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Open Physics

formerly Central European Journal of Physics

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Volume 16, Issue 1

Issues

Volume 13 (2015)

Exact solutions and conservation laws for the modified equal width-Burgers equation

Chaudry Masood Khalique
  • Corresponding author
  • International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences,North-West University Mmabatho Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, Republic of South Africa
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/ Innocent Simbanefayi
  • International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences,North-West University Mmabatho Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, Republic of South Africa
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Published Online: 2018-12-26 | DOI: https://doi.org/10.1515/phys-2018-0099

Abstract

In this paper we study the modified equal width-Burgers equation, which describes long wave propagation in nonlinear media with dispersion and dissipation. Using the Lie symmetry method in conjunction with the (G'/G)− expansion method we construct its travelling wave solutions. Also, we determine the conservation laws by invoking the new conservation theorem due to Ibragimov. As a result we obtain energy and linear momentum conservation laws.

Keywords: Modified equal width-Burgers equation; Lie point symmetries; exact solutions; conservation laws

PACS: 02.30.Jr; 04.20.Jb; 11.30.-j

1 Introduction

It is well-known that majority of the real-world physical phenomena are modeled by mathematical equations, especially nonlinear partial differential equations (NLPDEs). These phenomena include the problems from fluid mechanics, elasticity, plasma and optical fibers, relativity, gas dynamics, thermodynamics, and many more. In order to comprehend the understanding of such physical phenomena it is vital to look for exact solutions of the NLPDEs. During the last sixty years many scientists and mathematicians developed several effective and useful methods for obtaining exact solutions of NLPDEs. These include the simplified Hirota’s method [1, 2], the tanh-coth method [3], the sine-cosine method [4], the simplest equation method[5], the homogeneous balance method [6], the inverse scattering transform method [7], Hirota’s bilinear method [8], the (G'/G)−expansion method [9, 10], Riccati-Bernoulli sub-ordinary differential equation method [11], Jacobi elliptic function expansion method[12, 13], Kudryashov method [14, 15], the Lie symmetry method [16, 17, 18, 19, 20, 21], just to mention a few.

On the other hand conservation laws are very important in the study of NLPDEs and much research has been done on different methods of obtaining conservation laws. Conservation laws are essential in determining the extent of integrability of differential equations, development of numerical schemes, reduction and solutions of partial differential equations. See, for example [22, 23, 24, 25, 26, 27, 28, 29] and references therein.

The Burgers equation

ut+uuxvuxx=0,(1.1)

where v is a constant defining the kinematic viscosity models the turbulent flow in a channel and describes the effect of coupling between diffusion and convection processes on a fluid. It first appeared in academic circles in [30]. Over the years, various researchers have applied modifications to (1.1) and used many different methods to study the equation. For example, the equal width equation

utuuxutxx=0(1.2)

was first introduced in [31]. It describes amongst others, nonlinear dispersive waves such as shallow water waves and nonlinear waves in plasmas. Extensive work was done on Equation (1.2) in constructing numerical solutions [32, 33]. Arora et al. [34] performed the reduced differential transform method to find the numerical solution of the equal width wave equation and the exact analytical solution of the inviscid Burgers equation with initial conditions.

The modified equal width equation

ut+au2uxbutxx=0(1.3)

was studied in [35] and solitary wave solutions were obtained. Furthermore, using Quintic B-spline method the interactions through computer simulation were observed. In [36] the sine-cosine and the tanh methods were employed to obtain exact solutions of (1.3) and two of its variants, which included compactons, solitons, solitary patterns, and periodic solutions. Hasan [37] presented the numerical solution for (1.3) using Fourier spectral method that discretizes the space variable and Leap-frog method scheme for time dependence.

The generalised equal width equation

ut+aumuxbutxx=0(1.4)

was studied in [38] and its solitary wave solutions were obtained by a collocation method using quadratic B-spline at the midpoints.

In [39] the author introduced the modified equal width-Burgers (MEW-Burgers) equation

ut+αu2ux+ωuxxβutxx=0,(1.5)

which describes long wave propagation in nonlinear media with dispersion and dissipation. Here α, β are positive parameters and ω is a damping parameter. The bifurcation behaviour and an external periodic perturbation of the MEW-Burgers equation (1.5) was studied [39].

In this paper using the Lie symmetry method and the (G0/G)−expansion method we construct travelling wave solutions of (1.5). Furthermore, we derive the conservation laws by applying the new conservation theorem due to Ibragimov.

2 Travelling wave solutions of (1.5)

In this section we obtain travelling wave solutions of (1.5) by employing Lie symmetry analysis together with the (G'/G)−expansion method.

2.1 Lie point symmetries and reduction of (1.5)

We begin by determining the Lie point symmetries of (1.5). The vector field

X=ξ1(t,u,x)t+ξ2(t,u,x)x+η(t,u,x)u(2.6)

is a Lie point symmetry of (1.5) if and only if

X[3](ut+αu2ux+ωuxxβutxx)|(1.5)=0,(2.7)

where

X[3]=X+ζ1ut+ζ2ux+ζ22uxx+ζ122utxx

is the third prolongation of X [18]. Expanding (2.7), we obtain

ηt+2uxutξu2+2βuttxξt12μut,xξx1+2βutx2ξu1+βutxξxx2μuxξxx2+βuttξxx1+2μuxηxuμutξxx12βut,xηxu+αu2ηx+βξtxx1ut+βξtxx2ux+2βξtx1ut,x+2βuxxξtx2βutηxxuβηtuuux2βηtuuxx+βξxxu1ut2+ux3βξtuu2+2βξtxu2ux22βuxηtxu2μux2ξxu2μux3ξuu2+μux2ηuu+μuxxξt1+βuxxxξt2+2ξx2utξt2ux+ξu1ut2+μηxxβηtxx+βutuxξuxx2+3βuxuxxξtu2+βutux2ξtuu1+2βutuxξtxu1βutux2ηuuu+2αuuxη+βut2ux2ξuuu1+βuxxutξtu1+2uxutxβξtu1+4βututxξxu1+2βuxuttξxu12μutuxξxu1+2uxxutβξxu2+4uxβutxξxu2+3βux2utxξuu2βuxxutηuu2βutxηuu+βut2uxxξuu1+βux2uttξuu1μutux2ξuu1+αu2uxξx2+αu2uxξt1+2ξu2αu2ux2+ξu2βutuxxx+3ξu2βutxuxxαu2utξx1+2βuttxuxξu1+βuxxuttξu1+μuxxutξu12μuxutxξu1+utux3βξuuu2+2βut2uxξxuu1+2βutux2ξxuu22βutuxηxuu+3βutuxuxxξuu2+4βutuxutxξuu1+αu2utuxξu1=0.(2.8)

Splitting the above equation with respect to derivatives of u yields the system of ten overdetermined linear partial differential equations

ξx1=0,(2.9)ξu1=0,(2.10)ξt2=0,(2.11)ξu2=0,(2.12)ηuu=0,(2.13)ξxx22ηxu=0,(2.14)μξt1βηtu=0,(2.15)2ξx2βηxxu=0,(2.16)μηxxβηtxx+ηt+αu2ηx=0,(2.17)αu2ξx2+αu2ξt1+2αuη+2μηxuμξxx22βηtxu=0.(2.18)

Solving (2.9)–(2.18) for ξ1, ξ2 and η we obtain

ξ1=C1,ξ2=C2,η=0,

which results in two translation symmetries

X1=t,X2=x.

A linear combination of these Lie point symmetries, that is, X = X1 + cX2 produces the associated Lagrange equations

dt1=dxc=du0,(2.19)

which upon solving yields the two invariants z = xct and U = u, and hence the group-invariant solution u = U(z). Taking U and z as the new dependent and independent variables respectively, equation (1.5) is transformed into the third-order nonlinear ordinary differential equation (ODE)

βcU(z)cU(z)+ωU(z)+αU(z)2U(z)=0.(2.20)

2.2 Solution of (2.20) using (G'/G)−expansion method

In this subsection we use the (G'/G)−expansion method [9] to obtain solutions of (2.20). Firstly, we assume that

U(z)=i=0mAi(G(z)G(z))i(2.21)

is the solution to (2.20), where A0, A1, · · · Am are to be determined. The balancing procedure is used to find the value of m, a positive integer. In our case the balancing procedure yields m = 1, thus (2.21) becomes

U(z)=A0+A1(G(z)G(z)).(2.22)

Secondly, we substitute (2.22) into (2.20) and simultaneously use the second-order ODE

G+λG(z)+μG(z)=0,

where λ and μ are arbitrary constants. This yields

A1λμω+A1cμ2A1βcμ2αA02A1μA1βcλ2μ+(A1cλ8A1βcλμA1βcλ3+2A1μω+A1λ2ωαA02A1λ2αA0A12μ)(G(z)G(z))+(A1c7A1βcλ28A1βcμαA02A12αA0A12λαA13μ+3A1λω)(G(z)G(z))2+(2A1ω2αA0A12αA13λ12A1βcλ)(G(z)G(z))3(6A1+βc+αA13)(G(z)G(z))4=0.(2.23)

Collecting terms with like powers of (G'/G) and equating them to zero gives the overdetermined system of five algebraic equations

(G(z)G(z))4:αA13+6A1βc=0,(G(z)G(z))3:αA13λ+2αA0A12+12A1βcλ2Α1ω=0,(G(z)G(z))2:2αA0A12λ+αΑ13μ+αA02A1+7A1βcλ2+8A1βcμA1c3A1λω=0,(G(z)G(z))1:αA02Α1λ+2αΑ0Α12μ+A1βcλ3+8A1βcλμA1cλA1λ2ω2A1μω=0,(G(z)G(z))0:A1λμω+Α1cμαA02A1μA1βcλ2μ2A1βcμ2=0.

Using Mathematica, two solutions of the above system of algebraic equations are

Solution set 1

A02λω3α,A1=6ωλα,c=23λ(3μλ2).

Solution set 2

A0=2ωμα,A1=32ωαμ,c=4ωμ3.

Thus corresponding to solution set 1 above, we have the following three types of solutions for the MEW-Burgers (1.5):

Case 1.1 For M = λ2 − 4μ > 0, we obtain the hyperbolic function solution

u(t,x)=(1+3λ)2λω3α(Δ1Acosh(Δ1z)+Bsinh(Δ1z)Asinh(Δ1z)+Bcosh(Δ1z)λ2),(2.24)

where Δ1=M/2,z=x2(3μλ2)t/(3λ)and A and B are constants.

Case 1.2 For M = λ2 − 4μ < 0, we obtain the trigonometric function solution

u(t,x)=(1+3λ)2λω3α(Δ2Asinh(Δ2z)+Bcosh(Δ2z)Acos(Δ2z)+Bsin(Δ2z)λ2),(2.25)

where Δ2=M/2,z=x2(3μλ2)t/(3λ)and A and B are constants.

Case 1.3 For M = λ2 − 4μ = 0, we obtain the rational function solution

u(t,x)=(1+3λ)2λω3α(BBz+Aλ2),(2.26)

where z = x − 2(3μλ2)t/(3λ) and A and B are constants.

Similarly, considering the solution set 2 we obtain the following three types of solutions for the MEW-Burgers (1.5):

Case 2.1 For M = λ2 − 4μ > 0, we obtain the hyperbolic function solution

u(t,x)=(1+4βx)2μωα(λ2Δ1Acosh(Δ1z)+Bsinh(Δ1z)Asinh(Δ1z)+Bcosh(Δ1z)),

where Δ1=M/2,z=x+4μωt/3and A and B are constants.

Case 2.2 For M = λ2 − 4μ < 0, we obtain the trigonometric function solution

u(t,x)=(1+4βλ)2μωα(λ2Δ2Asin(Δ2z)+Bcos(Δ2z)Acos(Δ2z)+Bsin(Δ2z)),

where Δ2=M/2,z=x+4μωt/3and A and B are constants.

Case 2.3 For M = λ2 − 4μ = 0, we obtain the rational function solution

u(t,x)=(1+4βλ)2μωα(BBz+Aλ2),

where z=x+4μωt/3and A and B are constants.

The solution profile of (2.24) for λ = −0.09, A = 0, B = 1, μ = 0.01, ω = 0.03, α = 1 is presented in Figure 1, whereas the solution profile of (2.26) for λ = −0.8, A = 0.4, B = 3, μ = 0.2, ω = 0.01, α = 0.1 is given in Figure 2.

Profile of solution (2.24)
Figure 1

Profile of solution (2.24)

Profile of solution (2.26)
Figure 2

Profile of solution (2.26)

3 Conservation laws of (1.5)

In this section we derive conservation laws for the MEW-Burgers equation (1.5) using the new conservation theorem due to Ibragimov [40, 41].

We begin by determining the adjoint equation of (1.5), namely

Fut+αu2ux+ωuxxβutxx=0,

using the formula

Fδδu{v(ut+αu2ux+ωuxxβutxx)}=0,(3.27)

where δ/u is the Euler-Lagrange operator defined by

δδu=uDtutDxux+Dx2uxxDtDx2utxx(3.28)

with the total differential operators Dt and D x given by

Dt=t+utu+vtv+uttut+vttvt+utxux+vtxvx+,Dt=x+uxu+vxv+uxxux+vxxvx+uxtut+vxtvt+.

Thus Equation (3.27) becomes

Fvt+αu2vxωvxxβvtxx=0.(3.29)

The MEW-Burgers equation (1.5) together with its adjoint equation (3.29) have the Lagrangian

L=v(ut+αu2ux+ωuxxβutxx),(3.30)

which is equivalent to the second-order Lagrangian

L=vut+αvu2ux+ωvuxx+βvtuxx.(3.31)

We recall that the MEW-Burgers equation (1.5) admits two translation symmetries X1 = /∂t and X2 = /∂x. To obtain the conserved vectors corresponding to these two infinitesimal generators we use [40]

Ct=ξ1L+W1Lut+W2Lvt,(3.32)Cx=ξ2L+W1[LuxDxLuxx]+Dx(W1)Luxx,(3.33)

where W1 and W2 are the Lie characteristic functions. Let us first consider the infinitesimal generator X1 = / ∂t. It can be easily shown that the prolongation of the generator X1 to the derivatives involved in the MEW-Burgers equation (1.5) has the form / ∂t. In order to determine the value

of λ we use equation X1[3](F)=λ(F),which yields λ = 0. Since Dt(ξ1) = 0, we obtain η* = 0 and hence the operator admitted by the adjoint equation (3.29) is

Y=t.(3.34)

We now use (3.34) to compute the Lie characteristic functions W1 and W2, which in this case are W1 = −ut and W2 = −vt. Thus by using (3.32) and (3.33) the conserved vector for the system (1.5) and (3.29) corresponding to X1 is

C1t=αvu2ux+ωvuxx,C1x=ωvxut+βvtxutωvuxtαvu2utβuxtvt.

Similarly, we compute the conserved vector corresponding to X2. In this case W1 = −ux and W2 = −vx and the conserved vector thus rendered is

C2t=vuxβvxuxx,Cx2=vut+ωvxux+βvtxux.

Remark: It should be noted that the time translation symmetry gives us the energy conservation law whereas the space translation symmetry provides us with the linear momentum conservation law.

4 Conclusion

In this paper we obtained travelling wave solutions of the MEW-Burgers equation (1.5) The two translation symmetries were used to reduce the MEW-Burgers equation to an ordinary differential equation. The (G'/G)−expansion method was applied to the ordinary differential equation to obtain its solutions. Consequently travelling wave solutions were obtained for the MEW-Burgers equation. These solutions were hyperbolic, trigonometric and rational functions. Furthermore we derived two conservation laws using the new conservation theorem due to Ibragimov. These were the energy and the linear momentum conservation laws.

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About the article

Received: 2018-05-31

Accepted: 2018-07-02

Published Online: 2018-12-26


Citation Information: Open Physics, Volume 16, Issue 1, Pages 795–800, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0099.

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©2018 Chaudry Masood Khalique and Innocent Simbanefayi, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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