Exact Solutions of the Modified Nonlinear Burgers' Equation
BUBPHA JITSOM1,2..
ID
, SURATTANA SUNGNUL1,2..
ID ,
EKKACHAI KUNNAWUTTIPREECHACHAN1,2..
ID
1Department of Mathematics, Faculty of Applied Science
King Mongkut's University of Technology North Bangkok, Bangkok 10800,
THAILAND
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400,
THAILAND
Abstract: The expansion approach has been used to solve the modified nonlinear Burgers' equation, which has
a nonlinear convection term, a viscosity term, and a time-dependent term in its structure. In this paper, the main
focus is to find exact solutions of the modified nonlinear Burgers' equation. The (G′/G)-expansion method is one
of the methods used to find exact solutions of nonlinear problems. It requires an appropriate transform equation
to convert partial differential equations to ordinary differential equations, making it easier to find the solution. In
this work, we choose the traveling wave equation to covert the equation. The results show that the exact solutions
of the modified nonlinear Burgers' equation by the (G′/G)-expansion method are decreasing if traveling wave
parameter, ω, increases for the first case and the exact solution is increasing if ωis increasing for the second
case. It is observed that (G′/G)-expansion method is an advanced and easy tool for finding exact solutions of the
modified nonlinear Burgers' equation.
Key-Words: Exact solutions, (G′/G)-expansion method, modified nonlinear Burgers' equation, traveling wave
solutions, partial differential equation, Maple program.
Received: February 12, 2023. Revised: October 16, 2023. Accepted: November 23, 2023. Published: December 21, 2023.
1 Introduction
Burgers' equation has been widely used as a model
in many fields of science and engineering, for exam-
ples, fluid flow problem, traffic flow model and com-
puter network. Because of its important role, there are
many analytical and numerical solutions have been
developed to solve both linear and nonlinear Burgers'
equation, [1], [2], [3]. In the present, the methods for
seeking exact solutions of nonlinear partial differen-
tial equation are one of the methods widely used, such
as the extended tanh method, the (G′/G)-expansion
method, the symmetry method, and the first-integral
method, etc.
Some examples of researches which are related
to using the (G′/G)-expansion method to applied
with linear and nonlilear of partial differential equa-
tion are as follows. In 2008, [4], proposed the
(G′/G)-expansion method to solve nonlinear evolu-
tion equations such as the KdV equation, the mKdV
equation, the variant Boussinesq equations and the
Hirota-Satsuma equations. In addition, [5], proposed
a generalized (G′/G)-expansion method for finding
exact solutions of the (2+1)-dimensional Nizhnik-
Novikov-Vesselov equation, the (2+1)-dimensional
Broer-Kaup equation and the (2+1)-dimensional
Kadomstev-Petviashvili equation. In 2010, [6], pre-
sented an improved (G′/G)-expansion method for
seeking more general travelling wave solutions of
nonlinear evolution equations. Both, [7], [8], de-
veloped a generalized (G′/G)-expansion method to
solve nonlinear evolution equations. In 2014, [9],
proposed a new (G′/G)-expansion method to solve
a laws of Burgers' equation. In present, the (G′/G)-
expansion method is applied to solve numerous non-
linear evolution equations in mathematical physics,
[10], [11], [12], [13].
The nonlinear one-dimensional Burgers' equation
is given by
ut+uux=µuxx,(x, t)∈Ω,(1)
where uis unknown function, µis diffusion coeffi-
cient, xis position, tis time and Ωis a continuous
space-time domain.
The modified nonlinear Burgers' equation is writ-
ten in the form
ut+ (c+bu)ux=µuxx (x, t)∈Ω,(2)
where c,band µare arbitrary parameters.
In this work, the main goal is to find exact solu-
tions of the modified nonlinear Burgers' equation by
the (G′/G)-expansion method.
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2 The (G′/G)-expansion method
In this section, the (G′/G)-expansion method for
solving nonlinear partial differential equation is de-
scribed.
Suppose that a general nonlinear partial differntial
equation in the form:
P(u, ux, ut, uxx, utt, ...) = 0,(3)
where u=u(x, t)is an unknown function, Pis a
polynomial in uand its partial derivatives in which
the highest order derivatives and nonlinear terms are
involved. Next, we illustrate the main steps of the
(G′/G)-expansion method for seeking exact solu-
tions of a nonlinear partial differential equation as the
folowing steps.
Step 1: Suppose the function u=u(x, t)is in the
travelling form:
u(x, t) = U(ξ), ξ =x±ωt, (4)
where ωis the velocity of propagation. Then equation
(3) can be reduced to the following nonlinear ODE
with respect to variable ξby substituting u(x, t) =
U(ξ)in equation (4) into equation (3). We can rewrite
equation (3) in the following nonlinear ODE
R(U, U′, U′′ , ...) = 0,(5)
where Ris a polynomial of U(ξ)and its total deriva-
tives U′(ξ), U′′(ξ), ....
Step 2: We assume that the solution of equation (5)
can be expressed in the form
U(ξ) =
N
i=0
aiG′
Gi
,(6)
where ai(i= 1,2, ..., N)are constants to be deter-
mined, such that aN= 0 and G=G(ξ)satisfies the
following the second order linear ODE as
G′′(ξ) + λG′(ξ) + µG(ξ) = 0,(7)
where λand µare real constants.
Step 3: The positive integer Nin equation (6) can be
determined by considering the homogeneous balance
the highest-order derivatives and nonlinear terms in
equation (5), where the degree of the expressions will
be determined by formular
DUqdpU
(dξp)s=qN +s(N+p)(8)
Step 4: Substituting equation (6) with the value of N
that obtained in step 3 into equation (5) and collect-
ing all the coefficients of Gi(ξ)
G(ξ)i,(i=±1,±2, ...).
Setting these coefficients to zero, we can get a set of
the system of linear equations, which can be solved
using Maple program to find the values of ai(i=
0,1,2,3, ..., N), λ and µ.
Step 5: Substituting the values of ai, λ and µfrom
step 4 into equation (6), the exact solutions of equa-
tion (3) are obtained.
3 Main Results
We consider the following modified nonlinear Burg-
ers' equations [3]:
ut+ (c+bu)ux=µuxx,(x, t)∈Ω,(9)
where c,band µare arbitrary parameters. In this
work, nonlinear Burgers' equation of two cases is
studied as follows,
• Case I : ut+uux=1
4uxx,
• Case II : ut+1
2−uux=1
4uxx.
The exact solutions of both cases are solved by the
(G′/G)-expansion method.
3.1 Exact Solutions of Case I
Consider nonlinear Burgers' equation in the form,
ut+uux=1
4uxx.(10)
For Case I, we can use tranformation with equation
(5), then equation (10) becomes
U′(ξ)ω+U(ξ)U′(ξ)−1
4U′′(ξ),(11)
where balancing U(ξ)U′(ξ)with U′′(ξ)gives N= 1.
Therefore, we may choose
U(ξ) = a1G′
G+a0.(12)
Substituting equation (12) into (11), it yields a set of
algebraic equations for a0and a1. We obtian the fol-
lowing with the aid of Maple,
a0=−ω−1
4λ, a1=−1
2.(13)
Substituting equation (13) into (12), then we have
travelling wave solutions of equation (10).
Type A : if λ2−4µ > 0,
U1(ξ) = −1
4
αCsinh 1
2αξ+Dcosh 1
2αξ
Ccosh 1
2αξ+Dsinh 1
2αξ−ω,
(14)
where α=λ2−4µ,ξ=x±ωt,Cand Dare
arbitrary constants.
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The exact solutions of the modified nonlinear
Burgers' equation, as displayed in Figure 1, for the
respective values of λ= 2 and ω=−4,−1,1,4. At
the end of time t= 1, we found that the exact solu-
tion udecreases as ωis increasing for 0≤x≤1.
Additionally, if ωis positive, we can observe that the
exact solutions uincreases while tis increased.
Type B : if λ2−4µ < 0,
U2(ξ) = −1
4
β(−Csin (1
2βξ)+Dcos (1
2βξ))
Ccos (1
2βξ)+Dsin (1
2βξ)−1
2λ−ω,
(15)
where β=−λ2+ 4µ,ξ=x±ωt,Cand Dare
arbitrary constants.
In Figure 2, illustrates the graphs of exact solu-
tions of the modified nonlinear Burgers' equation for
λ= 0.8and ω=−4,−1,1,4, respectively. For
0≤x≤1, the exact solution uis decreasing while ω
is increasing at time t= 1. In addition, it was found
that when ωis positive, the exact solution udecreases
as tincreases.
Type C : if λ2−4µ= 0,
U3(ξ) = 1
2
D
Dξ +C−1
2λ−ω, (16)
where ξ=x±ωt,Cand Dare arbitrary constants.
The exact solutions of the modified nonlinear
Burgers' equation, as presented in Figure 3, for the
specific values of λ= 1 and ω=−2,−1,0,1. We
observed that for 0≤x≤1, the exact solution u
decreases as ωincreases at time t= 1. According
to Figure 3(c), the exact solution remains unchanged
for ω= 0. Additionally, it can be seen that the exact
solution udecreases as tincreases in the case where
ωis positive.
3.2 Exact Solutions of Case II
Consider the nonlinear Burgers' equation of the form
ut+1
2−uux=1
4uxx.(17)
We use tranformation with equation (5), then equation
(17) becomes
U′(ξ)ω+1
2U′(ξ)−U(ξ)U′(ξ)−1
4U′′(ξ),(18)
where balancing U(ξ)U′(ξ)with U′′(ξ)gives N= 1.
Therefore, we may choose
U(ξ) = a1G′
G+a0,(19)
(a) , λ = 2, ω =−4
(b) , λ = 2, ω =−1
(c) λ= 2, ω = 1
(d) λ= 2, ω = 4
Fig. 1: Exact solutions of Case I : Type A (λ2−4µ >
0)with µ= 1/4, C = 1, D = 2.
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(a) λ= 0.8, ω =−4
(b) λ= 0.8, ω =−1
(c) λ= 0.8, ω = 1
(d) λ= 0.8, ω = 4
Fig. 2: Exact solutions of Case I : Type B (λ2−4µ <
0)with µ= 1/4, C = 1, D = 2.
(a) λ= 1, ω =−2
(b) λ= 1, ω =−1
(c) λ= 1, ω = 0
(d) λ= 1, ω = 1
Fig. 3: Exact solutions of Case I : Type C (λ2−4µ=
0)with µ= 1/4, C = 1, D = 2.
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Substituting equation (12) into (11) yields a set of al-
gebraic equations for a0and a1. We obtian the fol-
lowing with the aid of Maple, then
a0=ω+1
2+1
4λand a1=1
2.(20)
Substituting equation (20) into (19) we have travel-
ling wave solutions of equation (17)
Type A : if λ2−4µ > 0,
U1(ξ) = −1
4
α(Csinh (1
2αξ)+Dcosh (1
2αξ))
Ccosh (1
2αξ)+Dsinh (1
2αξ)+ω+1
2,
(21)
where α=λ2−4µ,ξ=x±ωt,Cand Dare
arbitrary constants.
The exact solutions of the modified nonlinear
Burgers' equation, as displayed in Figure 4, for the
respective values of λ= 2 and ω=−4,−1,1,4. At
the end of time t= 1, we found that the exact solution
uincreases as ωis increasing for 0≤x≤1. Addi-
tionally, if ωis positive, then we can observe that the
exact solutions udecreases while tis increased.
Type B : if λ2−4µ < 0,
U2(ξ) = −1
4
β(−Csin (1
2βξ)+Dcos (1
2βξ))
Ccos (1
2βξ)+Dsin (1
2βξ)+ω+1
2,
(22)
where β=−λ2+ 4µ,ξ=x±ωt,Cand Dare
arbitrary constants.
Figure 5, illustrates the graphs of exact solutions of
the modified nonlinear Burgers' equation for λ= 0.8
and ω=−4,−1,1,4, respectively. For 0≤x≤1,
the exact solution uis increasing while ωis increas-
ing at time t= 1. In addition, it was found that if
ωis positive, then the exact solution udecreases as t
increases.
Type C : if λ2−4µ= 0,
U3(ξ) = 1
2
D
Dξ +C+ω+1
2λ(23)
where ξ=x±ωt,Cand Dare arbitrary constants.
The exact solutions of the modified nonlinear
Burgers' equation, as presented in Figure 6, for the
specific values of λ= 1 and ω=−2,−1,0,1. We
observed that for 0≤x≤1, the exact solution u
decreases as ωincreases at time t= 1. According to
Figure 6(c), the exact solution remains unchanged for
ω= 0. Additionally, we have found that the exact
solution udecreases as tincreases in the case where
ωis positive.
4 Discussion
The modified nonlinear Burgers' equation with a non-
linear convection term, a viscosity term, and a time-
dependent term in its structure has been solved by the
(a) , λ = 2, ω =−4
(b) , λ = 2, ω =−1
(c) λ= 2, ω = 1
(d) λ= 2, ω = 4
Fig. 4: Exact solutions of Case II : Type A (λ2−
4µ > 0)with µ= 1/4, C = 1, D = 2.
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(a) λ= 0.8, ω =−4
(b) λ= 0.8, ω =−1
(c) λ= 0.8, ω = 1
(d) λ= 0.8, ω = 4
Fig. 5: Exact solutions of Case II : Type B (λ2−4µ <
0)with µ= 1/4, C = 1, D = 2.
(a) λ= 1, ω =−2
(b) λ= 1, ω =−1
(c) λ= 1, ω = 0
(d) λ= 1, ω = 1
Fig. 6: Exact solutions Case II : Type C (λ2−4µ=
0)with µ= 1/4, C = 1, D = 2.
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expansion method. The (G′/G)-expansion method
is applied to these nonlinear equations. In the first
case, the (G′/G)-expansion method gives three types
of solutions, the exact solutions uis slightly decreas-
ing when ωis increasing for 0≤x≤1at time t= 1
as shown in Figure 1, Figure 2 and Figure 3. For Case
II, the exact solutions are shown in Figure 4, Figure
5 and Figure 6. It can be seen that the solutions are
increasing when ωis increasing for 0≤x≤1at time
t= 1. Moreover, we compare the behavior of exact
solutions uof three types in both cases. We found that
the solutions uis a little changed as time increased ex-
cept in cases of ω=−4and ω=−2, the solutions
are oscillate as shown in Figure 7 and Figure 8. It
is observed that the (G′/G)-expansion method is an
easy tool for finding exact solutions of the modified
nonlinear Burgers' equation.
5 Conclusion
The (G′/G)-expansion method is one of the methods
used to find exact solutions of nonlinear problems. It
requires the appropriate transform equation to convert
the PDEs to the ODEs, making it easy to find the so-
lutions. In this work, we choose the traveling wave
equation to covert the equation. There are many trans-
form equations which can be chosen. For the one of
complex equations, it may difficult to convert equa-
tions and have to convert several times to be able to
find the exact solution. In further, we can apply the
(G′/G)-expansion method for constructing traveling
wave solutions of other nonlinear partial differential
equation which arise in science and engineering prob-
lems.
Acknowledgment:
The authors would like to express their sincere
thanks to the anonymous referees for their time and
helpful comments.
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(a) Type A (λ2−4µ≥0), λ= 2 at x= 1
(b) Type B (λ2−4µ≤0), λ= 0.8at x= 1
(c) Type C (λ2−4µ= 0), λ= 1 at x= 1
Fig. 7: Exact solutions of Case I.
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(a) Type A (λ2−4µ≥0), λ= 2 at x= 1
(b) Type B (λ2−4µ≤0), λ= 0.8at x= 1
(c) Type C (λ2−4µ= 0), λ= 1 at x= 1
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Bubpha Jitsom: Conceptualization, investigation,
methodology, software, visualization, writing-
original draft and writing-review and editing.
Surattana Sungnul: Conceptualization, for-
mal analysis, investigation, methodology, su-
pervision, validation, visualization, writing-
original draft and writing-review and editing.
Ekkachai Kunnawuttipreechachan: Supervi-
sion, validation, writing-review and editing.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflicts of Interest
The authors have no conflict of interest to declare that
are relevant to the content of this article.
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(Attribution 4.0 International , CC BY 4.0)
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Creative Commons Attribution License 4.0
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