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 , the sine-cosine method , the simplest equation method, the homogeneous balance method , the inverse scattering transform method , Hirota’s bilinear method , the (G'/G)−expansion method [9, 10], Riccati-Bernoulli sub-ordinary differential equation method , 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(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 . 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(1.2)
was first introduced in . 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.  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(1.3)
was studied in  and solitary wave solutions were obtained. Furthermore, using Quintic B-spline method the interactions through computer simulation were observed. In  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  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(1.4)
was studied in  and its solitary wave solutions were obtained by a collocation method using quadratic B-spline at the midpoints.
In  the author introduced the modified equal width-Burgers (MEW-Burgers) equation(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 .
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(2.6)
is a Lie point symmetry of (1.5) if and only if(2.7)
is the third prolongation of X . Expanding (2.7), we obtain(2.8)
Splitting the above equation with respect to derivatives of u yields the system of ten overdetermined linear partial differential equations(2.9)(2.10)(2.11)(2.12)(2.13)(2.14)(2.15)(2.16)(2.17)(2.18)
Solving (2.9)–(2.18) for ξ1, ξ2 and η we obtain
which results in two translation symmetries
A linear combination of these Lie point symmetries, that is, X = X1 + cX2 produces the associated Lagrange equations(2.19)
which upon solving yields the two invariants z = x − ct 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)(2.20)
2.2 Solution of (2.20) using (G'/G)−expansion method
In this subsection we use the (G'/G)−expansion method  to obtain solutions of (2.20). Firstly, we assume that(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(2.22)
Secondly, we substitute (2.22) into (2.20) and simultaneously use the second-order ODE
where λ and μ are arbitrary constants. This yields(2.23)
Collecting terms with like powers of (G'/G) and equating them to zero gives the overdetermined system of five algebraic equations
Using Mathematica, two solutions of the above system of algebraic equations are
Solution set 1
Solution set 2
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(2.24)
where and A and B are constants.
Case 1.2 For M = λ2 − 4μ < 0, we obtain the trigonometric function solution(2.25)
where and A and B are constants.
Case 1.3 For M = λ2 − 4μ = 0, we obtain the rational function solution(2.26)
where z = x − 2(3μ − λ2)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
where and A and B are constants.
Case 2.2 For M = λ2 − 4μ < 0, we obtain the trigonometric function solution
where and A and B are constants.
Case 2.3 For M = λ2 − 4μ = 0, we obtain the rational function solution
where and 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.
3 Conservation laws of (1.5)
We begin by determining the adjoint equation of (1.5), namely
using the formula(3.27)
where δ/u is the Euler-Lagrange operator defined by(3.28)
with the total differential operators Dt and D x given by
Thus Equation (3.27) becomes(3.29)(3.30)
which is equivalent to the second-order Lagrangian(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 (3.32)(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 which yields λ = 0. Since Dt(ξ1) = 0, we obtain η* = 0 and hence the operator admitted by the adjoint equation (3.29) is(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
Similarly, we compute the conserved vector corresponding to X2. In this case W1 = −ux and W2 = −vx and the conserved vector thus rendered is
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.
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
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.
©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