/compile/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 8, No. 1, 2015, 50-63

ISSN 1307-5543 – www.ejpam.com

An Unconventional Splitting for Korteweg de Vries–Burgers

Equation

A. Aydin ∗

Department of Mathematics, Atılım University, 06836, Ankara, Turkey

Abstract. Numerical solutions of the Korteweg de Vries–Burgers (KdVB) equation based on splitting is

studied. We put a real parameter into a KdVB equation and split the equation into two parts. The real

parameter that is inserted into the KdVB equation enables us to play with the splitted parts. The real

parameter enables to write the each splitted equation as close to the Korteweg de Vries (KdV) equation

as we wish and as far from the Burgers equation as we wish or vice a versa. Then we solve the splitted

parts numerically and compose the solutions to obtained the integrator for the KdVB equation. Finally

we present some numerical experiments for the solution of the KdV, Burger’s and KdVB equations. The

numerical experiments shows that the new splitting gives feasible and valid results.

2010 Mathematics Subject Classifications: 65M06, 35Q53, 33F05

Key Words and Phrases: Korteweg de Vries–Burgers equations, splitting methods, finite differences

1. Introduction

In this paper we consider the Korteweg de Vries–Burgers (KdVB) equation derived by Su

and Gardner [17]

ut + ǫuux − νux x +µux x x = 0, x ∈ Ω, t ∈ [0, T], (1)

with the initial condition

u(x , 0) = f (x), x ∈ Ω, (2)

and the boundary condition

u(xL , t) = u(xR, t) = 0, t ∈ [0, T] (3)

where Ω = [xL , xR], ǫ, ν and µ are positive parameters with ǫνµ 6= 0 and the subscripts t

and x denote the differentiation with respect to time and space. It is one of the important

mathematical models which has many applications in science and engineering such as fluid

∗Corresponding author.

Email address: ayhan.aydin@atilim.edu.tr

http://www.ejpam.com 50 c© 2015 EJPAM All rights reserved.



A. Aydin / Eur. J. Pure Appl. Math, 8 (2015), 50-63 51

dynamics, plasma physics, nonlinear circuit theory and astrophysics (see, for example [18]

and reference therein). It is also a model equation for wave propagation in fluid-filled elastic

or viscoelastic tubes [3, 13].

The equation (1) is interesting because of several reason. First of all, it contains the nonlin-

earity (uux), dissipation (ux x) and dispersion (ux x x) terms. In particular, a progressive wave

solution of the KdVB equation (1) by use of hyperbolic tangent method have been presented

in [4]. When the dissipative parameter ν is varied, the solution is altered from an oscillatory

profile to a monotonic one [7] and the solution vectors end up behaving like traveling waves

for which the amplitudes are damped [21]. Also, as ν varied shock wave becomes more oscil-

latory [21]. Moreover, the choices ǫ 6= 0, ν 6= 0 and µ = 0 reduces the evolution equation (1)

to the Burgers’ equation

ut + ǫuux − νux x = 0, (4)

and the choices ǫ 6= 0, µ 6= 0 and ν = 0 lead the equation (1) to the Korteweg de Vries (KdV)

equation

ut + ǫuux +µux x x = 0, (5)

which has at least the following three invariants

I1 =

∫ ∞

∞

udx , I1 =

∫ ∞

∞

u2 dx , I3 =

∫ ∞

∞

�
u3 −

3µ

ǫ

�
u′
�2�

dx (6)

corresponding to conservations of mass, momentum and energy respectively. It is well known

that the Burgers’ equation (4) and the KdV equation (5) are exactly solvable and have traveling

wave solutions of the form

u(x , t) =
2k

ǫ
+

2νk

ǫ
tanh [k(x − 2kt)] (7)

and

u(x , t) =
2µk2

ǫ
sech2
�
k(x − 4µk2 t)
�

(8)

respectively, where k = ν/(10µ). The KdVB equation (1) has an exact solution [19]

u(x , t) = −
3ν2

25ǫµ

�
− sech2(k(x − x0)− c t) + 2 tanh(k(x − x0)− c t) + 2

�
(9)

where c = 3ν3/(125µ2). Since most of the nonlinear partial differential equations (PDEs)

that contain dissipation and/or dispersion does not have exact solution, developing some new

techniques for the numerical solution of these types of nonlinear PDEs are essential to under-

stand solution behavior. For the equation (1) many works have been done analytically and

numerically. In particular, a progressive wave solution of the equation KdVB equation by use

of hyperbolic tangent method(1) have been presented in [4]. By introducing a new potential

function and by using the hyperbolic tangent method and an exponential rational function

approach, a traveling wave solution to the KdVB equation (1) has been obtained in [5]. An

exact solution to the equation (1) is presented in [6] by using the first-integral method. The



A. Aydin / Eur. J. Pure Appl. Math, 8 (2015), 50-63 52

numerical investigation of the problem has been carried out by many authors. In paricular,

the numerical solution of the KdVB equation has been studied by using a radial basis functions

(RBFs) collocation (Kansa) method in [9] . The collocation method with quintic B-spline fi-

nite element has been used in [21] to simulate the solutions of the equation (1). In [2] KdVB

equation is solved numericaly by means of spectral collocation method. Variational iteration

method has been implemented in [16] to solve the KdVB equation. In [10], a finite differences

with variable mesh and the semi-analytic Adomian decomposition method are used to solve

KdVB equation. In [19] the KdVB equations is solved by using the local discontinuous Galerkin

method.

Splitting methods are used frequently to recapture the dynamics of different parts of dif-

ferential equations and applying appropriate numerical integrators for each part (see [11, 15]

and references therein). In [1] a KdVB type equation with fast and slow dynamics is solved by

using operator splitting. The KdVB equation can be splitted as KdV equation and the Burgers’s

equation or vice a versa. However, since the KdVB equation is a nonlinear diffusive dispersive

equation, the numerical solution of this equation should represent all qualitative behavior of

both the KdV equation and the Burger’s equation. In this paper, to capture this feature of the

KdVB equation a new kind of splitting is introduced. For this we put a real parameter into a

KdV-Burgers equation and split the vector field of the equations into two parts both of which

are KdVB equation. The real parameter that is inserted into the KdVB equation enables us

to play with the splitted parts. By varying the real parameter, we are able to write the KdVB

equation as close to the KdV equation as we wish and as far from the Burgers equation as we

wish or vice a versa. We solve then the splitted parts numerically and compose the solutions to

obtained the integrator for the KdVB equation. The rest of the paper is organized as follows:

In Section 2, we described the proposed unconventional splitting for the KdVB equation. The

numerical method for the proposed splitting is presented in Section 3. In Section 4 we tested

the performance of the splitting in KdV simulation, Burgers’s simulation and KdV-Burgers sim-

ulation. Finally the conclusion is given in Section 5.

2. An Unconventional Splitting for the KdVB Equation

Numerical solution of differential equations by using splitting methods has been frequently

used in the literature especially after introducing higher-order composition formulae [15, 20].

The main idea of the composition method is to split the vector field associated with the differ-

ential equation into sum of two or more pieces that can be solved exactly or integrated easily

than the original equation. In this section we will presents a new splitting to integrate the

KdVB equation (1) numerically. The KdVB equation can be splitted as KdV equation

ut +
1

2
ǫuux +µux x x = 0 (10)

and the Burgers’s equation

ut +
1

2
ǫuux − νux x = 0 (11)

or vice a versa. However, since the KdVB equation (1) is a nonlinear diffusive dispersive

equation, the numerical solution of this equation should represent all qualitative behavior



A. Aydin / Eur. J. Pure Appl. Math, 8 (2015), 50-63 53

of both the KdV equation and the Burger’s equation. To capture this feature we rewrite the

equation (1) as

ut +
1

2
ǫuux +

1

2
ǫuux − νux x +µux x x = 0,

and add then subtract the terms ναux x and µβux x x to obtain

ut +
1

2
ǫuux +

1

2
ǫuux − νux x +µux x x − ν(α−α)ux x +µ(β − β)ux x x = 0 (12)

where 0≤ α≤ 1 and 0≤ β ≤ 1 are real parameters. Then the equation (12) is rewritten as

ut +
1

2
ǫuux +

1

2
ǫuux − ν(1−α)ux x − ναux x +µ(1− β)ux x x +µβux x x = 0. (13)

Note that the equation (13) is reduced to the KdVB equation (1) for the choices of α = β = 0

and α= β = 1.

Now the new splitting method for the KdVB equation (13) is defined by solving first

ut +
1

2
ǫuux − ν(1−α)ux x +µβux x x = 0, (14)

followed by

ut +
1

2
ǫuux − ναux x +µ(1− β)ux x x = 0. (15)

Note that the equations (14) and (15) are both KdVB equations for 0< α,β < 1. With this

point of view the splitting (14) and (15) are different from the splittings (10) and (11).

The equations (14) and (15) are reduced to the Burgers’ equation and the KdV equation

for α = β = 0 respectively. In addition, the choices α= β = 1 reverse the situation that is the

equation (14) becomes the KdV equation and (15) becomes the Burgers’ equation. Therefore,

the choices α = β = 0 or α = β = 1 reduces the splitting (14) and (15) to the splittings (10)

and (11).

3. Numerical Method

In order to obtain numerical solutions of the KdVB equation (13), the space interval [xL , xR]

is discretized by the uniform N grid

x i = xL + ih, i = 1,2, . . . N

where the grid spacing h is given by h= (xL− xR)/N . The solution is assumed to be negligible

outside the interval [xL , xR]. Let Ui(t) denotes the approximate solution to the exact solution

u(x i , t). We discretize the space variable of the splitted equation (14) and (15) using the

central difference approximation and get the semi–discrete systems

d

d t
Ui +

ǫ

2
Ui

�
Ui+1 − Ui−1

2h

�
− ν(1−α)
�

Ui+1 − 2Ui + Ui+1

h2

�



A. Aydin / Eur. J. Pure Appl. Math, 8 (2015), 50-63 54

+µβ

�
Ui+2 − 2Ui+1 + 2Ui−1 − Ui−2

2h3

�
= 0 (16)

d

d t
Ui +

ǫ

2
Ui

�
Ui+1 − Ui−1

2h

�
− να
�

Ui+1 − 2Ui + Ui+1

h2

�

+µ(1− β)
�

Ui+2 − 2Ui+1 + 2Ui−1 − Ui−2

2h3

�
= 0 (17)

where i = 1,2, . . . N and

U−2 = U−1 = 0, UN+1 = UN+2 = 0.

The systems (16) and (17) can be written as

d

d t
Ui = F1(Ui)

d

d t
Ui = F2(Ui) (18)

where

F1(Ui) =aA(Ui)Ui + b1BUi + c1CUi (19)

F2(Ui) =aA(Ui)Ui + b2BUi + c2CUi (20)

respectively with

a = −
ǫ

4h
, b1 =

ν(1−α)
h2

, c1 = −
µβ

2h3
,

b2 =
να

h2
, c2 = −

µ(1− β)
2h3

and

A=





U2 0 · · · 0 0

0 U3 − U1 · · · 0 0
...

...
. . .

...
...

0 0 · · · −UN − UN−2 0

0 0 · · · 0 −UN−1





N×N

,

B =





−2 1 0

1 −2 1
...

. . .
. . .

1 −2 1

0 1 −2





N×N

C =





0 −2 1 0 0

2 0 −2 1 0

−1 2 0 −2 1
...

. . .
. . .

. . .
. . .

−1 2 0 −2 1

0 −1 2 0 −2

0 0 −1 2 0





.



A. Aydin / Eur. J. Pure Appl. Math, 8 (2015), 50-63 55

Because the right-hand sides of (18) are quadratic, the reflexive method of Kahan [14] can

be applied for the time discretization. Application of the reflexive method to the systems of

equations in (18) yields

�
I −
∆t

2
JF1
(Un)

��
Un+1 − Un
�
=∆t F1(U

n), (21)

�
I −
∆t

2
JF2
(Un)

��
Un+1 − Un
�
=∆t F2(U

n) (22)

respectively, where JF1
and JF2

denote the Jacobians of the right-hand sides of (21) and (22).

Denoting the solution Un+1 of (21) and (22) by fS1 and fS2 respectively, the numerical solution

for the KdVB equation (13) is obtained by the time reversible second–order Strang splitting

[8]

S (t) = exp

�
fS1

�
∆t

2

��
exp
�fS2 (∆t)
�

exp

�
fS1

�
∆t

2

��
. (23)

4. Numerical Experiment

In this section, some numerical example will be demonstrated to examine the robustness

and accuracy of the proposed splitting (14)-(15) by using (23). The discrete values of (1) will

be computed via the (23). The accuracy of the method (23) is measured by calculating the L2

and L∞ error norms of the solution defined by

L2 =‖U
n − un‖2 =

�
h

N∑

i=1

��U(x i, tn)− u(x i , tn)
��2
�1/2

L∞ =‖U
n − un‖∞ = maxi

��U(x i , tn)− u(x i , tn)
�� , i = 1,2, . . . , N .

4.1. KdV Simulation

To examine the performance of the proposed scheme for the KdV equation, we set the

parameter ν = 0 in (1). The performance of the splittings (14)-(15) with α = β = 0.5 by

using (23) will be check for two types of initial conditions namely a soliton solution and a

Maxwellian initial condition. Moreover, the conservation of the numerical scheme will be

examined by looking the invariants (6) where the integrals are approximated by trapezoidal

rule.

Experiment A

The KdV equation has an analytic solution of the form

u(x , t) = 3Csech2(Ax − Bt + D) (24)

where A= 1
2

p
ǫCµ−1 and B = ǫAC . Initial condition

u(x , 0) = 3Csech2(Ax + D) (25)



A. Aydin / Eur. J. Pure Appl. Math, 8 (2015), 50-63 56

is obtained from the exact solution (24). Boundary conditions are taken as u(0, t) = 0 and

u(2, t) = 0. We take ǫ = 1.0, µ= 4.84×10−4,C = 0.3,D = −6.0. For these parameters the ex-

act values of the invariants I1, I2 and I3 in (6) are 0.144597866741365, 0.0867592530989926

and 0.0468494613399944 respectively. To see the conservation of the invariants of the pro-

posed scheme, we performed a numerical experiment with the time length ∆t = 0.005 and

the space length ∆x = h = 0.001 up to time T = 3.0 in the region x ∈ [0,2]. Moreover, the

L2-errors and the L∞-errors are presented to see the accuracy of the proposed scheme. The re-

sults are shown in Table 1. We see that the scheme (23) preserves the invariants Ii , i = 1,2,3.

Moreover, L2 and L∞ are satisfactorily small.

Table 1: Errors in Conservation of Mass, Momentum and Energy and L2 and L∞ Errors.

Time I1 I2 I3 L2 L∞
0.0 1.1e-8 2.5e-13 -2.8e-6 0.0 0.0

1.0 -7.6e-7 -3.7e-10 -2.8e-6 7.7e-5 2.2e-4

2.0 -1.9e-6 -3.3e-10 -2.8e-6 1.5e-4 4.3e-4

3.0 -2.2e-6 -3.8e-10 -2.8e-6 2.3e-4 6.3e-4

To see whether the proposed numerical scheme exhibits the expected convergence rates in

space, we performed some further numerical experiment for various values of h and a fixed

value of ∆t. The rate of convergence is calculated by

rate of convergence ≈
ln(E(h2)/E(h1))

ln(h2/h1)

where E(h) is the L2 or L∞ error. We take ∆t = 5× 10−4 to minimize the temporal errors.

The results corresponding to the scheme (23) are shown in Table 2 for an decreasing space

length. We present the L2-errors and the L∞-errors for the terminating time t = 1 together

with the observed rates of convergence in each case. The computed convergence rates agree

well with the expected rates when the second order central difference approximation is used

for discretization in space direction.

Table 2: Rate of Convergence.

h L2 Order L∞ Order

0.1 0.417248 - 1.060321 -

0.02 0.023961 1.77 0.068023 1.70

0.01 0.005802 2.04 0.016483 2.04

0.005 0.001436 2.01 0.004183 1.98

0.0025 0.000358 2.00 0.001048 1.99



A. Aydin / Eur. J. Pure Appl. Math, 8 (2015), 50-63 57

Experiment B

Now we consider the KdV equation with the Maxwellian initial condition

u(x , 0) = exp(−x2), −15< x < 15 (26)

subject to the boundary conditions

u(−15, t) = u(15,0) = 0, t > 0. (27)

We take ǫ = 1.0 and consider for µ the set of parameters µ= 0.04,0.01,0.001,0.0005 and run

the simulation up to time t = 12 with∆t = 0.03, h= 0.02. The results are shown in Figure 1.

It is seen from the figure that Maxwellian initial condition exhibits rapidly oscillating wave

packets. For example, for µ= 0.04 one solitary wave with an oscillating tail is observed, while

for µ = 0.0005 twelve solitons are formed. This is an agreement with the earlier works [21]

and [19]. The conservation of the invariants for µ = 0.001 and µ = 0.0005 are presented in

Table 3.

−15 −10 −5 0 5 10 15
−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

u

(a) µ= 0.04

−2 0 2 4 6 8 10
−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

u

(b) µ= 0.01

−2 0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

u

(c) µ= 0.001

−2 0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

u

(d) µ= 0.0005

Figure 1: KdV simulation with Maxwellian initial condition at t = 12 for different values of µ.

(a) One soliton plus oscillating tail with µ = 0.04. (b) Three solitons with µ = 0.01. (c) Nine

solitons with µ= 0.001. (d) Twelve solitons with µ= 0.0005



A. Aydin / Eur. J. Pure Appl. Math, 8 (2015), 50-63 58

Table 3: Errors in Conservation of Mass, Momentum and Energy.

Time µ= 0.001 µ= 5× 10−4

I1 I2 I3 I1 I2 I3

3.0 1.7e-12 -1.3e-2 -4.3e-2 3.0e-15 -2.7e-2 -9.3e-2

6.0 1.7e-6 -1.6e-2 -5.1e-2 -8.7e-8 -3.4e-2 -1.1e-1

9.0 -9.7e-8 -1.6e-2 -5.1e-2 -2.4e-5 -3.5e-2 -1.1e-1

12.0 7.0e-6 -1.6e-2 -5.1e-2 -3.5e-5 -3.5e-2 -1.1e-1

4.2. Burgers’ Simulation

Now we set µ = 0 and examine the performance of the proposed splittings (14)-(15)

by using the composition (23) for the Burgers’ equation. Burgers’s equation has an analytic

solution of the form [12]

u(x , t) =
x/t

1+ (t/t0)exp(x2/4vt)
(28)

where t0 = exp(1/8v). Initial condition is obtained by evaluating the Eq. (28) at t = 1. The

homogenous boundary conditions u(xL , t) = u(xR, t) = 0, (t ≥ 1) are used. We set ǫ = 1.0,

1 ≤ t ≤ 5, ∆t = 0.02 and h = 0.01 for all simulations. On the interval 0 ≤ x ≤ 8 we used

h = 0.01 ν = 0.5,0.05,5 × 10−3 and on the interval 0 ≤ x ≤ 2 we used h = 0.0025 for

µ = 5× 10−4. The Figure 2 shows the development of the solution for different values of ν

for the KdVB equations (14) and (15) with α = β = 0.5 by using (23). In the Figure 2, the

top curve shows the solution at t = 1 while the bottom curve shows the solution at t = 4.0.

It can be seen that the amplitudes of the solutions decrease in time. Moreover, decreasing the

viscosity parameter results in shock waves. All the results reported for different values of ν

are in good agreement with the earlier work of [21] and [19]. The error norms are recorded

in Table 4.

Table 4: Errors Norms for Burgers’ Type Solution.

Time ν= 0.5 ν= 0.05 ν= 0.005 ν= 5× 10−4

L2 L∞ L2 L∞ L2 L∞ L2 L∞
1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

2.0 8.2e-2 6.0e-2 2.9e-2 4.1e-2 9.6e-3 4.0e-2 3.9e-3 6.2e-2

3.0 9.8e-2 6.5e-2 4.0e-2 5.0e-2 1.4e-2 5.3e-2 4.3e-3 5.2e-2

4.0 9.8e-2 6.2e-2 4.5e-2 5.1e-2 1.6e-2 5.8e-2 5.1e-3 5.6e-2

5.0 9.8e-2 5.8e-2 4.8e-2 5.1e-2 1.8e-2 6.0e-2 5.6e-3 5.8e-2



A. Aydin / Eur. J. Pure Appl. Math, 8 (2015), 50-63 59

0 1 2 3 4 5 6 7 8
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

x

u

(a) ν = 0.5

0 0.5 1 1.5 2 2.5 3
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

x

u

(b) ν= 0.05

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0

0.1

0.2

0.3

0.4

0.5

0.6

x

u

(c) ν= 5× 10−3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

u

(d) ν= 5× 10−4

Figure 2: Burgers’s Type Solutions at t = 1,2,3,4 (from top to bottom).

4.3. The KdV–Burger Type Solutions

Now we consider the new splitting (14) and (15) with α = 0.2 and β = 0.8. We recall

that for α = β = 0 the equation (14) reduces to Burgers’ equation and the equation (15)

reduces to the KdV equation. Therefore the composition (23) corresponds to Burger-KdV-

Burger composition for α = β = 0 which will be abbreviated by BKB simulation. However

for α = β = 1 the equation (14) reduces to the KdV equation and the equation (15) reduces

to the Burgers’ equations. Therefore the composition (23) corresponds to KdV-Burger-KdV

compositon for α = β = 1 which will be abbreviated by KBK simulation. For 0 < α,β < 1,

both fS1 and fS2 are KdVB equation; therefore the composition (23) will be abbreviated by

KdVB simulation for which we choose α= 0.2 and β = 0.8.

Experiment A

Now we consider the KdVB equation (1) with ε = 1.0. We pick the initial and boundary

conditions from the exact solution (9)

u(x , 0) = −
3ν2

25ǫµ

�
− sech2(kx) + 2 tanh(kx) + 2

�
. (29)



A. Aydin / Eur. J. Pure Appl. Math, 8 (2015), 50-63 60

We present the numerical results over the spatial interval 0 ≤ x ≤ 100 and temporal interval

0≤ t ≤ 100 with ∆x = 0.2 and ∆t = 0.1.

Table 5 represents the absolute errors define by

Absolute Error = |u(x i , t)− U(x i , t)|, i = 0,25,50,75,100

for various space values at time t = 100. From the table we see that the three kinds of splitting

gives approximately the same remarkable accuracy. These results are in good agrement with

the earlier work of [16].

Table 5: Absolute Errors at t = 100

x KdVB KBK BKB

α= 0.1, β = 0.8 α= β = 0 α= β = 1

ν= µ= 0.1 0.0 1.21× 10−6 1.22× 10−6 1.21× 10−6

25.0 6.63× 10−8 6.63× 10−8 6.63× 10−8

50.0 4.60× 10−10 4.60× 10−10 4.60× 10−10

75.0 2.90× 10−12 2.89× 10−12 3.00× 10−12

100.0 9.40× 10−10 9.33× 10−10 9.37× 10−10

ν= µ= 0.01 0.0 1.46× 10−8 1.47× 10−8 1.46× 10−8

25.0 1.02× 10−9 1.02× 10−9 1.02× 10−9

50.0 7.08× 10−12 7.08× 10−12 7.10× 10−12

75.0 4.77× 10−14 7.77× 10−14 4.77× 10−14

100.0 5.45× 10−15 4.83× 10−15 5.36× 10−15

Experiment B

In the last experiment we compute the numerical solution of Eq.(1) by using the scheme (23)

with the initial condition

u(x , 0) = 0.5

�
1− tanh

|x | − 25

5

�

and boundary conditions u(−50, t) = u(150, t) = 0. We choose ǫ = 0.2, µ = 0.1, ∆t = 0.4,

h = 0.5 and study the effect of increasing the viscosity and hence the dispersion term on the

solution. For this we run the scheme (23) for different values of ν. The results for shown in

Fig. 3. From the figure when we see that increase the viscosity ν, the solution of the KdVB

equation (1) behaves like the solution of the Burger’s equation (4) as in the work of Zaki [21].



A. Aydin / Eur. J. Pure Appl. Math, 8 (2015), 50-63 61

−50 0 50 100 150

0

0.5

1

1.5

2

2.5

x

u

(a) ν = 0.0

−50 0 50 100 150

0

0.5

1

1.5

2

2.5

x

u

(b) ν= 0.0001

−50 0 50 100 150
−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

u

(c) ν = 0.005

−50 0 50 100 150
−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

u

(d) ν= 0.01

−50 0 50 100 150
−0.2

0

0.2

0.4

0.6

0.8

1

x

u

(e) ν= 0.03

−50 0 50 100 150
−0.2

0

0.2

0.4

0.6

0.8

1

x

u

(f) ν= 0.05

−50 0 50 100 150
−0.2

0

0.2

0.4

0.6

0.8

1

x

u

(g) ν= 0.2

−50 0 50 100 150
−0.2

0

0.2

0.4

0.6

0.8

1

x

u

(h) ν= 0.4

Figure 3: KdV–Burgers’s Solutions at t = 800 with ǫ = 0.2, µ= 0.1.



REFERENCES 62

5. Conclusion

Since most of the nonlinear PDEs that contain dissipation and/or dispersion does not have

exact solution, developing some new techniques for the numerical solution of these types of

nonlinear PDEs are essential to understand solution behavior. In this paper, we considered a

numerical solution of the KdVB equation based on the splitting. The KdVB equation can be

splitted as KdV equation and the Burgers’s equation or vice a versa. However, since the KdVB

equation is a nonlinear diffusive dispersive equation, the numerical solution of this equation

should represent all qualitative behavior of both the KdV equation and the Burger’s equation.

To capture this feature a new kind of splitting, which is different from the splitting in the

literature, is introduced for KdVB equation. For this we introduce a real parameters α and β for

the KdVB equation and then split the equation into two parts where both splitted equations are

KdVB equation. The real parameters enable us to play with the splitted equations in the sense

that each splitted equation can be made arbitrarily close to the KdV equation or the Burger’s

equation. From the above tables and figures we conclude that the numerical results based on

the new splitting are feasible and valid. We see that the numerical results are not sensitive

for the added parameters namely playing with the real parameters α and β does not chance

the numerical results. The splitting method we introduce in this paper is less restrictive and

comprises the numerical solutions that exist in the literature. This splitting approach can be

extended to solve other types of equations such as nonlinear Klein-Gordon-Maxwell equation

and the Klein-Gordon-Schrödinger equation. It is an open question that whether playing with

such an added parameters chance the numerical results or not.

ACKNOWLEDGEMENTS The author thanks to Prof.Dr. Bülent Karasözen for his suggestion

during the preparation of this paper.

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