Symmetries,Symmetry Reductions and Exact Solutions to the Generalized Nonlinear Fractional Wave Equations?

Han-Ze Liu(劉漢澤), Zeng-Gui Wang(王增桂),Xiang-Peng Xin(辛祥鵬),and Xi-Qiang Liu(劉希強(qiáng))

School of Mathematical Sciences,Liaocheng University,Liaocheng 252059,China

1 Introduction

Clarkson,et al.[1]considered the symmetry reductions and exact solutions to a class of nonlinear heat equation ut=uxx+f(u).In Ref.[2],we studied the Lie group classifications of the generalized nonlinear wave equation and its partial difference schemes as follows

In the present paper,we consider the following time fractional generalized nonlinear wave equation:

where u=u(x,t)denotes the unknown function of the space variable x and time t,f=f(u)is a given arbitrary analytic function with respect to the unknown function u=u(x,t),the parameters a and b are all arbitrary constants,and b0.The fractional order α is a real number and ?αu/?tα:=u denotes the Riemann-Liouville(RL)fractional partial derivative with respect to t defined by

where the Euler gamma function Γ(z)defined by the integral

which converges in the right half of the complex plane Re(z)>0.[3?10]

Equation(2)is also called the generalized fractional Burgers’-heat(FBGH)equation.If α =1,then Eq.(2)reduces to the generalized nonlinear wave equation(1).In general,if 0<α<1,then Eq.(2)is the fractional partial differential equation(FPDE)and it includes a lot of important FPDEs as its special cases.For example,if f=0,then Eq.(2)becomes the following fractional Burgers’(FBG)equation

If a=0,then Eq.(2)reduces to the fractional nonlinear heat equation(FNLHE)

If a=0 and f=0,then Eq.(2)is the fractional heat equation(FHE)

We note first that such FPDEs differ greatly from its integer order(α=1)counterparts,especially,the fractional order α>0 affects the properties of the equations greatly.In what follows,we shall find how it affects the symmetries,symmetry reductions,and exact solutions to the equations.Moreover,for these FPDEs,there is no general method for dealing with exact explicit solutions as far as we know.By contrast,the integer-order partial differential equations(PDEs)are studied more extensively and thoroughly,a lot of systematic methods have been developed.[11?14]Relatively,the FPDEs are more complicated and studied rudimentarily.[3?10,15?19]Recently,Chen et al.[15]studied the symmetries and invariant solution to the nonlinear time-fractional diffusion convection equations,we considered the complete group classifications and symmetry reductions of the fractional KdV types of equations,[10]the point symmetries,symmetry reductions of the equations are obtained.However,the exact explicit solutions to the nonlinear fractional equations are not provided as far as we know.

The main purpose of this paper is to deal with the complete Lie group classification,symmetry reductions and exact solutions to the FPDEs.Then,we show that the fractional order case is compatible with the integer order case α=1 in the sense of Lie symmetry analysis method for the first time.

A rough description of this paper is as follows.First,we give complete Lie group classification of Eq.(2)in the fractional case 0<α<1.So,all of the point symmetries of the fractional equation are obtained with respect to their arbitrary parameters and analytic function f=f(u),the compatibility of the symmetry analysis for fractional and integer-order cases is verified simultaneously.Second,we deal with symmetry reductions of the generalized fractional Burgers’-heat equation(2)by the Erdélyi-Kober(E-K)fractional operator method,then the similarity reduction of the integer-order equation(1)is given as its special case for α=1.Then,the exact analytic solutions to the fractional equations are obtained finally.

2 Complete Lie Group Classification of Eq.(2)

In this section,we lucubrate the symmetries of the generalized fractional nonlinear wave equation(2)in the case 0<α<1,then all of the point symmetries of the other FPDEs such as Eqs.(3)–(5)are obtained accordingly.

We assume that the geometric vector field of an FPDE is as follows:

where the coefficient functions ξ(x,t,u), τ(x,t,u),and ?(x,t,u)are to be determined.

Then the vector field(6)generates a symmetry of Eq.(2)if and only if V satisfies the following Lie symmetry condition

where 0<α<1 and?=?αu/?tα?auux?buxx?f(u).Thus,the Lie group analysis method[10,15]for the FPDE leads to the following result:

(I) a=0.In this case,there are two subcases as follows:

(I-1)In particular,if f=0,then the vector field of Eq.(2)is

where the function q=q(x,t)satisfies Eq.(5).

(I-2)In general,if f0,then we have the following cases:

(I-2.1)If f=lu(l0 is an arbitrary constant),then the vector field of Eq.(2)is

(I-2.2) If f=lu2(l0 is an arbitrary constant),then the vector field of Eq.(2)is

(I-2.3)If f=luk(kl0 are arbitrary constants,and k1,2),then the vector field of Eq.(2)is

(I-2.4)If f=leku(kl0 are arbitrary constants),then the vector field of Eq.(2)is

(I-2.5) Except for the above subcases,that is,f is none of the above cases(I-2.1)–(I-2.4),then the vector field of Eq.(2)is Eq.(12)also.

(II) a0.In this case,there are two subcases as follows:

(II-1)In particular,if f=0,then the vector field of Eq.(2)is

(II-2)In general,if f0,then we have the following cases:

(II-2.1)If f=lu(l0 is an arbitrary constant),then the vector field of Eq.(2)is Eq.(12).

(II-2.2) If f=lu2(l0 is an arbitrary constant),then the vector field of Eq.(2)is Eq.(12)as well.

(II-2.3)If f=lu3(kl0 is an arbitrary constant),then the vector field of Eq.(3)is Eq.(13).

(II-2.4)If f=luk(kl0 are arbitrary constants,and k1,2),then the vector field of Eq.(2)is Eq.(12).

(II-2.5)If f=leku(kl0 are arbitrary constants),then the vector field of Eq.(2)is Eq.(12)also.

(II-2.6)Except for the above subcases,that is,f is none of the above cases(II-2.1)–(II-2.5),then the vector field of Eq.(2)is Eq.(12).

By means of the above complete Lie group classification of Eq.(2),we can give all of the point symmetries of the other fractional-order equations straightforwardly.For example,the symmetries of the fractional Burgers’equation(3)is Eq.(13).

The symmetries of the fractional nonlinear heat equation(4)are given by Eqs.(9)–(12),respectively,in terms of the arbitrary analytic function f=f(u).

The symmetries of the fractional heat equation(5)is Eq.(8).

In particular,if α=1/2,then the symmetries of the fractional heat equation

is

where b0 is an arbitrary constant,the function q=q(x,t)satisfies Eq.(14).

If α=1/3,then the symmetries of the fractional Burgers’equation

is

where ab0 are arbitrary constants.

From the above discussion,we can see that the point symmetries of the fractional-order equations are relatively fewer than its integer-order counterparts.There are two reasons as follows:The first is that the fractional order α∈(0,1)is an arbitrary parameter in our discussion,that is,for an arbitrary parameter α∈(0,1),Eq.(2)admits the point symmetries(8)–(13)under the conditions(I)–(II)in Theorem 2.1,respectively.The other reason is that the definition of R-L fractional derivative rather than integer-order derivative is nonlocal.

Remark 1It is worth noting that the Lie symmetry analysis method for dealing with symmetries of fractional and integer-order equations is unified(compare with Ref.[2]),and the former is the generalization of the latter.More importantly,the results of the two types of equations are compatible.For example,if α=1,then all the symmetries(13)of the fractional Burgers’equation(3)belong to the vector field of the general Burgers’equation ut=auux+buxxgiven in Ref.[2]exactly.So are the same as the other cases(see Ref.[2],p.205 for detail).

3 Symmetry Reductions and Exact Solutions to the FPDEs

In Sec.2,we obtained the complete group classifications of the fractional wave equation(2).In this section,we develop the power series method[14]for investigating symmetry reductions and exact solutions to Eq.(2)in the general case 0<α≤1.So,the symmetry reductions for the integer-order cases(α=1)are given as its special cases accordingly.

As an example,we consider the general fractional Burgers’equation(3).

where ξ=xt?α/2,0< α ≤ 1.

For reducing Eq.(3)to a fractional ordinary differential equation(FODE),we employ the following Erdélyi-Kober(E-K)fractional differential operatorof order α >0:[3?10]

is the Erdélyi-Kober fractional integral operator.

Now,we discuss the symmetry reduction in terms of the fractional order α.

In view of 0< α <1,then we have n=[α]+1=1 in Eq.(19).Thus,by the definition of R-L fractional derivative and the similarity transformation(18),we get

Then,let v=t/τ,and by the Erdélyi-Kober fractional differential operator(19),we have

where ?′=d?/dξ,the Erdélyi-Kober fractional differential operatorof order α is given by Eq.(19),a and0 are arbitrary constants.

So,we reduce the FBE(3)to the FODE(22)in the fractional case 0<α<1.We note that the result is valid for the case α =1 as well.In fact,if α =1,then Eq.(22)takes the form

where ?′=d?/dξ.Clearly,Eq.(23)is the same as the reduced equation by similarity reduction in integer-order(α=1)case(see Remark 2).

Now,we deal with the exact solutions to the fractional Burgers’equation(3)by the power series method.So,we suppose that Eq.(22)has a power series solution as follows:

where cn(n=0,1,2,...)are constants to be determined.

Then,substituting Eq.(24)into Eq.(22)and comparing coefficients,we have

where Γ(z)is the gamma function,[3?7]the fractional order 0<α<1.

Thus,for arbitrarily chosen constants r=c0and s=c1,in view of Eq.(25),we have

and so on.

Therefore,the other terms of the sequencecan be determined successively from Eq.(25)in a unique manner.This implies that for Eq.(22),there exists a power series solution(24)with the coefficients given by Eq.(25)in the fractional case 0<α<1.

So the general solution in the power series form of Eq.(22)can be written as follows:

Thus,we obtain the exact power series solution to Eq.(3)as follows:

where r=c0and s=c1are arbitrary constants,the other coefficients cn+2(n=0,1,2,...)are given by Eq.(25)successively.

Remark 2In particular,if α=1,then FBE(3)becomes the general Burgers’equation as follows

This equation admits a symmetry V4=x?x+2t?t? u?u(see Ref.[2],p.205),it is a particular case of V2=αx?x+2t?t? αu?uin Eq.(13)for α =1.For V4,we have the similarity transformation

where ξ=xt?1/2.Substituting Eq.(29)into Eq.(28),we get

Clearly,it is the same as Eq.(23).

4 Conclusion and Remarks

In this paper,the complete Lie group classification of the generalized factional nonlinear wave equation is presented,all of the point symmetries of the fractional equations are obtained.Then,the symmetry reductions and exact solutions to the equations are investigated.Especially,we develop the power series method for dealing with exact solutions to the fractional nonlinear equations,and the compatibility of the symmetries and symmetry reductions of the fractional and integer order equations are verified for the first time.Moreover,are there generalized symmetries and any other forms of exact solutions to the fractional systems?We hope to investigate it in the future.

Remark 3We would like to reiterate that the power series method is an unified approach to tackling exact power solutions to the fractional nonlinear equations,and the fractional order α affects the solutions to the fractional equations greatly.Now,we give a specific example.

If α =1/3,then the fractional Burgers’equation(3)becomes Eq.(17).From Eq.(22),we get the reduced FODE of Eq.(17)as follows

where ?′=d?/dξ,ξ=xt?1/6.

Thus,through the procedure in Sec.3,we can get the exact power series solution to Eq.(31),so the analytic solution to Eq.(17)is obtained immediately.

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