Research ArticleA Truncation Method for Solving the Time-FractionalBenjamin-Ono Equation

Mohamed R Ali

Department of Mathematics Benha Faculty of Engineering Benha University Benha Egypt

Correspondence should be addressed to Mohamed R Ali mohamedredabhitbuedueg

Received 12 December 2018 Revised 2 March 2019 Accepted 25 March 2019 Published 2 May 2019

Academic Editor Mustafa Inc

Copyright copy 2019 Mohamed R Ali 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

We deem the time-fractional Benjamin-Ono (BO) equation out of the RiemannndashLiouville (RL) derivative by applying the Liesymmetry analysis (LSA) By first using prolongation theorem to investigate its similarity vectors and then using these generators totransform the time-fractional BO equation to a nonlinear ordinary differential equation (NLODE) of fractional order we completethe solutions by utilizing the power series method (PSM)

1 Introduction

Lie symmetry method provides an effective tool for derivingthe analytic solutions of the nonlinear partial differentialequations (NLPDEs) [1ndash4] In recent years many authorshave studied the nonlinear fractional differential equations(NLFDEs) because these equations express many nonlinearphysical phenomena and dynamic forms in physics electro-chemistry and viscoelasticity [5ndash9]

Time-fractional NLDEs arise from classical NLPDEs byreplacing its time derivative with the fractional derivativeThe methods applied to derive the analytic solutions of NLF-PDEs are the exp-function the 1198661015840119866 expansion fractionalsu-equation Lie symmetry method and many more [10ndash19]

The one-dimensional Benjamin-Ono equation is consid-ered here as follows (see [20])

119906119905 + ℎ119906119909119909 + 119906119906119909 = 0 (1)

In fact the BO equation describes one-dimensional internalwaves in deep water We consider LSA for the analyticsolutions by using PS expansion for the time-fractional BOequation

119906120572119905 + ℎ119906119909119909 + 119906119906119909 = 0 0 lt 120572 lt 1 (2)

In division 2 of this paper some basic properties of theRiemannndashLiouville fractional derivative are shown firstly

and then the Lie group method for FPDEs is presentedIn division 3 the Lie group to the time-fractional BOequation (FBO) and the symmetry reductions are deter-mined In division 4 we derive anew arrangement of theFBO equation (2) via the PSM In division 5 we study theconvergence for the series solution We conclude our work indivision 6

2 Notations and Delineations

21 Description of Lie Symmetry Reduction Method for NLF-PDEs We present the principal notations and definitionsthat detecting the symmetries of the NLFPDEs

Here the time-fractional NLFPDEs are

120597120572119905 119906 = 119865 (119905 119909 119906 119906119909 119906119909119909 ) (3)

Suppose that the infinitesimal vector119883 has the form

119883 = 1205851 (119909 119905 119906) 120597120597119909 + 1205852 (119909 119905 119906) 120597120597119905 + 120578 (119909 119905 119906) 120597120597119906 (4)

The Lie group parameter of infinitesimal transformations [821 22] has the formula

HindawiJournal of Applied MathematicsVolume 2019 Article ID 3456848 7 pageshttpsdoiorg10115520193456848

2 Journal of Applied Mathematics

119909 = 119909 + 1205761205851 (119905 119909 119906) + 119874 (1205762) 119905 = 119905 + 1205761205852 (119905 119909 119906) + 119874 (1205762) 119906 = 119906 + 120576120578 (119905 119909 119906) + 119874 (1205762)

120597120572119906120597119905120572 = 120597120572119906120597119905120572 + 1205761205780120572 (119905 119909 119906) + 119874 (1205762) 120597119906120597119909 = 120597119906120597119909 + 120576120578119909 (119905 119909 119906) + 119874 (1205762) 12059721199061205971199092 = 12059721199061205971199092 + 120576120578119909119909 (119905 119909 119906) + 119874 (1205762)

(5)

where 1205851 1205852 and 120578 are considered as the infinitesimals ofthe transformationrsquos variables (119909 119905 119906) respectively and 120576ń1is considered as the group parameter we will take it to beequal to oneThe explicit expressions of 120578119909 and 120578119909119909 which weconsider as the prolongation of the infinitesimals are givenby

120578119909 = 119863119909 (120578) minus 119906119909119863119909 (1205851) minus 119906119905119863119905 (1205852) (6)

and

120578119909119909 = 119863119909 (120578119909) minus 119906119909119905119863119909 (1205851) minus 119906119909119909119863119905 (1205852) (7)

where119863119909 is in [8] assigned as

119863119909 = 120597120597119909 + 119906119909 120597120597119909 + 119906119909119909 120597120597119906119909 + (8)

Theorem 1 Equation (2) coincides with a one-parametergroup of transformations (5) with the infinitesimal generator Xif and only if the accompanying infinitesimal conditions holdstrue

119875119903(1205722) 119883(Δ)|Δ=0 = 0 (9)

whereΔ = 119863120572119905 119906minus119865(119905 119909 119906 119906119909 119906119909119909 ) and 119875119903 is the secondprolongation of the infinitesimal generator119883Definition 2 The prolonged vector is demonstrated by

119875119903(119899)119883 = 119883 + 119901sum119894=1

119902sum120572=1

120585120572119894 120597120597119906120572119894 + + 119901sum1198951=1

119901sum119895119899=1

119902sum120572=1

1205851205721198951 119895119899 1205971205971199061205721198951 119895119899(10)

where 119902 is the number of dependent variables 119901 is thenumber of independent variables 1205971205971199061205721198951 = 120597120597119906120572119909 and thePDE involves derivatives of up to the order 119899 The condition[21ndash23] is given by

1205852 (119905 119909 119906)|119905=0 = 0 (11)

Lemma 3 The function 119906 = 120579(119909 119905) is an invariant solution of(3) if and only if

(i) 1205852 (119909 119905 120579) 120579119905 + 1205851(119909 119905 120579) 120579119909 = 120578 (119909 119905 120579)Lemma 4 The 120572119905ℎ extended infinitesimal [24 25] for thefractional derivative part utilizing the RL definition with (11)is given by

1205780120572 = 120597120572120578120597119905120572 + (120578119906 minus 120572119863119905 (1205852)) 120597120572119906120597119905120572 minus 119906120597

120572120578119906120597119905120572 + 120583minus infinsum119899=1

(120572119899)119863119899119905 (1205851)119863120572minus119899119905 (119906119909) + infinsum119899=1

[(120572119899) 120597120572120578119906120597119905120572minus [( 120572

119899 + 1)119863119899+1119905 (1205852)]119863120572minus119899119905 (119906) (12)

where

120583 = infinsum119899=2

119899sum119898=2

119898sum119896=2

119896minus1sum119903=2

(120572119899)(119899119898)(

119896119903) 1119896

sdot 119905119899minus120572Γ (119899 + 1 minus 120572) [minus119906]119903 120597119898

120597119905119898 [119906119896minus119903] 120597119899minus119898+119896120597119905119899minus119898120597119906119896(13)

Remember that

(120572119899) = (minus1)119899minus1 120572Γ (119899 minus 120572)Γ (1 minus 120572) Γ (119899 + 1) (14)

3 Reduction of Time-FractionalBenjamin-Ono Equation

We use the LSA to find the similarity solution for 1D time-factional BO equation (1) Suppose that (2) is an invariantunder (5) so that we have

119906120572119905 + ℎ119906119909119909 + 119906 119906119909 = 0 (15)

Thus 119906(119909 119905) satisfies (2) Applying the second prolongationto (2) symmetry invariant equation is

1205780120572 + 119906120578119909 + ℎ120578119909119909 + 119906119909120578 = 0 (16)

Substituting the values from (6) (7) and (12) into (16) andisolating coefficients in partial derivatives regarding 119909 andpower of 119906 we have

(120572119899) 120597119899119905 120578119906 minus (120572

119899 + 1)119863119899+1119905 (1205852) = 0 119899 = 1 2 3 1205851199062 = 1205851199092 = 1205851199061 = 1205851199051 = 120578119906119906 = 0

Journal of Applied Mathematics 3

1205721205851199052 minus 21205851199091 = 0ℎ120578119909119909 minus 119906120597120572119905 120578119906 + 120597120572119905 120578 + 119906120578119909 = 01205851199091199091 minus 2120578119909119906 = 0

(17)

Solving the obtained determining equation we get

1205851 = 1198882 + 12057211990911988811205852 = 21199051198881120578 = minus1205721199061198881

(18)

where 1198881 and 1198882 are constants for simplicity We take theirvalues equal to one So (2) has two vector fields that cangenerate its infinitesimal symmetry These Lie vectors areconsidered as follows

1198831 = 120597120597119909 (19)

1198832 = 120572119909 120597120597119909 + 2119905 120597120597119905 minus 120572119906 120597120597119906 (20)

Case 1 For (19) we have

1198891199091 = 1198891199050 = 1198891199060 (21)

Solving this equation 119906 = 119891(119905) Putting 119906 = 119891(119905) into (1) weget

119863120572119905 119891 (119905) = 0 (22)

where 119906 = 1198861119905120572minus1Case 2 For1198832 in (20) we have

119889119909120572119909 = 1198891199052119905 = minus119889119906120572119906 (23)

This is the characteristic equation By solving it the resultingsimilarity variable in the form

1199111 = 119909119905minus12057221199112 = 119906119905minus1205722 (24)

The variables transformation is as follows

119906 = 119905minus1205722119891 (120585) 120585 = 119909119905minus1205722 (25)

where 119891(120585) is a function in one variable 120585 We use (25) totransform (2) into a fractional ODE

Theorem 5 Transformation (25) reduces (2) to the nonlinearFODE as follows

(1198751minus312057221205723120572 119891) (120585) + ℎ119891120585120585 + 119891119891120585 = 0 (26)

utilizing the Erdelyi-Kober (EK) fractional derivative operator[20]

(1198751205852120572120573 119891) (120585)= 119899minus1prod119895=0

(1205852 + 119895 minus 1120573 119889119889120585) (1198701205852+120572119899minus120572120573 119891) (120585) (27)

where

(1198701205852120572120573 119891) (120585)

=

1Γ (120572) intinfin

1(119906 minus 1)120572minus1 119906minus(1205852+120572)119891 (1205851199061120573) 119889119906 120572 gt 0

119891 (120585) 120572 = 0(28)

and

119899 = [120572] + 1 120572 = 119873120572 120572 isin 119873 (29)

Proof Utilizing the definition of the RL fractional derivativein (25) we get

120597120572119906120597119905120572 = 120597119899120597119905119899 [ 1Γ (119899 minus 120572)sdot int1199051(119905 minus 119904)119899minus120572minus1 119904minus1205722119891 (119909119904minus(1205722)) 119889119904]

119899 minus 1 lt 120572 lt 1 119899 = 1 2 (30)

Assume that V = 119905119904 119889119904 = minus(119905V2)119889V Thus (30) becomes

120597120572119906120597119905120572 = 120597119899120597119905119899 [119905119899minus31205722 1Γ (119899 minus 120572)sdot intinfin0(V minus 1)119899minus120572minus1 Vminus(119899+1minus31205722)119891 (120585V(1205722)) 119889V]

(31)

4 Journal of Applied Mathematics

Applying the EK fractional integral operator (28) in (31) wehave

120597120572119906120597119905120572 = 120597119899120597119905119899 [119905119899minus31205722 (1198701minus1205722119899minus1205722120572 119891) (120585)] (32)

For simplicity we consider= 119909 119905minus1205722120601 isin (0infin)We thus findthat

119905 120597120597119905120601 (120585) = 119905119909 (minus1205722 ) 119905minus1205722minus1 120601 (120585) = minus1205722 120585 120597120597120585120601 (120585) (33)

Hence we have

120597119899120597119905119899 [119905119899minus31205722 (1198701minus1205722119899minus1205722120572 119891) (120585)] = 120597119899minus1120597119905119899minus1 [ 120597120597119905sdot 119905119899minus31205722 (1198701minus1205722119899minus1205722120572 119891) (120585)]= 120597119899minus1120597119905119899minus1 [119905119899minus31205722minus1 (119899 minus 31205722 minus 1205722 120585 120597120597120585)sdot (1198701minus1205722119899minus1205722120572 119891) (120585)]

(34)

Repeating 119899 minus 1 times we have

120597119899120597119905119899 [119905119899minus31205722 (1198701minus1205722119899minus1205722120572 119891) (120585)] = 120597119899minus1120597119905119899minus1 [ 120597120597119905sdot 119905119899minus31205722 (1198701minus1205722119899minus1205722120572 119891) (120585)] = 120597119899minus1120597119905119899minus1 [119905119899minus31205722minus1 (119899minus 31205722 minus 1205722 120585 120597120597120585) (1198701minus1205722119899minus1205722120572 119891) (120585)]

= 119905minus1205722119899minus1prod119895=0

[(1 minus 31205722 + 119895 minus 1205722 120585 120597120597120585) (1198701minus1205722119899minus1205722120572 119891)sdot (120585)

(35)

Applying the EK fractional differential operator (27) in (35)we get

120597119899120597119905119899 [(119905119899minus1205723 (1198701minus1205722119899minus1205722120572 119891) (120585))]= 119905minus1205722 (1198751minus312057221205722120572 119891) (120585)

(36)

Substituting (36) into (32) we get

120597120572119906120597119905120572 = 119905minus1205722 (1198751minus312057221205722120572 119891) (120585) (37)

Thus (2) is reduced to a fractional-order ODE as follows

(1198751minus312057221205722120572 119891) (120585) + 119891119891120585 + ℎ119891120585120585 = 0 (38)

4 The Explicit Solution forthe Time-Fractional Benjamin-OnoEquation by Using PSM

The analytic solutions via PSM [26] are demonstrated Weassume that

119891 (120585) = infinsum119899=0

119886119899120601 (120585)119899 (39)

Differentiating (39) twice regarding 120585 we get1198911015840 (120585) = infinsum

119899=0

119899119886119899120601 (120585)119899minus1 (40)

and

11989110158401015840 (120585) = infinsum119899=0

119899 (119899 minus 1) 119886119899120601 (120585)119899minus2 (41)

Substituting (39) (40) and (41) into (38) we have

infinsum119899=0

Γ (2 minus 31205722 minus 1198991205722)Γ (2 minus 1205722 minus 1198991205722) 119886119899120601 (120585)119899

+ infinsum119899=0

119886119899120601 (120585)119899 infinsum119899=0

(119899 + 1) 119886119899+1120601 (120585)119899

+ ℎinfinsum119899=0

(119899 + 2) (119899 + 1) 119886119899+2120601 (120585)119899 = 0

(42)

Comparing coefficients in (42) when 119899 = 0 we obtain1198862 = minus12ℎ (Γ (2 minus 31205722)Γ (2 minus 1205722) 1198860 + 11988601198861) (43)

Journal of Applied Mathematics 5

When 119899 ge 1 the recurrence relations between the seriescoefficients are

119886119899+2 = minus12ℎ (119899 + 2) (119899 + 1) (Γ (2 minus 31205722 minus 1198991205722)Γ (2 minus 1205722 minus 1198991205722) 119886119899+ (119899 + 1) 119886119899119886119899+1)

(44)

Using (44) the series solution for (39) can be represented bysubstituting (43) and (44) into (39)

119891 (120585) = 1198860 + 1198861120585 + 11988621205852 + infinsum119899=1

119886119899+2120585119899+2 = 1198860 + 1198861120585

minus 12ℎ (Γ (2 minus 31205722)Γ (2 minus 1205722) 1198860 + 11988601198861) 1205852

minus infinsum119899=1

12ℎ (119899 + 2) (119899 + 1) (Γ (2 minus 31205722 minus 1198991205722)Γ (2 minus 1205722 minus 1198991205722) 119886119899+ (119899 + 1) 119886119899119886119899+1) 120585119899+2

(45)

Upon substitution using similarity variables in (25) thefollowing explicit solutions for (2) are

119906 (119909 119905) = 1198860119905minus1205722 + 1198861119909119905minus120572 minus 12ℎ (Γ (2 minus 31205722)Γ (2 minus 1205722) 1198860+ 11988601198861) 119905minus1205722 (119909119905minus1205722)2

minus infinsum119899=1

12ℎ (119899 + 2) (119899 + 1) Γ (2 minus 31205722 minus 1198991205722)Γ (2 minus 1205722 minus 1198991205722) 119886119899

+ 119899sum119896=0

119896sum119895=0

(119886119895+119886119896minus119895) 119886119899minus119896 (119899 + 1)(119909)119899+2

sdot 119905minus120572(2119899+120572120573)2120573

(46)

5 Convergence Analysis

To satisfy the convergence test there are many kinds of testsas the ratio the comparison and the quotient tests Theconvergence of the solution equation (46) will be presentedas follows We revamp (46) as follows

1003816100381610038161003816119886119899+21003816100381610038161003816 le (|Γ (2 minus 31205722 minus 1198991205722)||Γ (2 minus 1205722 minus 1198991205722)| 10038161003816100381610038161198861198991003816100381610038161003816minus 119899sum119896=0

119896sum119895=0

1003816100381610038161003816100381611988611989510038161003816100381610038161003816 10038161003816100381610038161003816119886119896minus11989510038161003816100381610038161003816 1003816100381610038161003816119886119899minus1198961003816100381610038161003816 minus 10038161003816100381610038161198861198991003816100381610038161003816)(47)

Equation (47) utilizing the Gamma function shows that|Γ(2 minus 31205722 minus 1198991205722)||Γ(2 minus 1205722 minus 1198991205722)| lt 1 for arbitrary119899 that1003816100381610038161003816119886119899+21003816100381610038161003816 le 119872(10038161003816100381610038161198861198991003816100381610038161003816 minus 119899sum

119896=0

119896sum119895=0

1003816100381610038161003816100381611988611989510038161003816100381610038161003816 10038161003816100381610038161003816119886119896minus11989510038161003816100381610038161003816 1003816100381610038161003816119886119899minus1198961003816100381610038161003816 minus 10038161003816100381610038161198861198991003816100381610038161003816) (48)

where119872 = max(|1198881| |1198882|) We now assume another form ofthe PSM

119861 (120585) = infinsum119899=0

119888119899120585119899 (49)

By comparing the two series we can observe that |119888119899| le 119886119899119899 = 0 1 Hence the series119861(120585) = suminfin119899=0 119888119899120585119899 is themajorantseries of (47) So we find that

119861 (120585) = 1198880 + 1198881120585+119872(infinsum

119899=0

1198881198991205852119861 (120585) + infinsum119899=0

119899sum119896=0

119896sum119895=0

119888119895119888119896minus119895119888119899minus119896 + infinsum119899=0

119888119899)sdot 120585119899+2

(50)

Consider an implicit functional system regarding 120585 as follows120573 (120585 119861)

= 119861 minus 1198880 minus 1198881120585 minus 11988821205852minus119872(1205852119861 (120585) + 2119861 (120585)2 + (1205852 minus 1198881120585 minus 31198880) 119861)

(51)

since 120573 is analytic in a neighborhood of (0 1198880) where120573(0 1198880) = 0 and (120597120597119861)120573(0 1198880) = 0 Then the series 119861(120585) =suminfin119899=0 120585119899 is analytic around (0 1198880) and this is verified utilizing[27] and the radius of convergence of this series belongs toa positive domain This shows that (46) converges around(0 1198880)6 Physical Performance of the Power SeriesTechnique for Eqs (46)

To have expressed and convenient conception of the physicalcharacteristic of the power series solution the 3D plots forthe explicit solution equations (46) is plotted in Figures 1ndash4 atℎ = 1 by utilizing appropriate parameter formsThe spectaclevision of the real portion of (46) can be visible in the 3D plotsproof in Figures 1 2 3 and 4 respectively

6 Journal of Applied Mathematics

2002

2001

2

1999

0 0

4 4

xt

Figure 1 3D plot of (46) with 1198860 = 19 1198861 = 17 1198862 = 077 120574 = 1120572 = 08 120573 = 2

u(x

t)

2002

2001

2

1999

0 0

4 4xt

Figure 2 3D plot of (46) with 1198860 = 39 1198861 = 17 1198862 = 087 120574 = 1120572 = 06 120573 = 1

7 Conclusions

Lie point symmetry properties of (1 + 1)-dimensional time-fractional Benjamin-Ono equation have been consideredwith the RiemannndashLiouville fractional derivativeThese sym-metries are used here to transform the FPDEs intoNLFODEsClosed-form solutions are determined by using PSM inthe last division The accuracy exhibits the assembly ofthe solution Considerable frames for the acquired explicitsolutions were approached

2002

2001

2

1999

0 0

4 4

xt

u(x t)

Figure 3 3D plot of (46) with 1198860 = 1 1198861 = 11 1198862 = 057 120574 = 1120572 = 05 120573 = 07u(x t)

2002

2001

2

1999

0 0

4 4

xt

Figure 4 3D plot of (46) with 1198860 = 1 1198861 = 1 1198862 = 04 120574 = 1120572 = 03 120573 = 04

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Applied Mathematics 7

Conflicts of Interest

The author declares that they have no conflicts of interest

References

[1] M Bruzon et al ldquoThe symmetry reductions of a turbulencemodelrdquo Journal of Physics A Mathematical and General vol 34no 18 p 3751 2001

[2] R Sadat and M Kassem ldquoExplicit Solutions for the (2+ 1)-dimensional jaulentndashmiodek equation using the integratingfactors method in an unbounded domainrdquo Mathematical ampComputational Applications vol 23 no 1 p 15 2018

[3] A Paliathanasis andMTsamparlis ldquoLie symmetries for systemsof evolution equationsrdquo Journal of Geometry and Physics vol124 pp 165ndash169 2018

[4] Y Y Zhang X Q Liu and G W Wang ldquoSymmetry reductionsand exact solutions of the (2+ 1)-dimensional JaulentndashMiodekequationrdquo Applied Mathematics and Computation vol 219 no3 pp 911ndash916 2012

[5] M Mirzazadeh and M Eslami ldquoExact solutions of theKudryashov-Sinelshchikov equation and nonlinear telegraphequation via the first integral methodrdquo Nonlinear AnalysisModelling and Control vol 17 no 4 pp 481ndash488 2012

[6] M Mirzazadeh M Eslami A H Bhrawy B Ahmed and ABiswas ldquoSolitons and other solutions to complex-valued Klein-Gordon equation in Φ-4 field theoryrdquo Applied Mathematics ampInformation Sciences vol 9 no 6 pp 2793ndash2801 2015

[7] ANazarzadehM Eslami andMMirzazadeh ldquoExact solutionsof somenonlinear partial differential equations using functionalvariablemethodrdquoPramanamdashJournal of Physics vol 81 no 2 pp225ndash236 2013

[8] 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

[9] M Mirzazadeh M Eslami and A H Arnous ldquoDark opticalsolitons of Biswas-Milovic equation with dual-power law non-linearityrdquo The European Physical Journal Plus vol 130 4 no 12015

[10] E Aksoy M Kaplan and A Bekir ldquoExponential rational func-tion method for space-time fractional differential equationsrdquoWaves in random and complex media propagation scatteringand imaging vol 26 no 2 pp 142ndash151 2016

[11] A Bekir andA C Cevikel ldquoNew exact travelling wave solutionsof nonlinear physical modelsrdquo Chaos Solitons and Fractals vol41 no 4 pp 1733ndash1739 2009

[12] M Eslami M Mirzazadeh B Fathi Vajargah and A BiswasldquoOptical solitons for the resonant nonlinear Schrodingerrsquosequation with time-dependent coefficients by the first integralmethodrdquo Optik - International Journal for Light and ElectronOptics vol 125 no 13 pp 3107ndash3116 2014

[13] M F El-Sabbagh R Zait and R M Abdelazeem ldquoNew exactsolutions of somenonlinear partial differential equations via theimproved exp-functionmethodrdquo IJRRAS vol 18 no 2 pp 132ndash144 2014

[14] Y Gurefe E Misirli A Sonmezoglu and M Ekici ldquoExtendedtrial equation method to generalized nonlinear partial differen-tial equationsrdquo Applied Mathematics and Computation vol 219no 10 pp 5253ndash5260 2013

[15] A H Khater M H Moussa and S F Abdul-Aziz ldquoInvariantvariational principles and conservation laws for some nonlinearpartial differential equations with constant coefficients - IrdquoChaos Solitons and Fractals vol 14 no 9 pp 1389ndash1401 2002

[16] Q Feng and F Meng ldquoTraveling wave solutions for fractionalpartial differential equations arising in mathematical physicsby an improved fractional Jacobi elliptic equation methodrdquoMathematical Methods in the Applied Sciences vol 40 no 10pp 3676ndash3686 2017

[17] R K Gazizov A A Kasatkin and S Y Lukashchuk ldquoContinu-ous transformation groups of fractional differential equationsrdquoVestnik Usatu vol 9 no 3 p 21 2007

[18] S Zhang and H Q Zhang ldquoFractional sub-equation methodand its applications to nonlinear fractional PDEsrdquo PhysicsLetters A vol 375 no 7 pp 1069ndash1073 2011

[19] B Zheng ldquoA new fractional Jacobi elliptic equation methodfor solving fractional partial differential equationsrdquoAdvances inDifference Equations vol 2014 no 1 p 228 2014

[20] A Neyrame A Roozi S S Hosseini and S M Shafiof ldquoExacttravelling wave solutions for some nonlinear partial differentialequationsrdquo Journal of King Saud University - Science vol 22 no4 pp 275ndash278 2010

[21] K B Oldham and J Spanier The Fractional Calculus Theoryand Applications of Differentiation and Integration to ArbitraryOrder vol 111 Academic Press New York NY USA LondonUK 1974

[22] B Ahmad S K Ntouyas and A Alsaedi ldquoOn a coupled systemof fractional differential equations with coupled nonlocal andintegral boundary conditionsrdquo Chaos Solitons amp Fractals vol83 pp 234ndash241 2016

[23] V S Kiryakova Generalized Fractional Calculus and Applica-tions CRC press Botan Roca Fl USA 1993

[24] G-WWang X-Q Liu andY-Y Zhang ldquoLie symmetry analysisto the time fractional generalized fifth-order KdV equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 9 pp 2321ndash2326 2013

[25] S S Ray and S Sahoo ldquoInvariant analysis and conservation lawsof (2+ 1) dimensional time-fractional ZKBBM equation in grav-ity water wavesrdquo Computers amp Mathematics with Applications2017

[26] G-W Wang and T-Z Xu ldquoInvariant analysis and exact solu-tions of nonlinear time fractional Sharma-Tasso-Olver equationby Lie group analysisrdquo Nonlinear Dynamics vol 76 no 1 pp571ndash580 2014

[27] M Inc A Yusuf A I Aliyu and D Baleanu ldquoLie symmetryanalysis explicit solutions and conservation laws for the space-time fractional nonlinear evolution equationsrdquo Physica AStatistical Mechanics and its Applications vol 496 pp 371ndash3832018

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

Journal of Applied Mathematics 3

1205721205851199052 minus 21205851199091 = 0ℎ120578119909119909 minus 119906120597120572119905 120578119906 + 120597120572119905 120578 + 119906120578119909 = 01205851199091199091 minus 2120578119909119906 = 0

(17)

Solving the obtained determining equation we get

1205851 = 1198882 + 12057211990911988811205852 = 21199051198881120578 = minus1205721199061198881

(18)

where 1198881 and 1198882 are constants for simplicity We take theirvalues equal to one So (2) has two vector fields that cangenerate its infinitesimal symmetry These Lie vectors areconsidered as follows

1198831 = 120597120597119909 (19)

1198832 = 120572119909 120597120597119909 + 2119905 120597120597119905 minus 120572119906 120597120597119906 (20)

Case 1 For (19) we have

1198891199091 = 1198891199050 = 1198891199060 (21)

Solving this equation 119906 = 119891(119905) Putting 119906 = 119891(119905) into (1) weget

119863120572119905 119891 (119905) = 0 (22)

where 119906 = 1198861119905120572minus1Case 2 For1198832 in (20) we have

119889119909120572119909 = 1198891199052119905 = minus119889119906120572119906 (23)

This is the characteristic equation By solving it the resultingsimilarity variable in the form

1199111 = 119909119905minus12057221199112 = 119906119905minus1205722 (24)

The variables transformation is as follows

119906 = 119905minus1205722119891 (120585) 120585 = 119909119905minus1205722 (25)

where 119891(120585) is a function in one variable 120585 We use (25) totransform (2) into a fractional ODE

Theorem 5 Transformation (25) reduces (2) to the nonlinearFODE as follows

(1198751minus312057221205723120572 119891) (120585) + ℎ119891120585120585 + 119891119891120585 = 0 (26)

utilizing the Erdelyi-Kober (EK) fractional derivative operator[20]

(1198751205852120572120573 119891) (120585)= 119899minus1prod119895=0

(1205852 + 119895 minus 1120573 119889119889120585) (1198701205852+120572119899minus120572120573 119891) (120585) (27)

where

(1198701205852120572120573 119891) (120585)

=

1Γ (120572) intinfin

1(119906 minus 1)120572minus1 119906minus(1205852+120572)119891 (1205851199061120573) 119889119906 120572 gt 0

119891 (120585) 120572 = 0(28)

and

119899 = [120572] + 1 120572 = 119873120572 120572 isin 119873 (29)

Proof Utilizing the definition of the RL fractional derivativein (25) we get

120597120572119906120597119905120572 = 120597119899120597119905119899 [ 1Γ (119899 minus 120572)sdot int1199051(119905 minus 119904)119899minus120572minus1 119904minus1205722119891 (119909119904minus(1205722)) 119889119904]

119899 minus 1 lt 120572 lt 1 119899 = 1 2 (30)

Assume that V = 119905119904 119889119904 = minus(119905V2)119889V Thus (30) becomes

120597120572119906120597119905120572 = 120597119899120597119905119899 [119905119899minus31205722 1Γ (119899 minus 120572)sdot intinfin0(V minus 1)119899minus120572minus1 Vminus(119899+1minus31205722)119891 (120585V(1205722)) 119889V]

(31)

4 Journal of Applied Mathematics

Applying the EK fractional integral operator (28) in (31) wehave

120597120572119906120597119905120572 = 120597119899120597119905119899 [119905119899minus31205722 (1198701minus1205722119899minus1205722120572 119891) (120585)] (32)

For simplicity we consider= 119909 119905minus1205722120601 isin (0infin)We thus findthat

119905 120597120597119905120601 (120585) = 119905119909 (minus1205722 ) 119905minus1205722minus1 120601 (120585) = minus1205722 120585 120597120597120585120601 (120585) (33)

Hence we have

120597119899120597119905119899 [119905119899minus31205722 (1198701minus1205722119899minus1205722120572 119891) (120585)] = 120597119899minus1120597119905119899minus1 [ 120597120597119905sdot 119905119899minus31205722 (1198701minus1205722119899minus1205722120572 119891) (120585)]= 120597119899minus1120597119905119899minus1 [119905119899minus31205722minus1 (119899 minus 31205722 minus 1205722 120585 120597120597120585)sdot (1198701minus1205722119899minus1205722120572 119891) (120585)]

(34)

Repeating 119899 minus 1 times we have

120597119899120597119905119899 [119905119899minus31205722 (1198701minus1205722119899minus1205722120572 119891) (120585)] = 120597119899minus1120597119905119899minus1 [ 120597120597119905sdot 119905119899minus31205722 (1198701minus1205722119899minus1205722120572 119891) (120585)] = 120597119899minus1120597119905119899minus1 [119905119899minus31205722minus1 (119899minus 31205722 minus 1205722 120585 120597120597120585) (1198701minus1205722119899minus1205722120572 119891) (120585)]

= 119905minus1205722119899minus1prod119895=0

[(1 minus 31205722 + 119895 minus 1205722 120585 120597120597120585) (1198701minus1205722119899minus1205722120572 119891)sdot (120585)

(35)

Applying the EK fractional differential operator (27) in (35)we get

120597119899120597119905119899 [(119905119899minus1205723 (1198701minus1205722119899minus1205722120572 119891) (120585))]= 119905minus1205722 (1198751minus312057221205722120572 119891) (120585)

(36)

Substituting (36) into (32) we get

120597120572119906120597119905120572 = 119905minus1205722 (1198751minus312057221205722120572 119891) (120585) (37)

Thus (2) is reduced to a fractional-order ODE as follows

(1198751minus312057221205722120572 119891) (120585) + 119891119891120585 + ℎ119891120585120585 = 0 (38)

4 The Explicit Solution forthe Time-Fractional Benjamin-OnoEquation by Using PSM

The analytic solutions via PSM [26] are demonstrated Weassume that

119891 (120585) = infinsum119899=0

119886119899120601 (120585)119899 (39)

Differentiating (39) twice regarding 120585 we get1198911015840 (120585) = infinsum

119899=0

119899119886119899120601 (120585)119899minus1 (40)

and

11989110158401015840 (120585) = infinsum119899=0

119899 (119899 minus 1) 119886119899120601 (120585)119899minus2 (41)

Substituting (39) (40) and (41) into (38) we have

infinsum119899=0

Γ (2 minus 31205722 minus 1198991205722)Γ (2 minus 1205722 minus 1198991205722) 119886119899120601 (120585)119899

+ infinsum119899=0

119886119899120601 (120585)119899 infinsum119899=0

(119899 + 1) 119886119899+1120601 (120585)119899

+ ℎinfinsum119899=0

(119899 + 2) (119899 + 1) 119886119899+2120601 (120585)119899 = 0

(42)

Comparing coefficients in (42) when 119899 = 0 we obtain1198862 = minus12ℎ (Γ (2 minus 31205722)Γ (2 minus 1205722) 1198860 + 11988601198861) (43)

Journal of Applied Mathematics 5

When 119899 ge 1 the recurrence relations between the seriescoefficients are

119886119899+2 = minus12ℎ (119899 + 2) (119899 + 1) (Γ (2 minus 31205722 minus 1198991205722)Γ (2 minus 1205722 minus 1198991205722) 119886119899+ (119899 + 1) 119886119899119886119899+1)

(44)

Using (44) the series solution for (39) can be represented bysubstituting (43) and (44) into (39)

119891 (120585) = 1198860 + 1198861120585 + 11988621205852 + infinsum119899=1

119886119899+2120585119899+2 = 1198860 + 1198861120585

minus 12ℎ (Γ (2 minus 31205722)Γ (2 minus 1205722) 1198860 + 11988601198861) 1205852

minus infinsum119899=1

12ℎ (119899 + 2) (119899 + 1) (Γ (2 minus 31205722 minus 1198991205722)Γ (2 minus 1205722 minus 1198991205722) 119886119899+ (119899 + 1) 119886119899119886119899+1) 120585119899+2

(45)

Upon substitution using similarity variables in (25) thefollowing explicit solutions for (2) are

119906 (119909 119905) = 1198860119905minus1205722 + 1198861119909119905minus120572 minus 12ℎ (Γ (2 minus 31205722)Γ (2 minus 1205722) 1198860+ 11988601198861) 119905minus1205722 (119909119905minus1205722)2

minus infinsum119899=1

12ℎ (119899 + 2) (119899 + 1) Γ (2 minus 31205722 minus 1198991205722)Γ (2 minus 1205722 minus 1198991205722) 119886119899

+ 119899sum119896=0

119896sum119895=0

(119886119895+119886119896minus119895) 119886119899minus119896 (119899 + 1)(119909)119899+2

sdot 119905minus120572(2119899+120572120573)2120573

(46)

5 Convergence Analysis

To satisfy the convergence test there are many kinds of testsas the ratio the comparison and the quotient tests Theconvergence of the solution equation (46) will be presentedas follows We revamp (46) as follows

1003816100381610038161003816119886119899+21003816100381610038161003816 le (|Γ (2 minus 31205722 minus 1198991205722)||Γ (2 minus 1205722 minus 1198991205722)| 10038161003816100381610038161198861198991003816100381610038161003816minus 119899sum119896=0

119896sum119895=0

1003816100381610038161003816100381611988611989510038161003816100381610038161003816 10038161003816100381610038161003816119886119896minus11989510038161003816100381610038161003816 1003816100381610038161003816119886119899minus1198961003816100381610038161003816 minus 10038161003816100381610038161198861198991003816100381610038161003816)(47)

Equation (47) utilizing the Gamma function shows that|Γ(2 minus 31205722 minus 1198991205722)||Γ(2 minus 1205722 minus 1198991205722)| lt 1 for arbitrary119899 that1003816100381610038161003816119886119899+21003816100381610038161003816 le 119872(10038161003816100381610038161198861198991003816100381610038161003816 minus 119899sum

119896=0

119896sum119895=0

1003816100381610038161003816100381611988611989510038161003816100381610038161003816 10038161003816100381610038161003816119886119896minus11989510038161003816100381610038161003816 1003816100381610038161003816119886119899minus1198961003816100381610038161003816 minus 10038161003816100381610038161198861198991003816100381610038161003816) (48)

where119872 = max(|1198881| |1198882|) We now assume another form ofthe PSM

119861 (120585) = infinsum119899=0

119888119899120585119899 (49)

By comparing the two series we can observe that |119888119899| le 119886119899119899 = 0 1 Hence the series119861(120585) = suminfin119899=0 119888119899120585119899 is themajorantseries of (47) So we find that

119861 (120585) = 1198880 + 1198881120585+119872(infinsum

119899=0

1198881198991205852119861 (120585) + infinsum119899=0

119899sum119896=0

119896sum119895=0

119888119895119888119896minus119895119888119899minus119896 + infinsum119899=0

119888119899)sdot 120585119899+2

(50)

Consider an implicit functional system regarding 120585 as follows120573 (120585 119861)

= 119861 minus 1198880 minus 1198881120585 minus 11988821205852minus119872(1205852119861 (120585) + 2119861 (120585)2 + (1205852 minus 1198881120585 minus 31198880) 119861)

(51)

since 120573 is analytic in a neighborhood of (0 1198880) where120573(0 1198880) = 0 and (120597120597119861)120573(0 1198880) = 0 Then the series 119861(120585) =suminfin119899=0 120585119899 is analytic around (0 1198880) and this is verified utilizing[27] and the radius of convergence of this series belongs toa positive domain This shows that (46) converges around(0 1198880)6 Physical Performance of the Power SeriesTechnique for Eqs (46)

To have expressed and convenient conception of the physicalcharacteristic of the power series solution the 3D plots forthe explicit solution equations (46) is plotted in Figures 1ndash4 atℎ = 1 by utilizing appropriate parameter formsThe spectaclevision of the real portion of (46) can be visible in the 3D plotsproof in Figures 1 2 3 and 4 respectively

6 Journal of Applied Mathematics

2002

2001

2

1999

0 0

4 4

xt

Figure 1 3D plot of (46) with 1198860 = 19 1198861 = 17 1198862 = 077 120574 = 1120572 = 08 120573 = 2

u(x

t)

2002

2001

2

1999

0 0

4 4xt

Figure 2 3D plot of (46) with 1198860 = 39 1198861 = 17 1198862 = 087 120574 = 1120572 = 06 120573 = 1

7 Conclusions

Lie point symmetry properties of (1 + 1)-dimensional time-fractional Benjamin-Ono equation have been consideredwith the RiemannndashLiouville fractional derivativeThese sym-metries are used here to transform the FPDEs intoNLFODEsClosed-form solutions are determined by using PSM inthe last division The accuracy exhibits the assembly ofthe solution Considerable frames for the acquired explicitsolutions were approached

2002

2001

2

1999

0 0

4 4

xt

u(x t)

Figure 3 3D plot of (46) with 1198860 = 1 1198861 = 11 1198862 = 057 120574 = 1120572 = 05 120573 = 07u(x t)

2002

2001

2

1999

0 0

4 4

xt

Figure 4 3D plot of (46) with 1198860 = 1 1198861 = 1 1198862 = 04 120574 = 1120572 = 03 120573 = 04

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Applied Mathematics 7

Conflicts of Interest

The author declares that they have no conflicts of interest

References

[1] M Bruzon et al ldquoThe symmetry reductions of a turbulencemodelrdquo Journal of Physics A Mathematical and General vol 34no 18 p 3751 2001

[2] R Sadat and M Kassem ldquoExplicit Solutions for the (2+ 1)-dimensional jaulentndashmiodek equation using the integratingfactors method in an unbounded domainrdquo Mathematical ampComputational Applications vol 23 no 1 p 15 2018

[3] A Paliathanasis andMTsamparlis ldquoLie symmetries for systemsof evolution equationsrdquo Journal of Geometry and Physics vol124 pp 165ndash169 2018

[4] Y Y Zhang X Q Liu and G W Wang ldquoSymmetry reductionsand exact solutions of the (2+ 1)-dimensional JaulentndashMiodekequationrdquo Applied Mathematics and Computation vol 219 no3 pp 911ndash916 2012

[5] M Mirzazadeh and M Eslami ldquoExact solutions of theKudryashov-Sinelshchikov equation and nonlinear telegraphequation via the first integral methodrdquo Nonlinear AnalysisModelling and Control vol 17 no 4 pp 481ndash488 2012

[6] M Mirzazadeh M Eslami A H Bhrawy B Ahmed and ABiswas ldquoSolitons and other solutions to complex-valued Klein-Gordon equation in Φ-4 field theoryrdquo Applied Mathematics ampInformation Sciences vol 9 no 6 pp 2793ndash2801 2015

[7] ANazarzadehM Eslami andMMirzazadeh ldquoExact solutionsof somenonlinear partial differential equations using functionalvariablemethodrdquoPramanamdashJournal of Physics vol 81 no 2 pp225ndash236 2013

[8] 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

[9] M Mirzazadeh M Eslami and A H Arnous ldquoDark opticalsolitons of Biswas-Milovic equation with dual-power law non-linearityrdquo The European Physical Journal Plus vol 130 4 no 12015

[10] E Aksoy M Kaplan and A Bekir ldquoExponential rational func-tion method for space-time fractional differential equationsrdquoWaves in random and complex media propagation scatteringand imaging vol 26 no 2 pp 142ndash151 2016

[11] A Bekir andA C Cevikel ldquoNew exact travelling wave solutionsof nonlinear physical modelsrdquo Chaos Solitons and Fractals vol41 no 4 pp 1733ndash1739 2009

[12] M Eslami M Mirzazadeh B Fathi Vajargah and A BiswasldquoOptical solitons for the resonant nonlinear Schrodingerrsquosequation with time-dependent coefficients by the first integralmethodrdquo Optik - International Journal for Light and ElectronOptics vol 125 no 13 pp 3107ndash3116 2014

[13] M F El-Sabbagh R Zait and R M Abdelazeem ldquoNew exactsolutions of somenonlinear partial differential equations via theimproved exp-functionmethodrdquo IJRRAS vol 18 no 2 pp 132ndash144 2014

[14] Y Gurefe E Misirli A Sonmezoglu and M Ekici ldquoExtendedtrial equation method to generalized nonlinear partial differen-tial equationsrdquo Applied Mathematics and Computation vol 219no 10 pp 5253ndash5260 2013

[15] A H Khater M H Moussa and S F Abdul-Aziz ldquoInvariantvariational principles and conservation laws for some nonlinearpartial differential equations with constant coefficients - IrdquoChaos Solitons and Fractals vol 14 no 9 pp 1389ndash1401 2002

[16] Q Feng and F Meng ldquoTraveling wave solutions for fractionalpartial differential equations arising in mathematical physicsby an improved fractional Jacobi elliptic equation methodrdquoMathematical Methods in the Applied Sciences vol 40 no 10pp 3676ndash3686 2017

[17] R K Gazizov A A Kasatkin and S Y Lukashchuk ldquoContinu-ous transformation groups of fractional differential equationsrdquoVestnik Usatu vol 9 no 3 p 21 2007

[18] S Zhang and H Q Zhang ldquoFractional sub-equation methodand its applications to nonlinear fractional PDEsrdquo PhysicsLetters A vol 375 no 7 pp 1069ndash1073 2011

[19] B Zheng ldquoA new fractional Jacobi elliptic equation methodfor solving fractional partial differential equationsrdquoAdvances inDifference Equations vol 2014 no 1 p 228 2014

[20] A Neyrame A Roozi S S Hosseini and S M Shafiof ldquoExacttravelling wave solutions for some nonlinear partial differentialequationsrdquo Journal of King Saud University - Science vol 22 no4 pp 275ndash278 2010

[21] K B Oldham and J Spanier The Fractional Calculus Theoryand Applications of Differentiation and Integration to ArbitraryOrder vol 111 Academic Press New York NY USA LondonUK 1974

[22] B Ahmad S K Ntouyas and A Alsaedi ldquoOn a coupled systemof fractional differential equations with coupled nonlocal andintegral boundary conditionsrdquo Chaos Solitons amp Fractals vol83 pp 234ndash241 2016

[23] V S Kiryakova Generalized Fractional Calculus and Applica-tions CRC press Botan Roca Fl USA 1993

[24] G-WWang X-Q Liu andY-Y Zhang ldquoLie symmetry analysisto the time fractional generalized fifth-order KdV equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 9 pp 2321ndash2326 2013

[25] S S Ray and S Sahoo ldquoInvariant analysis and conservation lawsof (2+ 1) dimensional time-fractional ZKBBM equation in grav-ity water wavesrdquo Computers amp Mathematics with Applications2017

[26] G-W Wang and T-Z Xu ldquoInvariant analysis and exact solu-tions of nonlinear time fractional Sharma-Tasso-Olver equationby Lie group analysisrdquo Nonlinear Dynamics vol 76 no 1 pp571ndash580 2014

[27] M Inc A Yusuf A I Aliyu and D Baleanu ldquoLie symmetryanalysis explicit solutions and conservation laws for the space-time fractional nonlinear evolution equationsrdquo Physica AStatistical Mechanics and its Applications vol 496 pp 371ndash3832018

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 Journal of Applied Mathematics

Applying the EK fractional integral operator (28) in (31) wehave

120597120572119906120597119905120572 = 120597119899120597119905119899 [119905119899minus31205722 (1198701minus1205722119899minus1205722120572 119891) (120585)] (32)

For simplicity we consider= 119909 119905minus1205722120601 isin (0infin)We thus findthat

119905 120597120597119905120601 (120585) = 119905119909 (minus1205722 ) 119905minus1205722minus1 120601 (120585) = minus1205722 120585 120597120597120585120601 (120585) (33)

Hence we have

120597119899120597119905119899 [119905119899minus31205722 (1198701minus1205722119899minus1205722120572 119891) (120585)] = 120597119899minus1120597119905119899minus1 [ 120597120597119905sdot 119905119899minus31205722 (1198701minus1205722119899minus1205722120572 119891) (120585)]= 120597119899minus1120597119905119899minus1 [119905119899minus31205722minus1 (119899 minus 31205722 minus 1205722 120585 120597120597120585)sdot (1198701minus1205722119899minus1205722120572 119891) (120585)]

(34)

Repeating 119899 minus 1 times we have

120597119899120597119905119899 [119905119899minus31205722 (1198701minus1205722119899minus1205722120572 119891) (120585)] = 120597119899minus1120597119905119899minus1 [ 120597120597119905sdot 119905119899minus31205722 (1198701minus1205722119899minus1205722120572 119891) (120585)] = 120597119899minus1120597119905119899minus1 [119905119899minus31205722minus1 (119899minus 31205722 minus 1205722 120585 120597120597120585) (1198701minus1205722119899minus1205722120572 119891) (120585)]

= 119905minus1205722119899minus1prod119895=0

[(1 minus 31205722 + 119895 minus 1205722 120585 120597120597120585) (1198701minus1205722119899minus1205722120572 119891)sdot (120585)

(35)

Applying the EK fractional differential operator (27) in (35)we get

120597119899120597119905119899 [(119905119899minus1205723 (1198701minus1205722119899minus1205722120572 119891) (120585))]= 119905minus1205722 (1198751minus312057221205722120572 119891) (120585)

(36)

Substituting (36) into (32) we get

120597120572119906120597119905120572 = 119905minus1205722 (1198751minus312057221205722120572 119891) (120585) (37)

Thus (2) is reduced to a fractional-order ODE as follows

(1198751minus312057221205722120572 119891) (120585) + 119891119891120585 + ℎ119891120585120585 = 0 (38)

4 The Explicit Solution forthe Time-Fractional Benjamin-OnoEquation by Using PSM

The analytic solutions via PSM [26] are demonstrated Weassume that

119891 (120585) = infinsum119899=0

119886119899120601 (120585)119899 (39)

Differentiating (39) twice regarding 120585 we get1198911015840 (120585) = infinsum

119899=0

119899119886119899120601 (120585)119899minus1 (40)

and

11989110158401015840 (120585) = infinsum119899=0

119899 (119899 minus 1) 119886119899120601 (120585)119899minus2 (41)

Substituting (39) (40) and (41) into (38) we have

infinsum119899=0

Γ (2 minus 31205722 minus 1198991205722)Γ (2 minus 1205722 minus 1198991205722) 119886119899120601 (120585)119899

+ infinsum119899=0

119886119899120601 (120585)119899 infinsum119899=0

(119899 + 1) 119886119899+1120601 (120585)119899

+ ℎinfinsum119899=0

(119899 + 2) (119899 + 1) 119886119899+2120601 (120585)119899 = 0

(42)

Comparing coefficients in (42) when 119899 = 0 we obtain1198862 = minus12ℎ (Γ (2 minus 31205722)Γ (2 minus 1205722) 1198860 + 11988601198861) (43)

Journal of Applied Mathematics 5

When 119899 ge 1 the recurrence relations between the seriescoefficients are

119886119899+2 = minus12ℎ (119899 + 2) (119899 + 1) (Γ (2 minus 31205722 minus 1198991205722)Γ (2 minus 1205722 minus 1198991205722) 119886119899+ (119899 + 1) 119886119899119886119899+1)

(44)

Using (44) the series solution for (39) can be represented bysubstituting (43) and (44) into (39)

119891 (120585) = 1198860 + 1198861120585 + 11988621205852 + infinsum119899=1

119886119899+2120585119899+2 = 1198860 + 1198861120585

minus 12ℎ (Γ (2 minus 31205722)Γ (2 minus 1205722) 1198860 + 11988601198861) 1205852

minus infinsum119899=1

12ℎ (119899 + 2) (119899 + 1) (Γ (2 minus 31205722 minus 1198991205722)Γ (2 minus 1205722 minus 1198991205722) 119886119899+ (119899 + 1) 119886119899119886119899+1) 120585119899+2

(45)

Upon substitution using similarity variables in (25) thefollowing explicit solutions for (2) are

119906 (119909 119905) = 1198860119905minus1205722 + 1198861119909119905minus120572 minus 12ℎ (Γ (2 minus 31205722)Γ (2 minus 1205722) 1198860+ 11988601198861) 119905minus1205722 (119909119905minus1205722)2

minus infinsum119899=1

12ℎ (119899 + 2) (119899 + 1) Γ (2 minus 31205722 minus 1198991205722)Γ (2 minus 1205722 minus 1198991205722) 119886119899

+ 119899sum119896=0

119896sum119895=0

(119886119895+119886119896minus119895) 119886119899minus119896 (119899 + 1)(119909)119899+2

sdot 119905minus120572(2119899+120572120573)2120573

(46)

5 Convergence Analysis

To satisfy the convergence test there are many kinds of testsas the ratio the comparison and the quotient tests Theconvergence of the solution equation (46) will be presentedas follows We revamp (46) as follows

1003816100381610038161003816119886119899+21003816100381610038161003816 le (|Γ (2 minus 31205722 minus 1198991205722)||Γ (2 minus 1205722 minus 1198991205722)| 10038161003816100381610038161198861198991003816100381610038161003816minus 119899sum119896=0

119896sum119895=0

1003816100381610038161003816100381611988611989510038161003816100381610038161003816 10038161003816100381610038161003816119886119896minus11989510038161003816100381610038161003816 1003816100381610038161003816119886119899minus1198961003816100381610038161003816 minus 10038161003816100381610038161198861198991003816100381610038161003816)(47)

Equation (47) utilizing the Gamma function shows that|Γ(2 minus 31205722 minus 1198991205722)||Γ(2 minus 1205722 minus 1198991205722)| lt 1 for arbitrary119899 that1003816100381610038161003816119886119899+21003816100381610038161003816 le 119872(10038161003816100381610038161198861198991003816100381610038161003816 minus 119899sum

119896=0

119896sum119895=0

1003816100381610038161003816100381611988611989510038161003816100381610038161003816 10038161003816100381610038161003816119886119896minus11989510038161003816100381610038161003816 1003816100381610038161003816119886119899minus1198961003816100381610038161003816 minus 10038161003816100381610038161198861198991003816100381610038161003816) (48)

where119872 = max(|1198881| |1198882|) We now assume another form ofthe PSM

119861 (120585) = infinsum119899=0

119888119899120585119899 (49)

By comparing the two series we can observe that |119888119899| le 119886119899119899 = 0 1 Hence the series119861(120585) = suminfin119899=0 119888119899120585119899 is themajorantseries of (47) So we find that

119861 (120585) = 1198880 + 1198881120585+119872(infinsum

119899=0

1198881198991205852119861 (120585) + infinsum119899=0

119899sum119896=0

119896sum119895=0

119888119895119888119896minus119895119888119899minus119896 + infinsum119899=0

119888119899)sdot 120585119899+2

(50)

Consider an implicit functional system regarding 120585 as follows120573 (120585 119861)

= 119861 minus 1198880 minus 1198881120585 minus 11988821205852minus119872(1205852119861 (120585) + 2119861 (120585)2 + (1205852 minus 1198881120585 minus 31198880) 119861)

(51)

since 120573 is analytic in a neighborhood of (0 1198880) where120573(0 1198880) = 0 and (120597120597119861)120573(0 1198880) = 0 Then the series 119861(120585) =suminfin119899=0 120585119899 is analytic around (0 1198880) and this is verified utilizing[27] and the radius of convergence of this series belongs toa positive domain This shows that (46) converges around(0 1198880)6 Physical Performance of the Power SeriesTechnique for Eqs (46)

To have expressed and convenient conception of the physicalcharacteristic of the power series solution the 3D plots forthe explicit solution equations (46) is plotted in Figures 1ndash4 atℎ = 1 by utilizing appropriate parameter formsThe spectaclevision of the real portion of (46) can be visible in the 3D plotsproof in Figures 1 2 3 and 4 respectively

6 Journal of Applied Mathematics

2002

2001

2

1999

0 0

4 4

xt

Figure 1 3D plot of (46) with 1198860 = 19 1198861 = 17 1198862 = 077 120574 = 1120572 = 08 120573 = 2

u(x

t)

2002

2001

2

1999

0 0

4 4xt

Figure 2 3D plot of (46) with 1198860 = 39 1198861 = 17 1198862 = 087 120574 = 1120572 = 06 120573 = 1

7 Conclusions

Lie point symmetry properties of (1 + 1)-dimensional time-fractional Benjamin-Ono equation have been consideredwith the RiemannndashLiouville fractional derivativeThese sym-metries are used here to transform the FPDEs intoNLFODEsClosed-form solutions are determined by using PSM inthe last division The accuracy exhibits the assembly ofthe solution Considerable frames for the acquired explicitsolutions were approached

2002

2001

2

1999

0 0

4 4

xt

u(x t)

Figure 3 3D plot of (46) with 1198860 = 1 1198861 = 11 1198862 = 057 120574 = 1120572 = 05 120573 = 07u(x t)

2002

2001

2

1999

0 0

4 4

xt

Figure 4 3D plot of (46) with 1198860 = 1 1198861 = 1 1198862 = 04 120574 = 1120572 = 03 120573 = 04

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Applied Mathematics 7

Conflicts of Interest

The author declares that they have no conflicts of interest

References

[1] M Bruzon et al ldquoThe symmetry reductions of a turbulencemodelrdquo Journal of Physics A Mathematical and General vol 34no 18 p 3751 2001

[2] R Sadat and M Kassem ldquoExplicit Solutions for the (2+ 1)-dimensional jaulentndashmiodek equation using the integratingfactors method in an unbounded domainrdquo Mathematical ampComputational Applications vol 23 no 1 p 15 2018

[3] A Paliathanasis andMTsamparlis ldquoLie symmetries for systemsof evolution equationsrdquo Journal of Geometry and Physics vol124 pp 165ndash169 2018

[4] Y Y Zhang X Q Liu and G W Wang ldquoSymmetry reductionsand exact solutions of the (2+ 1)-dimensional JaulentndashMiodekequationrdquo Applied Mathematics and Computation vol 219 no3 pp 911ndash916 2012

[5] M Mirzazadeh and M Eslami ldquoExact solutions of theKudryashov-Sinelshchikov equation and nonlinear telegraphequation via the first integral methodrdquo Nonlinear AnalysisModelling and Control vol 17 no 4 pp 481ndash488 2012

[6] M Mirzazadeh M Eslami A H Bhrawy B Ahmed and ABiswas ldquoSolitons and other solutions to complex-valued Klein-Gordon equation in Φ-4 field theoryrdquo Applied Mathematics ampInformation Sciences vol 9 no 6 pp 2793ndash2801 2015

[7] ANazarzadehM Eslami andMMirzazadeh ldquoExact solutionsof somenonlinear partial differential equations using functionalvariablemethodrdquoPramanamdashJournal of Physics vol 81 no 2 pp225ndash236 2013

[8] 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

[9] M Mirzazadeh M Eslami and A H Arnous ldquoDark opticalsolitons of Biswas-Milovic equation with dual-power law non-linearityrdquo The European Physical Journal Plus vol 130 4 no 12015

[10] E Aksoy M Kaplan and A Bekir ldquoExponential rational func-tion method for space-time fractional differential equationsrdquoWaves in random and complex media propagation scatteringand imaging vol 26 no 2 pp 142ndash151 2016

[11] A Bekir andA C Cevikel ldquoNew exact travelling wave solutionsof nonlinear physical modelsrdquo Chaos Solitons and Fractals vol41 no 4 pp 1733ndash1739 2009

[12] M Eslami M Mirzazadeh B Fathi Vajargah and A BiswasldquoOptical solitons for the resonant nonlinear Schrodingerrsquosequation with time-dependent coefficients by the first integralmethodrdquo Optik - International Journal for Light and ElectronOptics vol 125 no 13 pp 3107ndash3116 2014

[13] M F El-Sabbagh R Zait and R M Abdelazeem ldquoNew exactsolutions of somenonlinear partial differential equations via theimproved exp-functionmethodrdquo IJRRAS vol 18 no 2 pp 132ndash144 2014

[14] Y Gurefe E Misirli A Sonmezoglu and M Ekici ldquoExtendedtrial equation method to generalized nonlinear partial differen-tial equationsrdquo Applied Mathematics and Computation vol 219no 10 pp 5253ndash5260 2013

[15] A H Khater M H Moussa and S F Abdul-Aziz ldquoInvariantvariational principles and conservation laws for some nonlinearpartial differential equations with constant coefficients - IrdquoChaos Solitons and Fractals vol 14 no 9 pp 1389ndash1401 2002

[16] Q Feng and F Meng ldquoTraveling wave solutions for fractionalpartial differential equations arising in mathematical physicsby an improved fractional Jacobi elliptic equation methodrdquoMathematical Methods in the Applied Sciences vol 40 no 10pp 3676ndash3686 2017

[17] R K Gazizov A A Kasatkin and S Y Lukashchuk ldquoContinu-ous transformation groups of fractional differential equationsrdquoVestnik Usatu vol 9 no 3 p 21 2007

[18] S Zhang and H Q Zhang ldquoFractional sub-equation methodand its applications to nonlinear fractional PDEsrdquo PhysicsLetters A vol 375 no 7 pp 1069ndash1073 2011

[19] B Zheng ldquoA new fractional Jacobi elliptic equation methodfor solving fractional partial differential equationsrdquoAdvances inDifference Equations vol 2014 no 1 p 228 2014

[20] A Neyrame A Roozi S S Hosseini and S M Shafiof ldquoExacttravelling wave solutions for some nonlinear partial differentialequationsrdquo Journal of King Saud University - Science vol 22 no4 pp 275ndash278 2010

[21] K B Oldham and J Spanier The Fractional Calculus Theoryand Applications of Differentiation and Integration to ArbitraryOrder vol 111 Academic Press New York NY USA LondonUK 1974

[22] B Ahmad S K Ntouyas and A Alsaedi ldquoOn a coupled systemof fractional differential equations with coupled nonlocal andintegral boundary conditionsrdquo Chaos Solitons amp Fractals vol83 pp 234ndash241 2016

[23] V S Kiryakova Generalized Fractional Calculus and Applica-tions CRC press Botan Roca Fl USA 1993

[24] G-WWang X-Q Liu andY-Y Zhang ldquoLie symmetry analysisto the time fractional generalized fifth-order KdV equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 9 pp 2321ndash2326 2013

[25] S S Ray and S Sahoo ldquoInvariant analysis and conservation lawsof (2+ 1) dimensional time-fractional ZKBBM equation in grav-ity water wavesrdquo Computers amp Mathematics with Applications2017

[26] G-W Wang and T-Z Xu ldquoInvariant analysis and exact solu-tions of nonlinear time fractional Sharma-Tasso-Olver equationby Lie group analysisrdquo Nonlinear Dynamics vol 76 no 1 pp571ndash580 2014

[27] M Inc A Yusuf A I Aliyu and D Baleanu ldquoLie symmetryanalysis explicit solutions and conservation laws for the space-time fractional nonlinear evolution equationsrdquo Physica AStatistical Mechanics and its Applications vol 496 pp 371ndash3832018

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Submit your manuscripts atwwwhindawicom

Journal of Applied Mathematics 5

When 119899 ge 1 the recurrence relations between the seriescoefficients are

119886119899+2 = minus12ℎ (119899 + 2) (119899 + 1) (Γ (2 minus 31205722 minus 1198991205722)Γ (2 minus 1205722 minus 1198991205722) 119886119899+ (119899 + 1) 119886119899119886119899+1)

(44)

Using (44) the series solution for (39) can be represented bysubstituting (43) and (44) into (39)

119891 (120585) = 1198860 + 1198861120585 + 11988621205852 + infinsum119899=1

119886119899+2120585119899+2 = 1198860 + 1198861120585

minus 12ℎ (Γ (2 minus 31205722)Γ (2 minus 1205722) 1198860 + 11988601198861) 1205852

minus infinsum119899=1

12ℎ (119899 + 2) (119899 + 1) (Γ (2 minus 31205722 minus 1198991205722)Γ (2 minus 1205722 minus 1198991205722) 119886119899+ (119899 + 1) 119886119899119886119899+1) 120585119899+2

(45)

Upon substitution using similarity variables in (25) thefollowing explicit solutions for (2) are

119906 (119909 119905) = 1198860119905minus1205722 + 1198861119909119905minus120572 minus 12ℎ (Γ (2 minus 31205722)Γ (2 minus 1205722) 1198860+ 11988601198861) 119905minus1205722 (119909119905minus1205722)2

minus infinsum119899=1

12ℎ (119899 + 2) (119899 + 1) Γ (2 minus 31205722 minus 1198991205722)Γ (2 minus 1205722 minus 1198991205722) 119886119899

+ 119899sum119896=0

119896sum119895=0

(119886119895+119886119896minus119895) 119886119899minus119896 (119899 + 1)(119909)119899+2

sdot 119905minus120572(2119899+120572120573)2120573

(46)

5 Convergence Analysis

To satisfy the convergence test there are many kinds of testsas the ratio the comparison and the quotient tests Theconvergence of the solution equation (46) will be presentedas follows We revamp (46) as follows

1003816100381610038161003816119886119899+21003816100381610038161003816 le (|Γ (2 minus 31205722 minus 1198991205722)||Γ (2 minus 1205722 minus 1198991205722)| 10038161003816100381610038161198861198991003816100381610038161003816minus 119899sum119896=0

119896sum119895=0

1003816100381610038161003816100381611988611989510038161003816100381610038161003816 10038161003816100381610038161003816119886119896minus11989510038161003816100381610038161003816 1003816100381610038161003816119886119899minus1198961003816100381610038161003816 minus 10038161003816100381610038161198861198991003816100381610038161003816)(47)

Equation (47) utilizing the Gamma function shows that|Γ(2 minus 31205722 minus 1198991205722)||Γ(2 minus 1205722 minus 1198991205722)| lt 1 for arbitrary119899 that1003816100381610038161003816119886119899+21003816100381610038161003816 le 119872(10038161003816100381610038161198861198991003816100381610038161003816 minus 119899sum

119896=0

119896sum119895=0

1003816100381610038161003816100381611988611989510038161003816100381610038161003816 10038161003816100381610038161003816119886119896minus11989510038161003816100381610038161003816 1003816100381610038161003816119886119899minus1198961003816100381610038161003816 minus 10038161003816100381610038161198861198991003816100381610038161003816) (48)

where119872 = max(|1198881| |1198882|) We now assume another form ofthe PSM

119861 (120585) = infinsum119899=0

119888119899120585119899 (49)

By comparing the two series we can observe that |119888119899| le 119886119899119899 = 0 1 Hence the series119861(120585) = suminfin119899=0 119888119899120585119899 is themajorantseries of (47) So we find that

119861 (120585) = 1198880 + 1198881120585+119872(infinsum

119899=0

1198881198991205852119861 (120585) + infinsum119899=0

119899sum119896=0

119896sum119895=0

119888119895119888119896minus119895119888119899minus119896 + infinsum119899=0

119888119899)sdot 120585119899+2

(50)

Consider an implicit functional system regarding 120585 as follows120573 (120585 119861)

= 119861 minus 1198880 minus 1198881120585 minus 11988821205852minus119872(1205852119861 (120585) + 2119861 (120585)2 + (1205852 minus 1198881120585 minus 31198880) 119861)

(51)

since 120573 is analytic in a neighborhood of (0 1198880) where120573(0 1198880) = 0 and (120597120597119861)120573(0 1198880) = 0 Then the series 119861(120585) =suminfin119899=0 120585119899 is analytic around (0 1198880) and this is verified utilizing[27] and the radius of convergence of this series belongs toa positive domain This shows that (46) converges around(0 1198880)6 Physical Performance of the Power SeriesTechnique for Eqs (46)

To have expressed and convenient conception of the physicalcharacteristic of the power series solution the 3D plots forthe explicit solution equations (46) is plotted in Figures 1ndash4 atℎ = 1 by utilizing appropriate parameter formsThe spectaclevision of the real portion of (46) can be visible in the 3D plotsproof in Figures 1 2 3 and 4 respectively

6 Journal of Applied Mathematics

2002

2001

2

1999

0 0

4 4

xt

Figure 1 3D plot of (46) with 1198860 = 19 1198861 = 17 1198862 = 077 120574 = 1120572 = 08 120573 = 2

u(x

t)

2002

2001

2

1999

0 0

4 4xt

Figure 2 3D plot of (46) with 1198860 = 39 1198861 = 17 1198862 = 087 120574 = 1120572 = 06 120573 = 1

7 Conclusions

Lie point symmetry properties of (1 + 1)-dimensional time-fractional Benjamin-Ono equation have been consideredwith the RiemannndashLiouville fractional derivativeThese sym-metries are used here to transform the FPDEs intoNLFODEsClosed-form solutions are determined by using PSM inthe last division The accuracy exhibits the assembly ofthe solution Considerable frames for the acquired explicitsolutions were approached

2002

2001

2

1999

0 0

4 4

xt

u(x t)

Figure 3 3D plot of (46) with 1198860 = 1 1198861 = 11 1198862 = 057 120574 = 1120572 = 05 120573 = 07u(x t)

2002

2001

2

1999

0 0

4 4

xt

Figure 4 3D plot of (46) with 1198860 = 1 1198861 = 1 1198862 = 04 120574 = 1120572 = 03 120573 = 04

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Applied Mathematics 7

Conflicts of Interest

The author declares that they have no conflicts of interest

References

[1] M Bruzon et al ldquoThe symmetry reductions of a turbulencemodelrdquo Journal of Physics A Mathematical and General vol 34no 18 p 3751 2001

[2] R Sadat and M Kassem ldquoExplicit Solutions for the (2+ 1)-dimensional jaulentndashmiodek equation using the integratingfactors method in an unbounded domainrdquo Mathematical ampComputational Applications vol 23 no 1 p 15 2018

[3] A Paliathanasis andMTsamparlis ldquoLie symmetries for systemsof evolution equationsrdquo Journal of Geometry and Physics vol124 pp 165ndash169 2018

[4] Y Y Zhang X Q Liu and G W Wang ldquoSymmetry reductionsand exact solutions of the (2+ 1)-dimensional JaulentndashMiodekequationrdquo Applied Mathematics and Computation vol 219 no3 pp 911ndash916 2012

[5] M Mirzazadeh and M Eslami ldquoExact solutions of theKudryashov-Sinelshchikov equation and nonlinear telegraphequation via the first integral methodrdquo Nonlinear AnalysisModelling and Control vol 17 no 4 pp 481ndash488 2012

[6] M Mirzazadeh M Eslami A H Bhrawy B Ahmed and ABiswas ldquoSolitons and other solutions to complex-valued Klein-Gordon equation in Φ-4 field theoryrdquo Applied Mathematics ampInformation Sciences vol 9 no 6 pp 2793ndash2801 2015

[7] ANazarzadehM Eslami andMMirzazadeh ldquoExact solutionsof somenonlinear partial differential equations using functionalvariablemethodrdquoPramanamdashJournal of Physics vol 81 no 2 pp225ndash236 2013

[8] 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

[9] M Mirzazadeh M Eslami and A H Arnous ldquoDark opticalsolitons of Biswas-Milovic equation with dual-power law non-linearityrdquo The European Physical Journal Plus vol 130 4 no 12015

[10] E Aksoy M Kaplan and A Bekir ldquoExponential rational func-tion method for space-time fractional differential equationsrdquoWaves in random and complex media propagation scatteringand imaging vol 26 no 2 pp 142ndash151 2016

[11] A Bekir andA C Cevikel ldquoNew exact travelling wave solutionsof nonlinear physical modelsrdquo Chaos Solitons and Fractals vol41 no 4 pp 1733ndash1739 2009

[12] M Eslami M Mirzazadeh B Fathi Vajargah and A BiswasldquoOptical solitons for the resonant nonlinear Schrodingerrsquosequation with time-dependent coefficients by the first integralmethodrdquo Optik - International Journal for Light and ElectronOptics vol 125 no 13 pp 3107ndash3116 2014

[13] M F El-Sabbagh R Zait and R M Abdelazeem ldquoNew exactsolutions of somenonlinear partial differential equations via theimproved exp-functionmethodrdquo IJRRAS vol 18 no 2 pp 132ndash144 2014

[14] Y Gurefe E Misirli A Sonmezoglu and M Ekici ldquoExtendedtrial equation method to generalized nonlinear partial differen-tial equationsrdquo Applied Mathematics and Computation vol 219no 10 pp 5253ndash5260 2013

[15] A H Khater M H Moussa and S F Abdul-Aziz ldquoInvariantvariational principles and conservation laws for some nonlinearpartial differential equations with constant coefficients - IrdquoChaos Solitons and Fractals vol 14 no 9 pp 1389ndash1401 2002

[16] Q Feng and F Meng ldquoTraveling wave solutions for fractionalpartial differential equations arising in mathematical physicsby an improved fractional Jacobi elliptic equation methodrdquoMathematical Methods in the Applied Sciences vol 40 no 10pp 3676ndash3686 2017

[17] R K Gazizov A A Kasatkin and S Y Lukashchuk ldquoContinu-ous transformation groups of fractional differential equationsrdquoVestnik Usatu vol 9 no 3 p 21 2007

[18] S Zhang and H Q Zhang ldquoFractional sub-equation methodand its applications to nonlinear fractional PDEsrdquo PhysicsLetters A vol 375 no 7 pp 1069ndash1073 2011

[19] B Zheng ldquoA new fractional Jacobi elliptic equation methodfor solving fractional partial differential equationsrdquoAdvances inDifference Equations vol 2014 no 1 p 228 2014

[20] A Neyrame A Roozi S S Hosseini and S M Shafiof ldquoExacttravelling wave solutions for some nonlinear partial differentialequationsrdquo Journal of King Saud University - Science vol 22 no4 pp 275ndash278 2010

[21] K B Oldham and J Spanier The Fractional Calculus Theoryand Applications of Differentiation and Integration to ArbitraryOrder vol 111 Academic Press New York NY USA LondonUK 1974

[22] B Ahmad S K Ntouyas and A Alsaedi ldquoOn a coupled systemof fractional differential equations with coupled nonlocal andintegral boundary conditionsrdquo Chaos Solitons amp Fractals vol83 pp 234ndash241 2016

[23] V S Kiryakova Generalized Fractional Calculus and Applica-tions CRC press Botan Roca Fl USA 1993

[24] G-WWang X-Q Liu andY-Y Zhang ldquoLie symmetry analysisto the time fractional generalized fifth-order KdV equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 9 pp 2321ndash2326 2013

[25] S S Ray and S Sahoo ldquoInvariant analysis and conservation lawsof (2+ 1) dimensional time-fractional ZKBBM equation in grav-ity water wavesrdquo Computers amp Mathematics with Applications2017

[26] G-W Wang and T-Z Xu ldquoInvariant analysis and exact solu-tions of nonlinear time fractional Sharma-Tasso-Olver equationby Lie group analysisrdquo Nonlinear Dynamics vol 76 no 1 pp571ndash580 2014

[27] M Inc A Yusuf A I Aliyu and D Baleanu ldquoLie symmetryanalysis explicit solutions and conservation laws for the space-time fractional nonlinear evolution equationsrdquo Physica AStatistical Mechanics and its Applications vol 496 pp 371ndash3832018

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 Journal of Applied Mathematics

2002

2001

2

1999

0 0

4 4

xt

Figure 1 3D plot of (46) with 1198860 = 19 1198861 = 17 1198862 = 077 120574 = 1120572 = 08 120573 = 2

u(x

t)

2002

2001

2

1999

0 0

4 4xt

Figure 2 3D plot of (46) with 1198860 = 39 1198861 = 17 1198862 = 087 120574 = 1120572 = 06 120573 = 1

7 Conclusions

Lie point symmetry properties of (1 + 1)-dimensional time-fractional Benjamin-Ono equation have been consideredwith the RiemannndashLiouville fractional derivativeThese sym-metries are used here to transform the FPDEs intoNLFODEsClosed-form solutions are determined by using PSM inthe last division The accuracy exhibits the assembly ofthe solution Considerable frames for the acquired explicitsolutions were approached

2002

2001

2

1999

0 0

4 4

xt

u(x t)

Figure 3 3D plot of (46) with 1198860 = 1 1198861 = 11 1198862 = 057 120574 = 1120572 = 05 120573 = 07u(x t)

2002

2001

2

1999

0 0

4 4

xt

Figure 4 3D plot of (46) with 1198860 = 1 1198861 = 1 1198862 = 04 120574 = 1120572 = 03 120573 = 04

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Applied Mathematics 7

Conflicts of Interest

The author declares that they have no conflicts of interest

References

[1] M Bruzon et al ldquoThe symmetry reductions of a turbulencemodelrdquo Journal of Physics A Mathematical and General vol 34no 18 p 3751 2001

[2] R Sadat and M Kassem ldquoExplicit Solutions for the (2+ 1)-dimensional jaulentndashmiodek equation using the integratingfactors method in an unbounded domainrdquo Mathematical ampComputational Applications vol 23 no 1 p 15 2018

[3] A Paliathanasis andMTsamparlis ldquoLie symmetries for systemsof evolution equationsrdquo Journal of Geometry and Physics vol124 pp 165ndash169 2018

[4] Y Y Zhang X Q Liu and G W Wang ldquoSymmetry reductionsand exact solutions of the (2+ 1)-dimensional JaulentndashMiodekequationrdquo Applied Mathematics and Computation vol 219 no3 pp 911ndash916 2012

[5] M Mirzazadeh and M Eslami ldquoExact solutions of theKudryashov-Sinelshchikov equation and nonlinear telegraphequation via the first integral methodrdquo Nonlinear AnalysisModelling and Control vol 17 no 4 pp 481ndash488 2012

[6] M Mirzazadeh M Eslami A H Bhrawy B Ahmed and ABiswas ldquoSolitons and other solutions to complex-valued Klein-Gordon equation in Φ-4 field theoryrdquo Applied Mathematics ampInformation Sciences vol 9 no 6 pp 2793ndash2801 2015

[7] ANazarzadehM Eslami andMMirzazadeh ldquoExact solutionsof somenonlinear partial differential equations using functionalvariablemethodrdquoPramanamdashJournal of Physics vol 81 no 2 pp225ndash236 2013

[8] 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

[9] M Mirzazadeh M Eslami and A H Arnous ldquoDark opticalsolitons of Biswas-Milovic equation with dual-power law non-linearityrdquo The European Physical Journal Plus vol 130 4 no 12015

[10] E Aksoy M Kaplan and A Bekir ldquoExponential rational func-tion method for space-time fractional differential equationsrdquoWaves in random and complex media propagation scatteringand imaging vol 26 no 2 pp 142ndash151 2016

[11] A Bekir andA C Cevikel ldquoNew exact travelling wave solutionsof nonlinear physical modelsrdquo Chaos Solitons and Fractals vol41 no 4 pp 1733ndash1739 2009

[12] M Eslami M Mirzazadeh B Fathi Vajargah and A BiswasldquoOptical solitons for the resonant nonlinear Schrodingerrsquosequation with time-dependent coefficients by the first integralmethodrdquo Optik - International Journal for Light and ElectronOptics vol 125 no 13 pp 3107ndash3116 2014

[13] M F El-Sabbagh R Zait and R M Abdelazeem ldquoNew exactsolutions of somenonlinear partial differential equations via theimproved exp-functionmethodrdquo IJRRAS vol 18 no 2 pp 132ndash144 2014

[14] Y Gurefe E Misirli A Sonmezoglu and M Ekici ldquoExtendedtrial equation method to generalized nonlinear partial differen-tial equationsrdquo Applied Mathematics and Computation vol 219no 10 pp 5253ndash5260 2013

[15] A H Khater M H Moussa and S F Abdul-Aziz ldquoInvariantvariational principles and conservation laws for some nonlinearpartial differential equations with constant coefficients - IrdquoChaos Solitons and Fractals vol 14 no 9 pp 1389ndash1401 2002

[16] Q Feng and F Meng ldquoTraveling wave solutions for fractionalpartial differential equations arising in mathematical physicsby an improved fractional Jacobi elliptic equation methodrdquoMathematical Methods in the Applied Sciences vol 40 no 10pp 3676ndash3686 2017

[17] R K Gazizov A A Kasatkin and S Y Lukashchuk ldquoContinu-ous transformation groups of fractional differential equationsrdquoVestnik Usatu vol 9 no 3 p 21 2007

[18] S Zhang and H Q Zhang ldquoFractional sub-equation methodand its applications to nonlinear fractional PDEsrdquo PhysicsLetters A vol 375 no 7 pp 1069ndash1073 2011

[19] B Zheng ldquoA new fractional Jacobi elliptic equation methodfor solving fractional partial differential equationsrdquoAdvances inDifference Equations vol 2014 no 1 p 228 2014

[20] A Neyrame A Roozi S S Hosseini and S M Shafiof ldquoExacttravelling wave solutions for some nonlinear partial differentialequationsrdquo Journal of King Saud University - Science vol 22 no4 pp 275ndash278 2010

[21] K B Oldham and J Spanier The Fractional Calculus Theoryand Applications of Differentiation and Integration to ArbitraryOrder vol 111 Academic Press New York NY USA LondonUK 1974

[22] B Ahmad S K Ntouyas and A Alsaedi ldquoOn a coupled systemof fractional differential equations with coupled nonlocal andintegral boundary conditionsrdquo Chaos Solitons amp Fractals vol83 pp 234ndash241 2016

[23] V S Kiryakova Generalized Fractional Calculus and Applica-tions CRC press Botan Roca Fl USA 1993

[24] G-WWang X-Q Liu andY-Y Zhang ldquoLie symmetry analysisto the time fractional generalized fifth-order KdV equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 9 pp 2321ndash2326 2013

[25] S S Ray and S Sahoo ldquoInvariant analysis and conservation lawsof (2+ 1) dimensional time-fractional ZKBBM equation in grav-ity water wavesrdquo Computers amp Mathematics with Applications2017

[26] G-W Wang and T-Z Xu ldquoInvariant analysis and exact solu-tions of nonlinear time fractional Sharma-Tasso-Olver equationby Lie group analysisrdquo Nonlinear Dynamics vol 76 no 1 pp571ndash580 2014

[27] M Inc A Yusuf A I Aliyu and D Baleanu ldquoLie symmetryanalysis explicit solutions and conservation laws for the space-time fractional nonlinear evolution equationsrdquo Physica AStatistical Mechanics and its Applications vol 496 pp 371ndash3832018

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

Journal of Applied Mathematics 7

Conflicts of Interest

The author declares that they have no conflicts of interest

References

[1] M Bruzon et al ldquoThe symmetry reductions of a turbulencemodelrdquo Journal of Physics A Mathematical and General vol 34no 18 p 3751 2001

[2] R Sadat and M Kassem ldquoExplicit Solutions for the (2+ 1)-dimensional jaulentndashmiodek equation using the integratingfactors method in an unbounded domainrdquo Mathematical ampComputational Applications vol 23 no 1 p 15 2018

[3] A Paliathanasis andMTsamparlis ldquoLie symmetries for systemsof evolution equationsrdquo Journal of Geometry and Physics vol124 pp 165ndash169 2018

[4] Y Y Zhang X Q Liu and G W Wang ldquoSymmetry reductionsand exact solutions of the (2+ 1)-dimensional JaulentndashMiodekequationrdquo Applied Mathematics and Computation vol 219 no3 pp 911ndash916 2012

[5] M Mirzazadeh and M Eslami ldquoExact solutions of theKudryashov-Sinelshchikov equation and nonlinear telegraphequation via the first integral methodrdquo Nonlinear AnalysisModelling and Control vol 17 no 4 pp 481ndash488 2012

[6] M Mirzazadeh M Eslami A H Bhrawy B Ahmed and ABiswas ldquoSolitons and other solutions to complex-valued Klein-Gordon equation in Φ-4 field theoryrdquo Applied Mathematics ampInformation Sciences vol 9 no 6 pp 2793ndash2801 2015

[7] ANazarzadehM Eslami andMMirzazadeh ldquoExact solutionsof somenonlinear partial differential equations using functionalvariablemethodrdquoPramanamdashJournal of Physics vol 81 no 2 pp225ndash236 2013

[8] 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

[9] M Mirzazadeh M Eslami and A H Arnous ldquoDark opticalsolitons of Biswas-Milovic equation with dual-power law non-linearityrdquo The European Physical Journal Plus vol 130 4 no 12015

[10] E Aksoy M Kaplan and A Bekir ldquoExponential rational func-tion method for space-time fractional differential equationsrdquoWaves in random and complex media propagation scatteringand imaging vol 26 no 2 pp 142ndash151 2016

[11] A Bekir andA C Cevikel ldquoNew exact travelling wave solutionsof nonlinear physical modelsrdquo Chaos Solitons and Fractals vol41 no 4 pp 1733ndash1739 2009

[12] M Eslami M Mirzazadeh B Fathi Vajargah and A BiswasldquoOptical solitons for the resonant nonlinear Schrodingerrsquosequation with time-dependent coefficients by the first integralmethodrdquo Optik - International Journal for Light and ElectronOptics vol 125 no 13 pp 3107ndash3116 2014

[13] M F El-Sabbagh R Zait and R M Abdelazeem ldquoNew exactsolutions of somenonlinear partial differential equations via theimproved exp-functionmethodrdquo IJRRAS vol 18 no 2 pp 132ndash144 2014

[14] Y Gurefe E Misirli A Sonmezoglu and M Ekici ldquoExtendedtrial equation method to generalized nonlinear partial differen-tial equationsrdquo Applied Mathematics and Computation vol 219no 10 pp 5253ndash5260 2013

[15] A H Khater M H Moussa and S F Abdul-Aziz ldquoInvariantvariational principles and conservation laws for some nonlinearpartial differential equations with constant coefficients - IrdquoChaos Solitons and Fractals vol 14 no 9 pp 1389ndash1401 2002

[16] Q Feng and F Meng ldquoTraveling wave solutions for fractionalpartial differential equations arising in mathematical physicsby an improved fractional Jacobi elliptic equation methodrdquoMathematical Methods in the Applied Sciences vol 40 no 10pp 3676ndash3686 2017

[17] R K Gazizov A A Kasatkin and S Y Lukashchuk ldquoContinu-ous transformation groups of fractional differential equationsrdquoVestnik Usatu vol 9 no 3 p 21 2007

[18] S Zhang and H Q Zhang ldquoFractional sub-equation methodand its applications to nonlinear fractional PDEsrdquo PhysicsLetters A vol 375 no 7 pp 1069ndash1073 2011

[19] B Zheng ldquoA new fractional Jacobi elliptic equation methodfor solving fractional partial differential equationsrdquoAdvances inDifference Equations vol 2014 no 1 p 228 2014

[20] A Neyrame A Roozi S S Hosseini and S M Shafiof ldquoExacttravelling wave solutions for some nonlinear partial differentialequationsrdquo Journal of King Saud University - Science vol 22 no4 pp 275ndash278 2010

[21] K B Oldham and J Spanier The Fractional Calculus Theoryand Applications of Differentiation and Integration to ArbitraryOrder vol 111 Academic Press New York NY USA LondonUK 1974

[22] B Ahmad S K Ntouyas and A Alsaedi ldquoOn a coupled systemof fractional differential equations with coupled nonlocal andintegral boundary conditionsrdquo Chaos Solitons amp Fractals vol83 pp 234ndash241 2016

[23] V S Kiryakova Generalized Fractional Calculus and Applica-tions CRC press Botan Roca Fl USA 1993

[24] G-WWang X-Q Liu andY-Y Zhang ldquoLie symmetry analysisto the time fractional generalized fifth-order KdV equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 9 pp 2321ndash2326 2013

[25] S S Ray and S Sahoo ldquoInvariant analysis and conservation lawsof (2+ 1) dimensional time-fractional ZKBBM equation in grav-ity water wavesrdquo Computers amp Mathematics with Applications2017

[26] G-W Wang and T-Z Xu ldquoInvariant analysis and exact solu-tions of nonlinear time fractional Sharma-Tasso-Olver equationby Lie group analysisrdquo Nonlinear Dynamics vol 76 no 1 pp571ndash580 2014

[27] M Inc A Yusuf A I Aliyu and D Baleanu ldquoLie symmetryanalysis explicit solutions and conservation laws for the space-time fractional nonlinear evolution equationsrdquo Physica AStatistical Mechanics and its Applications vol 496 pp 371ndash3832018

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