Cite this Article
Wang Gang-Wei, Kara A. H.. Group Analysis, Fractional Explicit Solutions and Conservation Laws of Time Fractional Generalized Burgers Equation. Communications in Theoretical Physics, 2018, 69(1): 5  
Permissions
Group Analysis, Fractional Explicit Solutions and Conservation Laws of Time Fractional Generalized Burgers Equation
Wang Gang-Wei1, , Kara A. H.2
School of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang, 050061, China
School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa

 

† Corresponding author. E-mail: wanggangwei@heuet.edu.cn pukai1121@163.com

Abstract
Abstract

The generalized fractional Burgers equation is studied in this paper. Using the classical Lie symmetry method, all of the vector fields and symmetry reduction of the equation with nonlinearity are constructed. In particular, an exact solution is provided by using the ansatz method. In addition, other types of exact solution are obtained via the invariant subspace method. Finally, conservation laws for this equation are derived.

PACS: ;02.30.Jr;;11.30.−j;;05.45.Yv;;02.20.Sv;
Keyword:time fractional generalized Burgers equation;Lie symmetry method;explicit solutions;conservation laws
1 Introduction

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 a fractional order nonlinear Burgers equation of the form

where is function of x and t, , while a and b are constants. a is the coefficient of the convection term while b is the coefficient of viscous term. If b = 0, Eq. (1) is the generalized form of the inviscid BE. The parameters m and n are all positive integers. In the special case when , Eq. (1) reduces to the case of the regular BE. If only n = 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 gets

3 Fractional Soliton Solution

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

where, A and B are the free parameters, p will be fixed further. Now we plug Eq. (15) into Eq. (1), one can get

Substituting Eqs. (17)–(20) into Eq. (1), one can have

Considering the exponents mp + p + 1 and np + 2, or the exponents mp + p − 1 and np, one can get

And then equating the exponents and , we arrive at

That is to say,

so,

Also, one can get the constraint conditions

Letting different coefficients of , , to zero, one can yield

And then, solve them, one can get

In this way, finally, one can get the exact solution

3.1 Numerical Simulations

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

Figure 1 shows the solution while α = 1 is fixed. Figure 2 expresses the solution for varying α at t = 2. Figure 3 shows the the solution for varying α at x = 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 case m = n = 1. It is clear that it has an invariant subspace , as . This means that Eq. (1) has an exact solution in the following form

Here, and are need to be fixed. Substitute Eq. (34) into Eq. (1) and let the different powers of x to zero, one can get

Solving Eq. (36), one can get

and then solve Eq. (35), one can get

It is clear that we must require , for using this method. At last, we get the exact solution for this case

It should be noted that, if α = 1, this equation has a rational solution

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 operator Tt is[15–16]

where J is the integral[15–16]

The operator Tx is given by

For any generator X and any solution of of Eq. (1), we have the following equation

This equality immediately generates the conservation law

Using Eqs. (42) and (44), we have components of conserved vectors

where i = 1,2 and functions Wi are

For example, , we have given by

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 if α = 1 and n = m = 1, the transformed equations can be reduced to integer order ones. The case , 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.

Reference
[1] Burgers J. M. Proc. Acad. Sci. Amsterdam 43 1940 2
[2] Vaganan B. M. Senthilkumaran M. Nonlinear Anal. Real. World Appl. 9 2008 2222
[3] Jawad A. J. M. Petkovic M. D. Biswas A. Appl. Math. Comput. 216 2010 3370
[4] Ray S. S. Commun. Nonlinear Sci. Numer. Simul. 14 2009 1295
[5] He J. H. Comput. Methods Appl. Mech. Eng. 167 1998 57
[6] Momani S. Odibat Z. Phys. Lett. A 365 2007 345
[7] Gazizov R. K. Kasatkin A. A. Comput. Math. Appl. 66 2013 576
[8] Gazizov R. K. Kasatkin A. A. Lukashchuk S. Yu. Vestnik, USATU 9 2007 125 (in Russian)
[9] Sahadevan S. Bakkyaraj T. J. Math. Anal. Appl. 393 2012 341
[10] Wang G. W. Liu X. Q. Zhang Y. Y. Commun. Nonlinear Sci. Numer. Simulat. 18 2013 2321
[11] Wang G. W. Xu T. Z. Feng T. Plos One 9 2014 e88336
[12] Wang G. W. Xu T. Z. Nonlinear Dyn. 76 2014 571
[13] Wang G. W. Xu T. Z. Nonlinear Analysis: Model. 20 2015 570
[14] Wang G. W. Xu T. Z. Bound. Value Probl. 232 2013 1
[15] Wang G. W. Kara A. H. Fakhar K. Nonlinear Dyn. 82 2015 281
[16] Lukashchuk S. Y. Nonlinear Dyn. 80 2015 791
[17] Wang X. B. Tian S. F. Qin C. Y. Zhang T. T. Europhys. Lett. 114 2016 20003
[18] Chen C. Jiang Y. L. Commun. Theor. Phys. 68 2017 295
[19] Podlubny I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications San Diego Academic Press 1999
[20] Kiryakova V. S. Generalised Fractional Calculus and Applications Harlow, England Longman Scientific & Technical 1994 388