Group Analysis,Fractional Explicit Solutions and Conservation Laws of Time Fractional Generalized Burgers Equation

Gang-Wei Wang(王崗偉) and A.H.Kara

1School of Mathematics and Statistics,Hebei University of Economics and Business,Shijiazhuang,050061,China

2School of Mathematics,University of the Witwatersrand,Private Bag 3,Wits 2050,Johannesburg,South Africa

1 Introduction

a fractional order nonlinear Burgers equation of the form

It is known that the classical Burgers equation(BE)is one of the better known of the nonlinear evolution equations.It comes up in various areas of sciences,in particular in physical fields.This equation can be derived from Navier-Stokes equation.Various methods have been used in the literatures to investigate these types equations(see Refs.[1–3]and the references therein).

In addition,fractional differential equations(FDEs)have been paid much attention in many fields.Therefore,how to get exact solutions of FDEs becomes the key step to better under the real world.Some methods have been developed to deal with FDEs,such as the adomian decomposition method,[4]variational iteration method,[5]homotopy perturbation method,[6]invariant subspace method,[7]symmetry analysis,[8?18]and other approaches.

In Ref.[8],the authors studied nonlinear anomalous diffusion equations with time fractional derivatives.They derived Lie point symmetries of these equations and obtained exact solutions of the nonlinear anomalous diffusion equations.In Ref.[9],the authors considered time fractional generalized Burgers and Korteweg-de Vries equations and derived their Lie point symmetries.In Refs.[10–11],fifth-order KdV FDEs are studied using the Lie symmetry analysis.One of the authors and his co-author studied fractional STO equation,perturbed Burgers equation,and KdV equation.[12?14]In Refs.[15–16],the authors studied the conservation laws of fractional equation.In view of the importance of FDEs,it is necessary for us to look for more properties of FDEs.

This paper is continuation of our previous work.[10?15]In this paper,we apply the Lie group method to analyze

whereu(x,t)is function ofxandt,0<α<1,whileaandbare constants.ais the coefficient of the convection term whilebis the coefficient of viscous term.Ifb=0,Eq.(1)is the generalized form of the inviscid BE.The parametersmandnare all positive integers.In the special case whenm=n=1,Eq.(1)reduces to the case of the regular BE.If onlyn=1,it collapses to the case of Ref.[17].

The paper is organized as follows.In Sec.2,symmetries of the generalized fractional Burgers equation are constructed.In Sec.3,exact solution is derived by using the ansatz method.Also,we get other types exact solutions by using Invariant subspace method.In Sec.4,conservation laws are constructed.Finally,the conclusion is presented.

2 Lie Symmetry Analysis of Time Fractional Generalized Burgers Equation

In this section,we will study the invariance properties of the time fractional generalized Burgers equation with full nonlinearity.

We employ Lie symmetry method to Eq.(1),one getsTheorem 1The time fractional generalized Burgers equation with full nonlinearity(1)has the following vector fields

From the symmetry equation,one can have

Applying Lie symmetry method,and equating various powers of derivatives ofuto zero,we derived the corresponding determining equations for Eq.(1)

is the Erd′elyi-Kober fractional integral operator.

3 Fractional Soliton Solution

In this section,we aim to construct the solitons solution of Eq.(1). First,we give the following transformation[3]

3.1 Numerical Simulations

This section shows some numerical solutions of the time fractional generalized Burgers with full nonlinearity.We consider the case whenm=n=2,b=1,a=3,A=1,B=1/2.

Figure 1 shows the solution whileα=1 is fixed.Figure 2 expresses the solution for varyingαatt=2.Figure 3 shows the the solution for varyingαatx=2.

Fig.1 Plots of exact solution with respect to α=1.

Fig.2 Plots with respect to varying α at t=2.

Fig.3 Plots with respect to varying α at x=2.

4 Exact Solution via Invariant Subspace Method

In this section,we consider exact solution via the invariant subspace method[7]for the special casem=n=1.It is clear that it has an invariant subspaceW2= 1,x,as.This means that Eq.(1)has an exact solution in the following form

5 Conservation Laws

In the present section,we will deal with conservation laws for Eq.(1).We can write the conservation laws of form

Here we direct write the operatorTtis[15?16]

6 Conclusion

In this article,the time fractional generalized Burgers equation with nonlinearity is considered.Its Lie point symmetries were also derived.It was shown that the underlying symmetry algebra is two dimensional.Note that ifα=1 andn=m=1,the transformed equations can be reduced to integer order ones.The casen=1,m=1,of this equation was considered in Ref.[17].In particular,some exact solutions were constructed.It is worth investigating more exact solutions and studying nonlocal symmetries and nonlocal conservation laws for fractional partial differential equation like the one discussed in this paper.

[1]J.M.Burgers,Proc.Acad.Sci.Amsterdam 43(1940)2.

[2]B.M.Vaganan and M.Senthilkumaran,Nonlinear Anal.Real.World Appl.9(2008)2222.

[3]A.J.M.Jawad,M.D.Petkovic,and A.Biswas,Appl.Math.Comput.216(2010)3370.

[4]S.S.Ray,Commun.Nonlinear Sci.Numer.Simul.14(2009)1295.

[5]J.H.He,Comput.Methods Appl.Mech.Eng.167(1998)57.

[6]S.Momani and Z.Odibat,Phys.Lett.A 365(2007)345.

[7]R.K.Gazizov and A.A.Kasatkin,Comput.Math.Appl.66(2013)576.

[8]R.K.Gazizov,A.A.Kasatkin,and S.Yu.Lukashchuk,Vestnik,USATU 9(2007)125(in Russian).

[9]S.Sahadevan and T.Bakkyaraj,J.Math.Anal.Appl.393(2012)341.

[10]G.W.Wang,X.Q.Liu,and Y.Y.Zhang,Commun.Nonlinear Sci.Numer.Simulat.18(2013)2321.

[11]G.W.Wang,T.Z.Xu,and T.Feng,Plos One 9(2014)e88336.

[12]G.W.Wang and T.Z.Xu,Nonlinear Dyn.76(2014)571.

[13]G.W.Wang and T.Z.Xu,Nonlinear Analysis:Model.20(2015)570.

[14]G.W.Wang and T.Z.Xu,Bound.Value Probl.232(2013)1.

[15]G.W.Wang,A.H.Kara,and K.Fakhar,Nonlinear Dyn.82(2015)281.

[16]S.Y.Lukashchuk,Nonlinear Dyn.80(2015)791.

[17]X.B.Wang,S.F.Tian,C.Y.Qin,and T.T.Zhang,Europhys.Lett.114(2016)20003.

[18]C.Chen and Y.L.Jiang,Commun.Theor.Phys.68(2017)295.

[19]I.Podlubny,Fractional Differential Equations:An Introduction to Fractional Derivatives,Fractional Differential Equations,to Methods of Their Solution and Some of Their Applications,Academic Press,San Diego(1999).

[20]V.S.Kiryakova,Generalised Fractional Calculus and Applications,Longman Scienti fic&Technical,Harlow,England(1994)p.388.