Kamyar Hosseini,Peyman Mayeli,Ahmet Bekir,and Ozkan Guner

1Department of Mathematics,Rasht Branch,Islamic Azad University,Rasht,Iran

2Young Researchers and Elite Club,Lahijan Branch,Islamic Azad University,Lahijan,Iran

3Department of Mathematics and Computer,Art-Science Faculty,Eskisehir Osmangazi University,Eskisehir,Turkey

4Department of International Trade,Faculty of Economics and Administrative Sciences,Cankiri Karatekin University,Cankiri,Turkey

1 Introduction

Fractional differential equations are mathematical models which are appeared in the vast areas of science and engineering and great attention has been directed toward them over the last few decades.To be more speci fic,FDEs are the generalizations of classical differential equations which play a signi ficant role in the mentioned areas.Fortunately,it is possible to establish a traveling wave transformation for a fractional differential equation which can convert it to an ordinary differential equation(ODE)of integer order such that the resulting ODE can be easily solved using a variety of robust methods.[1?15]

One of well-designed methods which may be employed to solve nonlinear fractional differential equations is the exp(??(ε))method. This method has a fairly great performance in handling nonlinear FDEs.For example,Mohyud-Din and Ali[16]adopted the exp(??(ε))method to construct the solitary wave solutions of the fractional generalized Sawada-Kotera equation;and Zahran[17]established the exact solutions of some nonlinear FDEs using the exp(??(ε))method.

Another mathematical method which is truly robust to solve nonlinear FDEs is a modi fied form of Kudryashov method. Fundamental of this method is described in detail in the next sections,so here just some applications of this technique are reviewed.Korkmaz[18]utilized the modi fied Kudryashov method to construct the exact solutions of a family of the conformable time-fractional Benjamin-Bona-Mahony equations;and Hosseiniet al.[19]constructed a series of new exact solutions of the conformable time-fractional Klein-Gordon equations using the modi fied Kudryashov method.More articles may be found in Refs.[20–28].

In this paper,the exp(??(ε))-expansion and modi fied Kudryashov methods are adopted to obtain the exact solutions of the DDCFDR equation as[29]

which is a model arising in the applied science.For the awareness of the reader,this equation has been solved by Guner and Bekir[29]via the exp-function method.

The rest of this work is as follows:In Sec.2,we define the conformable fractional derivative and list some of its properties.In Sec.3,we explain the ideas of the exp(??(ε))-expansion and modi fied Kudryashov methods.In Sec.4,we employ the methods to solve the densitydependent conformable space-time fractional diffusionreaction equation.Finally,we give a brief conclusion in Sec.5.

2 Conformable Fractional Derivative

There are various de finitions for the fractional derivatives.Among these,the conformable fractional derivative has gained a special interest during the last years.Theαth order of the conformable fractional derivative offcan be de fined as[30]

wheret>0 andα∈(0,1].The physical and geometrical interpretations of the conformable derivative have been given in Ref.[31].

A series of the properties of conformable derivative may be listed as follows[30,32]

3 Description of the Methods

Using the transformation

3.1 Exp(??(ε))Method

We look for an explicit solution for the Eq.(2)as the following form

where the constantsan,n=0,1,2,...,Nare determined later,Nis a positive number,which is computed by the technique of homogeneous balance,and?(ε)is an explicit function that satis fies the following ODE

By inserting Eq.(3)into Eq.(2)with the help ofMAPLEand equating the coefficients of like powers of exp(??(ε)),we will gain an algebraic system for obtainingan’s,l1,andl2.Setting the results into Eq.(3),at the end yields the exact solutions of Eq.(1).

3.2 Modi fied Kudryashov Method

The initial steps of the modi fied Kudryashov method are as before.With the same transformation,the original fractional differential equation can be converted to a nonlinear ODE,which its solution is supposed to be in the form

where the constantsan,n=0,1,2,...,Nare determined later,Nis a positive number which is computed by the technique of homogeneous balance,andQ(ε)=1/(1+daε)is an explicit function that satis fies the following ODE

By inserting Eq.(4)into Eq.(2)with the help ofMAPLEand equating the coefficients of like powers ofQ(ε),we will get an algebraic system for obtainingan’s,l1,andl2.Setting the results into Eq.(4),at the end gives new exact solutions of Eq.(1).

4 DDCFDR Equation and Its Exact Solutions

In thissection,the exactsolutionsofdensitydependent conformable space-time fractional diffusionreaction equation will be extracted using the exp(??(ε))-expansion and modi fied Kudryashov methods.Some of the solutions for the mentioned equation are new and have been reported for the first time.

Using the transformation

the DDCFDR equation can be reduced to a nonlinear ODE as

4.1 Applying the exp(??(ε))Method

From the technique of homogeneous balance,we findN=1.This offers a series as the following form

By inserting Eq.(6)into Eq.(5)with the help ofMAPLEand equating the coefficients of like powers of exp(??(ε)),we will gain an algebraic system for obtainingan’s,l1,andl2as follows

After solving the above system,we find

Case 1

Now,the following exact solutions for the DDCFDR equation can be extracted

4.2 Applying the Modi fied Kudryashov Method

It is obvious thatN=1.Therefore,a series can be derived as follows

By inserting Eq.(7)into Eq.(5)with the help ofMAPLEand equating the coefficients of like powers ofQ(ε),we will get an algebraic system for obtainingan’s,l1,andl2as

wheredis a constant.

RemarkThe correctness of the solutions listed in the present work has been veri fied using theMAPLEpackage.

5 Conclusion

The exact solutions of density-dependent conformable space-time fractional diffusion-reaction equation have been successfully extracted using the newly well-organized techniques called the exp(??(ε))-expansion and modi fied Kudryashov methods.Although the both methods result in a number of exact solutions for the governing model,the modi fied Kudryashov method has some clear advantages over the exp(??(ε))method.For example

(i)The modi fied Kudryashov method provides more straightforward solution procedure.

(ii)The modi fied Kudryashov method considers an arbitrary constanta=1 as the base of the exponential function;therefore,this method can generate new exact solutions of FDEs.

(iii)The modi fied Kudryashov method can be easily applied to handle the high order differential equations as illustrated by Zayed and Alurr fi.[21]

Accordingly,it is fair to say that the modi fied Kudryashov method can be considered as one of the best techniques to extract new exact solutions of FDEs.

[1]M.Mirzazadeh,M.Eslami,D.Milovic,and A.Biswas,Optik 125(2014)5480.

[2]M.Younis and S.T.R.Rizvi,Optik 126(2015)5812.

[3]Q.Zhou,M.Ekici,A.Sonmezoglu,and M.Mirzazadeh,Optik 127(2016)6277.

[4]K.Hosseini,Z.Ayati,and R.Ansari,Optik 145(2017)85.

[5]M.Ekici,M.Mirzazadeh,M.Eslami,et al.,Optik 127(2016)10659.

[6]K.Hosseini and P.Gholamin,Differ.Equ.Dyn.Syst.23(2015)317.

[7]A.Zerarka,S.Ouamane,and A.Attaf,Appl.Math.Comput.217(2010)2897.

[8]X.H.Wu and J.H.He,Chaos,Solitons&Fractals 38(2008)903.

[9]C.K.Kuo,Optik 147(2017)128.

[10]M.Kaplan,A.Bekir,and M.Naci Ozer,Opt.Quantum Electron.49(2017)266.

[11]M.Kaplan,Opt.Quantum Electron.49(2017)312.

[12]S.Saha Ray and S.Singh,Commun.Theor.Phys.67(2017)197.

[14]M.Fazli Aghdaei and H.Adibi,Opt.Quantum Electron.49(2017)316.

[15]A.Biswas and R.T.Alqahtani,Optik 147(2017)72.

[16]S.T.Mohyud-Din and A.Ali,Fundam.Inform.151(2017)173.

[17]E.H.M.Zahran,Int.J.Comput.Appl.109(2015)12.

[18]A.Korkmaz,arXiv:1611.07086v2[nlin.SI]3Dec.(2016).

[19]K.Hosseini,P.Mayeli,and R.Ansari,Optik 130(2017)737.

[20]R.S.Saha,Chin.Phys.B 25(2016)040204.

[21]E.M.E.Zayed and K.A.E.Alurr fi,World J.Model.Simul.11(2015)308.

[22]K.Hosseini,A.Bekir,and R.Ansari,Optik 132(2017)203.

[23]K.Hosseini,E.Yazdani Bejarbaneh,A.Bekir,and M.Kaplan,Opt.Quantum Electron.49(2017)241.

[24]K.Hosseini and R.Ansari,Waves Random Complex Media 27(2017)628.

[25]K.Hosseini,A.Bekir,and R.Ansari,Opt.Quantum Electron.49(2017)131.

[26]A.Korkmaz and K.Hosseini,Opt.Quantum Electron.49(2017)278.

[27]W.Chen,Chaos,Solitons&Fractals 28(2006)923.

[28]W.Chen,H.Sun,X.Zhang,and D.Koroak,Comput.Math.Appl.59(2010)1754.

[29]¨O.G¨uner and A.Bekir,Int.J.Biomath.8(2015)155003.

[30]R.Khalil,M.Al-Horani,A.Yousef,and M.Sababheh,J.Comput.Appl.Math.264(2014)65.

[31]D.Zhao and M.Luo,Calcolo 54(2017)903.

[32]M.Eslami and H.Rezazadeh,Calcolo 53(2016)475.