified shearflow Part IrdquoComputers andMathematics with Applications vol70 no 4 pp 345ndash352 2015

[19] A R Seadawy ldquoIon acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equa-tion in quantum plasmardquoMathematical Methods in the AppliedSciences vol 40 no 5 pp 1598ndash1607 2017

[20] A Yokus H M Baskonus T A Sulaiman and H BulutldquoNumerical simulation and solutions of the two-componentsecond order KdV evolutionary systemrdquoNumericalMethods forPartial Differential Equations vol 34 no 1 pp 211ndash227 2018

[21] J Liu M Eslami H Rezazadeh and M Mirzazadeh ldquoRationalsolutions and lump solutions to a non-isospectral and gen-eralized variable-coefficient KadomtsevndashPetviashvili equationrdquoNonlinear Dynamics vol 95 no 2 pp 1027ndash1033 2019

[22] N Zhang and T-C Xia ldquoA new negative discrete hierarchyand its N-fold darboux transformationrdquo Communications ineoretical Physics vol 68 no 6 pp 687ndash692 2017

[23] H M Baskonus T A Sulaiman and H Bulut ldquoNovel complexandhyperbolic forms to the strainwave equation inmicrostruc-tured solidsrdquo Optical and Quantum Electronics vol 50 no 142018

[24] B G Konopelchenko and VG Dubrovsky ldquoSome new inte-grable nonlinear evolution equations in (2+1) dimensionsrdquoPhysics Letters A vol 102 no 1-2 pp 15ndash17 1984

[25] H M Baskonus and H Bulut ldquoOn the complex structures ofKundu-Eckhaus equation via improved Bernoulli sub-equationfunction methodrdquo Waves in Random and Complex Media vol25 no 4 pp 720ndash728 2015

[26] FDusunceli ldquoSolutions for the drinfeld-sokolov equation usingan IBSEFM methodrdquo MSU Journal of Science vol 6 no 1 pp505ndash510 2018

[27] Z Sheng ldquoFurther Improved F-expansion method and newexact solutions of KD equationrdquo Chaos Solitons and Fractalsvol 32 no 4 pp 1375ndash1383 2007

[28] A-M Wazwaz ldquoNew kinks and solitons solutions to the(2+1) dimensional KD equationrdquo Mathematical and ComputerModelling vol 45 no 3-4 pp 473ndash479 2007

[29] SKumarAHama andABiswas ldquoSolutions ofKDequation byTravelingwave hypothesis and lie symmetry approachrdquoAppliedMathematics and Information Sciences vol 8 no 4 pp 1533ndash1539 2014

[30] L Song and H Zhang ldquoNew exact solutions for theKonopelchenko-Dubrovsky equation using an extendedRiccati equation rational expansion method and symboliccomputationrdquo Applied Mathematics and Computation vol 187no 2 pp 1373ndash1388 2007

[31] W-G Feng and C Lin ldquoExplicit exact solutions for the (2+1)dimensional KD equationrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 298ndash302 2009

[32] A Bekir ldquoApplications of the extended tanhmethod for couplednonlinear evolution equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1748ndash17572008

[33] T Xia Z Lu and H Zhang ldquoSymbolic computation andnew families of exact soliton-like solutions of Konopelchenko-Dubrovsky equationsrdquo Chaos Solitons and Fractals vol 20 no3 pp 561ndash566 2004

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

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

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Advances in Mathematical Physics

minus4 minus2 2 4x

minus0010

minus0005

0005

0010

minus4 minus2 2 4x

minus0010

minus0005

0005

0010Im[v]Im[u]

x5

minus50

x

5

minus50

Im[v]Im[u]

t

0

1

2

3

t

0

1

2

3

minus0005

0000

0005

minus0005

0000

0005

Figure 6The 3D and 2D surfaces of the analytical solution (26) (imaginer part) by considering the values 1198871 = 03 1198861 = 02 119896 = 16 119888 = 08119889 = 001 119910 = 15 120598 = 09 and 119905 = 0007 for 2119863

new traveling wave solutions to the nonlinear KD equationsWhen we compare our results with some of the existingresults in the literature we observe that the results reportedin this study especially (25) and (26) are new complex andexponential solutions All the reported solutions in this studyverify the Konopelchenko-Dubrovsky equation Our resultsmight be useful in explaining the physical meaning of variousnonlinear models arising in the field of nonlinear sciencesIBSEFM is powerful and efficient mathematical tool that canbe used to handle various nonlinear mathematical models

The performance of this method is effective and reliableIn our research these traveling wave solutions have notsubmitted to the literature in advance

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

Part of this article is presented in 3rd International Con-ference on Computational Mathematics and EngineeringSciences (CMES 2018)

Conflicts of Interest

The author declares that he has no conflicts of interest

Authorsrsquo Contributions

The author read and approved the final manuscript

Acknowledgments

This work is supported by MAUBAP18MMF020

References

[1] E M E Zayed G M Moatimid and A G Al-Nowehy ldquoThegeneralized kudryashovmethod and its applications for solvingnonlinear PDEs in mathematical physicsrdquo Scientific Journal ofMathematics Research vol 5 no 2 pp 19ndash39 2015

[2] CWang ldquoDynamic behavior of travelingwaves for the Sharma-Tasso-Olver equationrdquo Nonlinear Dynamics vol 85 no 2 pp1119ndash1126 2016

[3] T A Nofal ldquoSimple equation method for nonlinear partial dif-ferential equations and its applicationsrdquo Journal of the EgyptianMathematical Society vol 24 no 2 pp 204ndash209 2016

[4] Y Shang ldquoThe extended hyperbolic functionmethod and exactsolutions of the long-short wave resonance equationsrdquo ChaosSolitons and Fractals vol 36 no 3 pp 762ndash771 2008

Advances in Mathematical Physics 9

[5] L Wazzan ldquoA modified tanh-coth method for solving thegeneral Burgers-Fisher and the Kuramoto-Sivashinsky equa-tionsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 6 pp 2642ndash2652 2009

[6] Y-J Ren S-T Liu andH-Q Zhang ldquoA new generalized algebramethod and its application in the (2+1)-dimensional boiti-leon-pempinelli equationrdquoChaos Solitons and Fractals vol 32 no 5pp 1655ndash1665 2007

[7] C Cattani ldquoHarmonic wavelet solutions of the Schrodingerequationrdquo International Journal of Fluid Mechanics Researchvol 30 no 5 pp 463ndash472 2003

[8] C Cattani and Y Y Rushchitskii ldquoCubically nonlinear elasticwaves wave equations and methods of analysisrdquo InternationalApplied Mechanics vol 39 no 10 pp 1115ndash1145 2003

[9] A-MWazwaz andM SMehanna ldquoA variety of exact travellingwave solutions for the (2+1)-dimensional boiti-leon-pempinelliequationrdquo Applied Mathematics and Computation vol 217 no4 pp 1484ndash1490 2010

[10] W-G Feng K-M Li Y-Z Li and C Lin ldquoExplicit exactsolutions for (2+1)-dimensional boiti-leon-pempinelli equa-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2013ndash2017 2009

[11] M Eslami A Neyrame and M Ebrahimi ldquoExplicit solutionsof nonlinear (2+1)-dimensional dispersive long wave equationrdquoJournal of King SaudUniversity - Science vol 24 no 1 pp 69ndash712012

[12] M Eslami andH Rezazadeh ldquoThefirst integralmethod forWu-Zhang system with conformable time-fractional derivativerdquoCalcolo vol 53 no 3 pp 475ndash485 2016

[13] S T R Rizvi and K Ali ldquoJacobian elliptic periodic travelingwave solutions in the negative-index materialsrdquo NonlinearDynamics vol 87 no 3 pp 1967ndash1972 2017

[14] H M Baskonus T A Sulaiman and H Bulut ldquoOn the novelwave behaviors to the coupled nonlinear Maccarirsquos system withcomplex structurerdquoOptik vol 131 pp 1036ndash1043 2017

[15] H M Baskonus T A Sulaiman and H Bulut ldquoNew solitarywave solutions to the (2+1)-dimensional CalogerondashBogoyav-lenskiindashSchiff and the KadomtsevndashPetviashvili hierarchy equa-tionsrdquo Indian Journal of Physics vol 91 no 10 pp 1237ndash12432017

[16] C Cattani and A Ciancio ldquoHybrid two scales mathematicaltools for active particles modelling complex systems withlearning hiding dynamicsrdquo Mathematical Models and Methodsin Applied Sciences vol 17 no 2 pp 171ndash187 2007

[17] M H Heydari M R Hooshmandasl G B Loghmani and CCattani ldquoWavelets Galerkin method for solving stochastic heatequationrdquo International Journal of Computer Mathematics vol93 no 9 pp 1579ndash1596 2016

[18] A R Seadawy ldquoFractional solitary wave solutions of the non-linear higher-order extended KdV equation in a stratified shearflow Part IrdquoComputers andMathematics with Applications vol70 no 4 pp 345ndash352 2015

[19] A R Seadawy ldquoIon acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equa-tion in quantum plasmardquoMathematical Methods in the AppliedSciences vol 40 no 5 pp 1598ndash1607 2017

[20] A Yokus H M Baskonus T A Sulaiman and H BulutldquoNumerical simulation and solutions of the two-componentsecond order KdV evolutionary systemrdquoNumericalMethods forPartial Differential Equations vol 34 no 1 pp 211ndash227 2018

[21] J Liu M Eslami H Rezazadeh and M Mirzazadeh ldquoRationalsolutions and lump solutions to a non-isospectral and gen-eralized variable-coefficient KadomtsevndashPetviashvili equationrdquoNonlinear Dynamics vol 95 no 2 pp 1027ndash1033 2019

[22] N Zhang and T-C Xia ldquoA new negative discrete hierarchyand its N-fold darboux transformationrdquo Communications ineoretical Physics vol 68 no 6 pp 687ndash692 2017

[23] H M Baskonus T A Sulaiman and H Bulut ldquoNovel complexandhyperbolic forms to the strainwave equation inmicrostruc-tured solidsrdquo Optical and Quantum Electronics vol 50 no 142018

[24] B G Konopelchenko and VG Dubrovsky ldquoSome new inte-grable nonlinear evolution equations in (2+1) dimensionsrdquoPhysics Letters A vol 102 no 1-2 pp 15ndash17 1984

[25] H M Baskonus and H Bulut ldquoOn the complex structures ofKundu-Eckhaus equation via improved Bernoulli sub-equationfunction methodrdquo Waves in Random and Complex Media vol25 no 4 pp 720ndash728 2015

[26] FDusunceli ldquoSolutions for the drinfeld-sokolov equation usingan IBSEFM methodrdquo MSU Journal of Science vol 6 no 1 pp505ndash510 2018

[27] Z Sheng ldquoFurther Improved F-expansion method and newexact solutions of KD equationrdquo Chaos Solitons and Fractalsvol 32 no 4 pp 1375ndash1383 2007

[28] A-M Wazwaz ldquoNew kinks and solitons solutions to the(2+1) dimensional KD equationrdquo Mathematical and ComputerModelling vol 45 no 3-4 pp 473ndash479 2007

[29] SKumarAHama andABiswas ldquoSolutions ofKDequation byTravelingwave hypothesis and lie symmetry approachrdquoAppliedMathematics and Information Sciences vol 8 no 4 pp 1533ndash1539 2014

[30] L Song and H Zhang ldquoNew exact solutions for theKonopelchenko-Dubrovsky equation using an extendedRiccati equation rational expansion method and symboliccomputationrdquo Applied Mathematics and Computation vol 187no 2 pp 1373ndash1388 2007

[31] W-G Feng and C Lin ldquoExplicit exact solutions for the (2+1)dimensional KD equationrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 298ndash302 2009

[32] A Bekir ldquoApplications of the extended tanhmethod for couplednonlinear evolution equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1748ndash17572008

[33] T Xia Z Lu and H Zhang ldquoSymbolic computation andnew families of exact soliton-like solutions of Konopelchenko-Dubrovsky equationsrdquo Chaos Solitons and Fractals vol 20 no3 pp 561ndash566 2004

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

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

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Advances in Mathematical Physics 9

[5] L Wazzan ldquoA modified tanh-coth method for solving thegeneral Burgers-Fisher and the Kuramoto-Sivashinsky equa-tionsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 6 pp 2642ndash2652 2009

[6] Y-J Ren S-T Liu andH-Q Zhang ldquoA new generalized algebramethod and its application in the (2+1)-dimensional boiti-leon-pempinelli equationrdquoChaos Solitons and Fractals vol 32 no 5pp 1655ndash1665 2007

[7] C Cattani ldquoHarmonic wavelet solutions of the Schrodingerequationrdquo International Journal of Fluid Mechanics Researchvol 30 no 5 pp 463ndash472 2003

[8] C Cattani and Y Y Rushchitskii ldquoCubically nonlinear elasticwaves wave equations and methods of analysisrdquo InternationalApplied Mechanics vol 39 no 10 pp 1115ndash1145 2003

[9] A-MWazwaz andM SMehanna ldquoA variety of exact travellingwave solutions for the (2+1)-dimensional boiti-leon-pempinelliequationrdquo Applied Mathematics and Computation vol 217 no4 pp 1484ndash1490 2010

[10] W-G Feng K-M Li Y-Z Li and C Lin ldquoExplicit exactsolutions for (2+1)-dimensional boiti-leon-pempinelli equa-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2013ndash2017 2009

[11] M Eslami A Neyrame and M Ebrahimi ldquoExplicit solutionsof nonlinear (2+1)-dimensional dispersive long wave equationrdquoJournal of King SaudUniversity - Science vol 24 no 1 pp 69ndash712012

[12] M Eslami andH Rezazadeh ldquoThefirst integralmethod forWu-Zhang system with conformable time-fractional derivativerdquoCalcolo vol 53 no 3 pp 475ndash485 2016

[13] S T R Rizvi and K Ali ldquoJacobian elliptic periodic travelingwave solutions in the negative-index materialsrdquo NonlinearDynamics vol 87 no 3 pp 1967ndash1972 2017

[14] H M Baskonus T A Sulaiman and H Bulut ldquoOn the novelwave behaviors to the coupled nonlinear Maccarirsquos system withcomplex structurerdquoOptik vol 131 pp 1036ndash1043 2017

[15] H M Baskonus T A Sulaiman and H Bulut ldquoNew solitarywave solutions to the (2+1)-dimensional CalogerondashBogoyav-lenskiindashSchiff and the KadomtsevndashPetviashvili hierarchy equa-tionsrdquo Indian Journal of Physics vol 91 no 10 pp 1237ndash12432017

[16] C Cattani and A Ciancio ldquoHybrid two scales mathematicaltools for active particles modelling complex systems withlearning hiding dynamicsrdquo Mathematical Models and Methodsin Applied Sciences vol 17 no 2 pp 171ndash187 2007

[17] M H Heydari M R Hooshmandasl G B Loghmani and CCattani ldquoWavelets Galerkin method for solving stochastic heatequationrdquo International Journal of Computer Mathematics vol93 no 9 pp 1579ndash1596 2016

[18] A R Seadawy ldquoFractional solitary wave solutions of the non-linear higher-order extended KdV equation in a stratified shearflow Part IrdquoComputers andMathematics with Applications vol70 no 4 pp 345ndash352 2015

[19] A R Seadawy ldquoIon acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equa-tion in quantum plasmardquoMathematical Methods in the AppliedSciences vol 40 no 5 pp 1598ndash1607 2017

[20] A Yokus H M Baskonus T A Sulaiman and H BulutldquoNumerical simulation and solutions of the two-componentsecond order KdV evolutionary systemrdquoNumericalMethods forPartial Differential Equations vol 34 no 1 pp 211ndash227 2018

[21] J Liu M Eslami H Rezazadeh and M Mirzazadeh ldquoRationalsolutions and lump solutions to a non-isospectral and gen-eralized variable-coefficient KadomtsevndashPetviashvili equationrdquoNonlinear Dynamics vol 95 no 2 pp 1027ndash1033 2019

[22] N Zhang and T-C Xia ldquoA new negative discrete hierarchyand its N-fold darboux transformationrdquo Communications ineoretical Physics vol 68 no 6 pp 687ndash692 2017

[23] H M Baskonus T A Sulaiman and H Bulut ldquoNovel complexandhyperbolic forms to the strainwave equation inmicrostruc-tured solidsrdquo Optical and Quantum Electronics vol 50 no 142018

[24] B G Konopelchenko and VG Dubrovsky ldquoSome new inte-grable nonlinear evolution equations in (2+1) dimensionsrdquoPhysics Letters A vol 102 no 1-2 pp 15ndash17 1984

[25] H M Baskonus and H Bulut ldquoOn the complex structures ofKundu-Eckhaus equation via improved Bernoulli sub-equationfunction methodrdquo Waves in Random and Complex Media vol25 no 4 pp 720ndash728 2015

[26] FDusunceli ldquoSolutions for the drinfeld-sokolov equation usingan IBSEFM methodrdquo MSU Journal of Science vol 6 no 1 pp505ndash510 2018

[27] Z Sheng ldquoFurther Improved F-expansion method and newexact solutions of KD equationrdquo Chaos Solitons and Fractalsvol 32 no 4 pp 1375ndash1383 2007

[28] A-M Wazwaz ldquoNew kinks and solitons solutions to the(2+1) dimensional KD equationrdquo Mathematical and ComputerModelling vol 45 no 3-4 pp 473ndash479 2007

[29] SKumarAHama andABiswas ldquoSolutions ofKDequation byTravelingwave hypothesis and lie symmetry approachrdquoAppliedMathematics and Information Sciences vol 8 no 4 pp 1533ndash1539 2014

[30] L Song and H Zhang ldquoNew exact solutions for theKonopelchenko-Dubrovsky equation using an extendedRiccati equation rational expansion method and symboliccomputationrdquo Applied Mathematics and Computation vol 187no 2 pp 1373ndash1388 2007

[31] W-G Feng and C Lin ldquoExplicit exact solutions for the (2+1)dimensional KD equationrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 298ndash302 2009

[32] A Bekir ldquoApplications of the extended tanhmethod for couplednonlinear evolution equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1748ndash17572008

[33] T Xia Z Lu and H Zhang ldquoSymbolic computation andnew families of exact soliton-like solutions of Konopelchenko-Dubrovsky equationsrdquo Chaos Solitons and Fractals vol 20 no3 pp 561ndash566 2004

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

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

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

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

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom