Applied Mathematics, 2011, 2, 739-749
doi:10.4236/am.2011.26098 Published Online June 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
Septic B-Spline Collocation Method for the Numerica l
Solution of the Modified Equal Width Wave Equation
Turabi Geyikli, Seydi Battal Gazi Karakoc
Department of Mathematics, Faculty of Education, Inonu University, Malatya, Turkey
E-mail: tgeyikli@inonu.edu.tr, sbgk44@mynet.com
Received March 25, 2011; revised April 20, 2011; accepted April 23, 2011
Abstract
Numerical solutions of the modified equal width wave equation are obtained by using collocation method
with septic B-spline finite elements with three different linearization techniques. The motion of a single soli-
tary wave, interaction of two solitary waves and birth of solitons are studied using the proposed method.
Accuracy of the method is discussed by computing the numerical conserved laws error norms L2 and L. The
numerical results show that the present method is a remarkably successful numerical technique for solving
the MEW equation. A linear stability analysis shows that this numerical scheme, based on a Crank Nicolson
approximation in time, is unconditionally stable.
Keywords: Finite Element, Septic B-Spline, Collocation, Soliton, Solitary Waves
1. Introduction
This study is concerned with the numerical solution, us-
ing septic B-spline functions in collocation method, of
the modified equal width wave (MEW) equation, which
was introduced by Morrison et al. [1] as a model equa-
tion to describe the nonlinear dispersive waves. Few
analytical solutions of the MEW equation are known.
Thus numerical solutions of the MEW equation can be
important and comparison between analytic solution can
be made. Many methods have been proposed to solve the
EW and MEW equation. Gardner and Gardner solved the
EW equation with the Galerkin’s method using cubic
B-splines as a trial and test function [2,3] and a Pet-
rov-Galerkin method using quadratic B-spline element
[4]. Zaki considered the solitary wave interactions for the
MEW equation by Petrov-Galerkin method using quintic
B-spline finite elements [5] and obtained the numerical
solution of the EW equation by using least-squares
method [6]. Wazwaz investigated the MEW equation and
two of its variants by the tanh and the sine-cosine meth-
ods [7]. Esen applied a lumped Galerkin method based
on quadratic B-spline finite element has been used for
solving the EW and MEW equation [8,9]. Saka proposed
algorithms for the numerical solution of the MEW equa-
tion using quintic B-spline collocation method [10]. A
solution based on a collocation method incorporated cu-
bic B-splines is investigated by Dağ and Saka [11].
Variational iteration method is introduced to solve the
MEW equation by Junfeng Lu [12]. Hamdi et al. [13]
derived exact solitary wave solutions of the generalized
EW equation using Maple software. D. J. Evans and K.
R. Raslan [14] studied the generalized EW equation by
using collocation method based on quadratic B-splines to
obtain the numerical solutions of a single solitary waves,
and the birth of solitons.
The modified equal width wave equation which is as a
model for non-linear dispersive waves, considered here
has the normalized form [1]
2
3
tx xxt
UUU U
0
 (1)
with the physical boundary conditions as
, where is time and
0U
x
t
x
is the space coordi-
nate,
is a positive parameter. For this study bound-
ary conditions are chosen

 
 
,0, ,0,
,0, ,0,
,0, ,0
xx
xxx xxx
Uat Ubt
Uat Ubt
Uat Ubt


,
(2)
and the initial condition as
,0, Uxfxaxb

where
f
x is a localized disturbance inside the con-
sidered interval.
T. GEYIKLI ET AL.
740
2. Septic B-Spline Collocation Method
The interval
,ab is partitioned into uniformly sized
finite elements by the knots i
x
such that
01 N
ax xxb
 and . The septic
1ii
hx x

B-splines
i
, (i= 3(1) N + 3), at the knots i
x
are
defined over the interval
,ab as [15],










7
443
77
43 3
77 7
43 22
777 7
43211
777 7
74321
,,
8, ,
828, ,
82856,,
1( )8()28( )56(),,,
iii
ii i
ii ii
ii i ii
iii i iii
i
xxx xx
xxxxx x x
xxxxxxx xx
xxxxxxxxxxx
xxx xxxxxxxxx
h
x

 
 
  
 


 

 






777
43 21
77
43 2
7
434
828, ,
8, ,
,,
0o
ii i
ii i
iii
xxx xxxxx
xx xxxxx
x xxxx
 
 

 
 

2
1
1
,
,
,
,
i
i
i
2
3
,
,
,
therwise.
i
i
(3)
The set of splines,
 
32 3
,,,
N
x
x
 
 
x forms
a basis for functions defined over
,ab . The numerical
solution to takes the form

,
N
Uxt
,Uxt
i
 
3
3
,
N
Ni
i
Uxtt x


(4)
where i
are unknown, time dependent quantities to be
determined from the boundary and collocation conditions
and
i
are septic B-spline. Each septic B-spline
covers 8 elements thus each element
1
,
ii
x
x is cov-
ered by 8 splines. A typical finite interval
1
,
ii
x
x is
mapped to the interval [0, 1] by a local coordinate trans-
formation defined by i
hxx
, 01
. Therefore
septic B-splines (3) in terms of
over [0, 1] can be
given as
234567
3
23567
2
23456
1
467
23456
1
1721 3535217
120 39250428084427
1191 171531566531510510521
2416 168056014035
1191 1715315665315105105
i
i
i
i
i


7
 


 
 
  
 
7
23567
2
234567
3
7
4
35
120 392504280844221
1 72135352177
.
i
i
i

  

 
 
(5)
Since all splines apart from
 
32 3
,,,
ii i
x
x
 
 
x are zero over the element [0,
1] . For the problem the finite elements are identified
with the interval
1
,
ii
x
x. Using the nodal values
,,
iii
UUU
 and i
U
 are given in terms of the parameter
i
by:



32 112
321123
2
321 123
3
321 123
12011912416 1191120,
756245245 56,
42241580 1524,
210819198,
iiiiiii i
iiiii ii
iiiiiiii
iiiiiii
U
hU
hU
hU
3

 

 
 

 

 
  
 

(6)
and the variation of over the element
U
1
,
ii
x
x is
given by
3
3
N
ii
i
U

We now identify the collocation points with the knots
and use (6) to evaluate i and its space derivatives in
(1). This leads to a set of ordinary differential equations
f the form
U
o
Copyright © 2011 SciRes. AM
T. GEYIKLI ET AL.
Copyright © 2011 SciRes. AM
741

1
32 112332112
321 123
2
12011912416 1191120215624524556
42241580 15240,
iiiiiiiiii iii
h
iiiiiii
Z
h
 

 
 
  

 
3i
(7)
where

2
32 1123
12011912416 1191120.
iiiiiiii
Z
 
 

If time parameters i
and its time derivatives i
in (7) are discretized by the Crank-Nicolson formula and usual
finite difference approximation, respectively:

1
1
1,
2
nn
nn
ii
t

 
We obtain a recurrence relationship between two time levels n and n + 1 relating two unknown parameters 1n
i
, n
i
111111
132231 4516273
73 6251 4312213
nnnnnn
ii ii ii ii ii ii ii
nnnnnnn
iiiiiiii iiiiii
1n
 

 
 


(8)
where
 



12 3
45 6
7
2
1, 1205624, 119124515,
241680, 119124515, 1205624,
10,1,,,
21 42
,.
2
iiii ii
iiiii
ii
EZ MEZMEZM
M
EZ MEZM
EZ MiN
EtM
hh
 
 
 
 
 
 
For the first linearization, we suppose that the quantity
in the non-linear term
U2
x
UU to be locally constant.
This is equivalent to assuming that in (7) all i
are
equal to a local constant. Furthermore, we can write the
nonlinear term
2
x
x
U UUUU (9)
and apply the Rubin and Graves [16] linearization tech-
nique

111
nnnnnn
x
xx
UUU UUUUU


x
(10)
to the
x
UU term in (8) and we can also apply the
Caldwell and Smith [17] linearization technique


11
1
2
nnnn
xx
UUUUU U

1
x
(11)
to the
x
UU term in (8). The system (8) consists of
linear equation in unknowns
2 3
. To obtain a unique solution to
this system we need 6 additional constraints. These are
obtained from the boundary conditions and can be used
to eliminate
1N

7N
1
32
,,
3. Initial State
To start evolution of the vector of parameters n
, 0
can be determined from the boundary conditions and the
initial condition
,0Ux. So we can rewrite approxima-
tion (4) for the initial condition
 
3
3
,00 ,
N
Ni
i
Ux x


i
where parameters 0
i
will be determined. To determine
the parameters
000 00
3223
,,,,
NN
 
 
,0
N
Ux

, we require
the initial numerical approximation to satisfy
the following conditions:
1) it must agree with the exact initial condition
,0Ux at the knots i
x
.
2) the first, second and third derivatives of the ap-
proximate initial condition agree with those of the exact
initial conditions at both ends of the range. These two
conditions can be expressed as:
T
, ,
NN
 
 
32
,,


and 12
,,
NN N
 
 
 
,0,0, 0,
,0,0 0,
,0,0 0,
,0,0 0.
NNi
Nx Nx
Nxx Nxx
Nxxx Nxxx
Ux UxiN
Ua Ub
Ua Ub
Ua Ub




(12)
3

 
T
,
N
from the
set (2) which then becomes a matrix equation for the
unknowns
of the form 1N
1
01
,,

δ
nn
A
B
. The matrices
A
and are septa-dia-
gonal matrices and so are easily
solved by septa-diagonal algorithm.
B

1N 
1N
The above conditions lead to 0
K
b
matrix equa-
tion, which is solved by using a variant of Thomas algo-
rithm.
T. GEYIKLI ET AL.
742
1536 271276824
82731210568.5 10479610063.51
818181 81
960096597195768 96474120 1
818181 81
1120119124161191 1201
96474 195768965979600
1120 81 818181
10063.5 104796210568.582731
181 818181
247682712 1536
K
 

T
0
0122 1
,, , ,,,
NNN
 

δ
and




T
01 1
,0 ,,0 ,,,0 ,,0
NN
bUx UxUxUx
4. Stability Analysis
The stability analysis will be based on the von Neumann
theory in which the growth factor of a typical Fourier
mode
ˆe,
nn ijkh
j

(13)
where is the mode number and the element size,
is determined for a linearisation of the numerical scheme.
Substituting the Fourier mode (13) into the linearised
recurrence relationship (8) shows that the growth factor
for mod is
k
k
h
aib
gaib
where





 
1208 403397 5cos
245cos21cos3
245sin56sin 2sin 3.
iii
aM Mhk
MhkMhk
bEZhkEZhkEZhk

 

The modulus of
is 1 therefore the linearised
scheme is unconditionally stable.
5. Numerical Examples and Results
All computations were executed on a pentium 4PC in the
Fortran code using double precision arithmetic. The
conservation properties of (1) will be examined by cal-
culating the lowest three invariants given as




11
2
2
2
21
4
4
31
d,
d
d,
bNn
j
J
a
bn
Nn
xj
J,
x
j
a
bNn
j
J
a
CUxhU
CUUxhU U
CUxh U


 


which correspond to mass, momentum and energy re-
spectively [5]. The accuracy of the method is measured
by both the error norm

2
220
,
N
exact exact
Nj
j
J
LU UhUU
 
N
and the error norm

max .
exact exact
NjN
j
jU U

To implement the method, three test problems: motion
interaction of two solitary
ellian initial condition will be con-
sidered.
6. Motion of Single Solitary Wave
tion
LU U
of a single solitary wave,
waves and the maxw
For this problem, we consider Equation (1) with the
boundary conditions 0U as x and the ini-
ial condit
0
,0 sec.UxAhkxx
This problem has an exact solution of the form

,sUxtkx
which represents t
0
ecAhxvt
he motion of a single solitary wave
with amplitude
A
, here the wave velocity 22vA and
1k
. For this problem the analytical values of the
invariants are [5]
22 4
12 3
π22 4
, , .
33
A
AkAA
CC C
kk
 
k
The analytical values of invariants are obtained from
(1) 1
0.7853982C23
, 0.1666667, 0.0052083C C

For the numerical simulation of the motion of a single
wave, we have used the parameters 0.1h
.
solitary
,
0
0.05, 1, 30,tx
 0.25A through the inter-
Copyright © 2011 SciRes. AM
T. GEYIKLI ET AL.743
val 0. The computations are done until time
his lenght we find error norms
08x
20 and in tt2
L, L
s. In
the
thi
and
and numerical
norms ob
table that the
ntly small
ost
creasing time as expected. Amplitude is at
0.249999
0t
which is located at , while it is
at
30x0.249922
20t
which is located at . The absolute
difference in amplitudes at times and
30.6
0t
x
20t
is
5
07.7 1
so that there is a little change between ampli-
tudes.
invariants at various
tained usinnt meth
error norm
12 3
ble 1 we compare the values of the invariants and
error
three different approximation and the results of [5,14,
18,19] at different times. We can easily see from s
s 2Lare obtained suf-
ficie and the quantities in the variants remain
constant during the computer run for the t
second linearization techn we can not say the
same for the third linearization technique. For the first
and second linearization, the numerical values of invari-
ants are 123
0.7853966, 0.1667641, 0.0052083CCC
and for the third linearization numerical values of invari-
ants are 123
0.7855405,0.16676 ,0.0052144CCC
at the 20t. Figure 1 shows that the proposed method
perform the motion of propagation of a solitary wave
satisfactorily, which moved to the right at a constant
speed and preserved its amplitude and shape with in-
,,CC C
g the prese
L and
iques but
time
od with
the
firs
Ta
alm
41
7. Interaction of Two Solitary Waves
For this problem, we consider (1) with boundary condi-
tions as , interaction of two positive
solitary waves is studied by using the initial condition
0Ux


2
1
,0 sec.
jj
j
UxAhkx x



where 1.k
We first used the parameters
0.1, 0.025,ht
12
1, 1, 0.5,AA
 12
15,x 30x
through the in-
terval 080x
which is used by Zaki [5]. These pa-
rameters provide solitary waves of magnitudes 1 and 0.5
t
Table 1. In
1
C
variant
Lineerization
s and error n
2
C
orms for single solitary waves.
3
C 3
210L 3
10L
0 0.7853966 0.1666664 0.0052083 0.0000000 0.0000000
5 0.7853966 0.166666
10 0.7853966 0.1666664
First
4 0.0052083 0.0000979 0.0000622
0.0052083 0.0002113 0.0001368
03432 0.0002251
1666664 0.0052083 0.0004969 0.0003309
15
20
20 [5]
20 [14]
20 [18]
20 [19]
0.7853966 0.
Second
0.7853966 0.1666664 0.0052083 0.00
0 0.7853966 0.1666664 0.0052083 0.0000000 0.0000000
5 0.7853966 0.1666664 0.0052083 0.0000972 0.0000627
10 0.7853966 0.1666664 0.0052083 0.0002102 0.00001378
15 0.7853966 0.1666664 0.0052083 0.0003419 0.0002272
20 0.7853966 0.1666664 0.0052083 0.0004957 0.0003331
0 0.7853966 0.1666664 0.0052083 0.0000000 0.0000000
5 0.7854325 0.1666908 0.0052098 0.0237333 0.0228190
10 0.7854685 0.1667152 0.0052114 0.0480311 0.0454089
15 0.7855045 0.1667397 0.0052129 0.0734307 0.0678603
20
Third
0.7855405 0.1667641 0.0052144 0.1004249 0.0900579
0.785397 0.166667 0.005210 0.0034500 0.0020300
0.7849545 0.1664765 0.0051995 0.2498925 0.2905166
0.7853977 0.1664735 0.0052083 0.2692812 0.2569972
0.1958878 0.1744330
Copyright © 2011 SciRes. AM
T. GEYIKLI ET AL.
744
Figure 1. The motion of a single solitary wave.
and peak positions of them are located at and 30.
The analytical invariants are [l4]
15x



112
22
212
44
312
π4.7123889,
83.3333333,
3
41.4166667.
3
Calculation is carried out with t
CAA
CAA
CAA



he time step
and space step over the region
0. The experimen from
0.025t
08x
0.1h
nt was ru 0t
to
to allow the interaction to t ke place.
nteraction
hat at
of thalle
e
55t
shows the i
can be seen t
is on the left
The larger
inc
lapping pro
and waves
a
of two positive solitary
the wave with
wave with sm
Figure 2
waves. It
larger amplitude
r amplitude.
as tim
5t
e second
wave catches up with the smaller one
reases. Interaction started at about time 25t, over-
cesses occurred between times25t and 40
started to resume their original shapes after
time 40t. For the first and second linearization tech-
niques at 55t, the amplitude of larger waves is
1.000149 at the point 44.4x whereas the amplitude
of the smaller one is 0.507317 at the point 34.6x
. It
is found that the absolute difference in amplitude is
3
7.3 10
for the smaller wave and 0.149 3
10
for the
larger wave for this algorithm. For the third linearization
technique at 55t, the amplitude of larger waves is
0.99593 t the point 44.7x whereas the amplitude
of the smaller one is 0.507477 at the point 34.6x
3 a
. It
is found that the abso erence in amplitude is
3
7.410
for the smaller wave and 4 10 for the
larger wave for this algorithm. In Table 2 we compares
values invariants of the two solitary waves with results
from first, second and third linearization. We see from
the Table 2 that for the first and second linearization
s, all 3 invariaconserved by less than
lute diff
3
techniquents are
5
9.9 10
that the conservatio
Table 2. Invaria
during the experiment. Thus we have found
n quantities are satisfactorily constant
nts for interaction of two solitary wave.
12
1, 0.5AA
tLineerization 1
C 2
C 3
C
043.3333294 3.7123733 1.416664
54.7123660 3.3333183 1.4166532
154.7123494 3.3332959 1.4166308
254.7123331 3.3332741 1.4166083
354.7123243 3.3335335 1.4165818
454.7123127 3.3332470 1.4165824
First
4.7122960 3.3332247 1.4165605
04.7123733 3.3333294 1.4166643
55
154.7123602 892 1.4166241
254.71234 3.333203 1.41657
Third
54.7123696 3.3333160 1.4166509
3.3332
49 694
354.7123274 3.3332088 1.4165317
454.7123380 3.3332270 1.4165618
55
Second
4.7123291 3.3332012 1.4165363
04.7123733 3.3333294 1.4166643
54.7318586 3.3857414 1.4689535
154.7738238 3.5004328 1.5869329
254.8195659 3.6284390 1.7245060
354.8528439 3.7197153 1.8236340
454.9055278 3.8734565 2.0013048
55 4.9694524 4.0643034 2.2347812
Copyright © 2011 SciRes. AM
T. GEYIKLI ET AL.745
Figure 2. Interaction of two solitary waves at different times.
with the proposed algorithm.
We have also studied the interaction of two solitary
waves with the following parameters:
allow the interaction to take place. Figure 3 shows the
development of the solitary wave interaction.
As is seen from the Figure 3, at a wave with
the negative amplitude is on the left of another wave
with the positive amplitude. The larger wave with the
negative amplitude catches up with the smaller one with
1
, 115x
,
ime step
x
55
230x
0.02t
150. The
0t
, , together with t
t from
12A
5 and space ste
xper ime n
21A,
p h
was run
= 0.1 in the range 0
0 to
e tt
to
Copyright © 2011 SciRes. AM
T. GEYIKLI ET AL.
746
Figure 3. Interaction of two solitary waves at different times.
the positive amplitude as the time increases. For the first
linearization technique at, the amplitude of the
smaller wave is the point
55t
5 at 0.97469 52.5x
,
whereas the amr one is
at the point e seco
technique athe sm
is at the point , whereas the ampli-
arger on at the point
plitude of t
122.8
, the am
he large
. For th
plitude of t
nd linea
1.989036
rization
aller wave
x
55t
0.972778
tude of the l
122.8
52.5x
e is 1.986701
x
. It is found th difference in am-at the absolute
plitudes is 1
0.25310
1
13310
, for the smaller
1 0.272 10
wave and 0.
, 1
10910
0. for the larger wave,
respectively.
Copyright © 2011 SciRes. AM
T. GEYIKLI ET AL.747
The analytical invariants by using Equation (1) can be
found as ,
he invari-
wellian initial condition breaks up into more solitary
waves which were drawn in Figures 4(c)-(f) at time
12
3.1415927, 13.3333333CC 
66667 . Table 3 lists the values of t
ants of the two solitary waves with amplitudes
322.66C
12A
,
d for the
the computer
21A.
inva
It can be seen that the values obtaine
riants are satisfactorily constant during
run.
8. The Maxwellian Initial Condition
As a last study, we consider here is the numerical solu-
tion of the Equation (1) with the Maxwellian initial con-
dition

2
,0 e
x
Ux
(14)
with the boundary conditions


20,20,20,20, 0.
xx
UtUtUtUt
As it is known, Maxwellian initial condition (14) the
behavior of the solution, deends on the values of p
.
So we have considered various values for
. For
ases
the
first linearization technique the computations are carried
out for the c1,0.5,0.1,0.05,0.02
and 0.005
which are used in the earlier papers [5,14]. When
1, 0.5
is used as shown Figures 4(a) and (b) at time
12t the Maxwellian initial condition does not cause
development into a clean solitary wave. However with
smaller valu0.1
es of ,0.05,0.02
and 0.005 Max-
Table 3. Invariants for interaction of two solitary wave.
12
2, 1AA
t Lineerization 1
C 2
C 3
C
0 3.141573913.3332981 22.6665313
5 3.137332413.3219118 22.6210653
15 3.122709713.2806152 22.4483653
.3589494
3.089663813.1922565 22.0973499
0 3.1415739 13
13.3196543 22.6120524
15 3.132507613.2805526 22.463773
3.113721013.1745553 8880
25 3.114333713.2581754 22
First
35 3.106033413.2359744 22.2706624
45 3.097810613.2140028 22.1834701
55
.3332981 22.6665313
5 3.1391704
3
25 3.127738813.2535359 22.3561477
35 3.123013813.2268810 22.2501653
45 3.118341613.2005566 22.1457584
55
Second
22.042
12t
. The numerical conserved quantities with
1,0.5,0.1,0.05,0.02
and are given in Tabl e 4.
It is observed that the obtained values of the invariants
remain almost constant during the computer run.
Table 4. Invariants for Maxwellian initial condition, differ-
ent µ.
0.005
t
1
C 2
C 3
C
0 1.772454 2.506607 0.886227
3 1.772972 2.506836 0.886561
6 1.775116 2.512628 0.890240
2.514942 0.891795
12
1
1.776698 2.515136 0.891967
3
9 1.772451 1.879967 0.886223
0 1.772454 1.46 0.886226
3 1.7724 1.37850.88614
6
9
0.1
0.05
0
9 1.721.1
12 1.710490 1.162646 0.737751
9 1.776365
0 1.772454 1.879971 0.886227
1.772452 1.879970 0.886225
6 1.772451 1.879968 0.886224
0.5
12 1.772450 1.879966 0.886222
3786
2091 7
1.772368 1.378507 0.886054
1.772316 1.378424 0.885933
12 1.772264 1.378340 0.885813
0 1.772454 1.315980 0.886227
3 1.772266 1.315654 0.885789
6 1.771976 1.315150 0.884947
9 1.771685 1.314648 0.884107
12 1.771396 1.314147 0.883270
1.772454 1.278380 0.886227
3 1.770834 1.275265 0.880933
6 1.768546 1.271572 0.874402
9 1.766186 1.266707 0.864400
12
0.02
1.763931 1.262351 0.855782
0 1.772454 1.259581 0.886227
3 1.757684 1.254254 0.928815
6 1.738212 1.227138 0.880527
2397 71836 0.714373
0.005
Copyright © 2011 SciRes. AM
T. GEYIKLI ET AL.
Copyright © 2011 SciRes. AM
748
(a) b)
(
(c) (d)
(e) (f)
Figure 4. Maxwellian initial condition, state at time t = 12 (a) µ = 1, (b) µ = 0.5, (c) µ = 0.1, (d) µ = 0.05, (e) µ = 0.02, (f) µ =
0.005.
T. GEYIKLI ET AL.749
9. Conclusions
In this study, a numerical solution of the MEW equation
based on the septic B-spline finite element has been pre-
sented with three different linearization techniques.
Three test problems are worked out to examine the per-
formance of the algorithms. The performance and accu-
racy of the method were demonstrated by calculating the
error norms and on the motion of a single
solitary wave. For the first and second linearization tech-
niques, the error norms are sufficiently small and the
invariants are satisfactorily constant in all computer run.
The obtained results from the first and the second lin-
earization techniques are almost the same and the com-
puted results show that the present method is a remarka-
bly successful numerical
quation and can also be efficiently applied to other
types of non-linear problem.
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