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Investigation of Some Explicit Exact Solution of the Damped Forced
KDV Burger Equation by Modified Exp(-��(��))-Expansion Method

Ranjan Barman

Department of Mathematics, Dinhata College, Cooch Behar, 736135, India.

DOI: https://doi.org/10.51244/IJRSI.2025.1208004125

Received: 04 Oct 2025; Accepted: 11 Oct 2025; Published: 24 October 2025

ABSTRACT

This work presented the some explicit exact solution of the damped forced KdV-Burger equation with variable
coefficients. We have successfully applied the exp(-��(��))-expansion method with modification to obtain the
generalized explicit exact solution of the damped forced KdV-Burger’s equation. The obtained solution
contains the hyperbolic function and trigonometric function. The dynamic behavior of the solution is
demonstrated graphically in three dimensional and two dimensional space.

Keywords: Exp(-��(��))-expansion method, variable of separation, KdV-Burger’s equation,forcing term,
variable coefficients etc.

INTRODUCTION

The Korteweg-de Vries-Burgers (KdV-Burgers) equation has garnered significant attention over the past three
decades due to its relevance in various physical contexts. These include the propagation of undular bores in
shallow water [1], flow of liquids with gas bubbles [2], wave propagation in elastic tubes filled with viscous
fluids [3], and nonlinear plasma waves with dissipative effects [4-6]. The equation also finds applications in
crystal lattice theory, ferroelectricity, nonlinear circuit theory, and turbulence [7-12]. The standard form of the
KdV-Burgers equation is

ut + A(t)uux + B(t)uxxx + C(t)uxx = 0 (1.1)
It combines the Burgers equation [13] and the KdV equation [14], incorporating nonlinearity, dispersion, and
dissipation terms. Its validity has been demonstrated in specific physical problems, such as wave propagation
in liquid-filled elastic tubes [15], making it a fundamental model for understanding complex wave phenomena.
In general it is known that particle interactions in a medium often lead to damping effects. Various phenomena
can cause damping in dynamical systems, such as resonant energy exchange between particles and electrostatic
waves in plasma environments [16-17]. Experimental studies on space plasma have shown that externally
applied damping significantly influences wave propagation. External forces can also arise in specific
situations, like flowing water over bottom topography or waves generated by moving ships [18-19].
Considering these factors, the focus is on a KdV-Burger’s equation with external forcing and damping terms,
which is presented as a model to study these complex dynamics

ut + A(t)uux + B(t)uxxx + C(t)uxx + D(t)u = H(t) (1.2)

where A= non-linearity coefficients, B= dispersion coefficients, C=dissipation coefficients, D=damping
coefficients and H=forcing term. In recent years, a wide range of effective methods has been proposed to solve
nonlinear equations. These include the tanh function method [20], symmetry reduction method [21], extended
tanh method [22], sine-cosine method [23], homogeneous balance method [24], F-expansion method [25], exp-
function method [26-27], modified simple equation method [28-29], first integral method [30], extended trial
equation method [31], (G'/G)-expansion method[32], (G'/G, 1/G)-expansion method [33-34], soliton solution
method [35], and auxiliary equation method [36], among others. These methods have been successfully applied
to various nonlinear problems, providing valuable insights and solutions to complex equations.

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In this work we obtained the generalized exact solution of the damped forced KdV-Burger’s equation with
variable coefficients with the help of modified exp(−��(��))-expansion method. The obtain solution contains
hyperbolic function and trigonometric function. The dynamics behavior of the solution is demonstrated
graphically in three dimensional and two-dimensional space.

2.Preliminary of Exp(-��(��))-expansion method

Given a non-linear partial differential equation in general form

��(��, ����, ���� , ������ , ������ , ������ , …………… . . ) = 0 (2.1)

where �� = ��(��, ��), �� is a polynomial of �� and its derivatives and the subscripts means for the partial
derivatives. The method involves the following steps.

Step I: First we make the transformation �� = �� + ��(��), where ��(��) is an unknown function and
��(��, ��) = ��(��).Under this transformation ,Eq.(2.1) convert into an ordinary differential equation

R(u, u′, u′′, u′′′ , …………… . . ) = 0 (2.2)

where R is a polynomial of u and its derivatives and the subscripts means for ordinary derivatives w.r.t. ��.

Step II: Now we consider the solution of Eq.(2.2) in the form of

��(��) = ∑ ci(exp (−φ(ξ)))
in

i=0 (2.3)

where ����,(i=0,1…,n) are constant to be determined and ��(��) satisfies the ordinary differential equation. N is a
positive integer determined by the homogeneous balance principle between the nonlinear term and the highest
order derivatives in Eq. (2.1)

φ′(ξ) = exp(−φ(ξ)) + μexp(φ(ξ)) + λ (2.4)

When �� ≠ 0,△= ��2 − 4�� > 0,

φ(ξ) = ln

(

−√△ tanh (

√△
2 (ξ + a)) − λ

2μ

)

(2.5)

Step III: Now substitute Eq. (2.3) in Eq.(2.2) along with Eq. (2.4) and then collect the coefficients of different
power of exp (−φ(ξ)) and equating to zero. This provides a system of algebraic equations. Solving these
system for ����,, ��, �� and substitute it in Eq. (2.3) along with Eq.(2.5), we get a complete exact solution for the
Eq.(2.1).

3. Some exact solution by Exp(-��(��))-expansion method

Here we obtain some exact solution of the Eq.(1.2) with the help of this method. Under the submission of
u(x, t) = p(t)v(x, t) + q(t) into the Eq. (1.1), we get

vt + A(t)p(t)vvx + A(t)q(t)vx + B(t)vxxx + C(t)vxx = 0 (3.1)

with the choices p(t) = k1e
−∫D(t)dt, q(t) = e∫D(t)dt ∫(H(t)e−∫D(t)dt) dt ,where k1 is an integrating constant.

The variable separation solution of Eq. (1.2) is investigated by using the Exp(-φ(ξ))- expansion method with
variable separation transformation. Using the transformation �� = �� + ��(��) in Eq. (3.1), we get the following
ODE

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(��′(��) + ��(��)��(��))��′ + ��(��)��(��)����′ + ��(��)��′′ + ��(��)��′′′ = 0 (3.2)

The expression for the solution of Eq. (3.2) is given by

v(ξ) =c0(x, t) + c1(x, t) exp(−φ(ξ)) + c2(x, t)(exp (−φ(ξ)))
2 , (3.3)

Now inserting the Eq.(3.3) into Eq. (3.2) and collecting the corresponding coefficients of exp(−φ(ξ)), we
obtained a system of algebraic equation and by symbolic calculation, we get the following parameter values

c2 = −
12B(t)

A(t)k1e
−∫D(t)dt

, c1 =
12(C(t)−5λB(t))
5A(t)k1e

−∫D(t)dt

c0 =
−(25B2λ2+200B2μ−30BCλ+25ABq−C2+25Bg′)

25A(t)B(t)k1e
−∫D(t)dt

, (3.4)

with the condition ��(��) = 5√λ2 − 4μ B(t) and g(t) is a test function. Then the variable separation solution of
the Eq.(1) is given by

u = ��1e
−∫D(t)dt(c0 −

2μc1

√△tanh(
√△
2
(ξ+a))+λ

−
4μc2

(√△tanh(
√△
2
(ξ+a))+λ)

2)+e∫D(t)dt ∫(H(t)e−∫D(t)dt) dt

(3.5)

where k1 and �� are arbitrary constants. This solution (3.5) is clearly a singular non-traveling wave solution.
Now we will introduced the different test function ��(��) to get different type of solution of Eq.(1.2).

Case I: Set ��(��) = tanh(��).Then from Eq.(3.5),we get exact solution of Eq.(1.2) as

u = −
25B2λ2+200B2μ−30BCλ+25ABq−C2+25B sech(t)2

25AB
−

24μ(C(t)−5λB(t))

5A(√△tanh(
√△
2
(ξ+a))++λ)

+
48μ2B

A(√△tanh(
√△
2
(ξ+a))+λ)

2 +

e∫D(t)dt ∫(H(t)e−∫D(t)dt) dt (3.6)

Fig.1 presented the dynamic behavior of the exact solution (3.6).It shown that shock nature of the solution is
observed and a soliton in the background of shock is clearly observed. Clearly a dominance of ��(��) = tanh (��)
is observed on the solution. Fig.1(b) shown the two-dimensional view of the exact solution (3.6) with �� =
−0.5 which show a soliton-shock nature which is surely observed in the contour plot of the solution (3.6).

(a) (b) (c)

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Fig. 1: Three dimensional profile, two dimensional profile and contour plot of the exact solution (3.9) is presented with
the following numeric values of the parameters ��(��) = −1.5, ��(��) = 0.1, ��(��) = 1,��(��) = sin(��) , �� = 3, �� = 1, ��1 =
1, �� =2

Case II: Set ��(��) = cos(��).Then from Eq.(3.5),we get exact solution of Eq.(1.2) as

u = −
25B2λ2+200B2μ−30BCλ+25ABq−C2+25Bcos(t)

25AB
−

24μ(C(t)−5λB(t))

5A(√△tanh(
√△
2
(ξ+a))++λ)

+
48μ2B

A(√△tanh(
√△
2
(ξ+a))+λ)

2 +

e∫D(t)dt ∫(H(t)e−∫D(t)dt) dt (3.7)

Fig.2(a) presented the three dimensional dynamic behavior of the solution (3.7).It shown the shock nature of
the solution (3.7) in the periodic background ��(��) = cos (��). Obviously a dominance of ��(��) = cos (��) is
observed in the solution. Fig.2(b) depicted the two dimensional profile of this solution with �� = −1.5 and
Fig.2(c) is the contour plot of the solution (3.7) shown the strong periodic behavior the solution with shock
nature.

(a) (b) (c)

Case III: Set ��(��) = sech(��).Then from Eq.(3.5),we get exact solution of Eq.(1.1) as

u = −
25B2λ2+200B2μ−30BCλ+25ABq−C2−25B sech(t)tanh (��)

25AB
−

24μ(C(t)−5λB(t))

5A(√△tanh(
√△
2
(ξ+a))++λ)

+
48μ2B

A(√△tanh(
√△
2
(ξ+a))+λ)

2 +

e∫D(t)dt ∫(H(t)e−∫D(t)dt) dt (3.8)

Fig.3(a) presented the three dimensional dynamic behavior of the exact solution (3.8).It shown the shock
nature with dominance ��(��) = sech (��) of the solution (3.9).Fig.3(b) depicted the two dimensional
visualization of this solution with �� = −1.5 which shown peak in one side and deep in other side. Fig.2(c) is
the contour plot of the solution (3.8) which ensures our observation.

Fig. 2: Three dimensional profile, two dimensional profile and contour plot of the exact
solution (3.9) is presented with the following numeric values of the parameters ��(��) =
−1.5, ��(��) = 0.1, ��(��) = 1,��(��) = sin(��) , �� = 3, �� = 1, ��1 = 1, �� = 2

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(a) (b) (c)

Case IV: Set ��(��) = Sec (��).Then from Eq.(3.5),we get exact solution of Eq.(1.2) as

u = −
25B2λ2+200B2μ−30BCλ+25ABq−C2−25B sec(t)tan (��)

25AB
−

24μ(C(t)−5λB(t))

5A(√△tanh(
√△
2
(ξ+a))++λ)

+
48μ2B

A(√△tanh(
√△
2
(ξ+a))+λ)

2 +

e∫D(t)dt ∫(H(t)e−∫D(t)dt) dt (3.9)

Fig.3(a) demostrated the three dimensional dynamic behavior of the exact solution (3.8).It shown the singular
periodic exact solution with dominance ��(��) = sec(��).Fig.3(b) depicted the two dimensional visualization of
this solution with �� = 1 which shown singular shock periodic. Fig.2(c) is the contour plot of the solution (3.8)
which ensures our observation

(a) (b) (c)

Fig. 3: 3D profile,2D profile and contour plot of the exact solution (3.8) is presented with
the following numeric values of the parameters ��(��) = −1.5, ��(��) = 0.1, ��(��) =
1,��(��) = sin(��) , �� = 3, �� = 1, ��1 = 1, �� = 2.

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Fig. 4: 3D profile,2D profile and contour plot of the exact solution (3.9) is presented with the following
numeric values of the parameters ��(��) = −1.5, ��(��) = 0.1, ��(��) = 1,��(��) = sin(��) , �� = 3, �� = 1, ��1 =
1, �� = 2

CONCLUSION

The Exp(-φ(ξ))-expansion method, with modifications, was successfully applied to the damped forced KdV-
Burger's equation with variable coefficients, resulting in the derivation of generalized exact solutions. The
effectiveness of this method was demonstrated through its ability to yield precise solutions for this complex
equation. Given its success, the Exp(-φ(ξ))-expansion method holds promise for solving other nonlinear partial
differential equations, offering a powerful tool for researchers to explore and analyze various nonlinear
phenomena in physics and mathematics. Its potential applications span a wide range of fields, including fluid
dynamics, plasma physics, and nonlinear optics, where nonlinear PDEs play a crucial role in modeling real-
world systems.

Conflicts of interest

The authors declare that they have no conflicts of interests

The authors declare that they have no conflicts of interests

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