Ghulam Ishaq Khan Institute of Engineering Sciences and Technology NUMERICAL SOLUTION OF BOUNDARY-VALUE AND INITIAL-BOUNDARY-VALUE PROBLEMS USING SPLINE FUNCTIONS By Fazal-i-Haq Supervised by Professor Dr. Syed Ikram Abbas Tirmizi Co-Supervisor Dr. Siraj-ul-Islam This thesis is submitted in partial fulfillment of The requirement of Degree of Doctor of Philosophy (PhD) in Engineering Sciences May 2009 Faculty of Engineering Sciences GIK Institute of Engineering Sciences and Technology, Topi, Pakistan.
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Ghulam Ishaq Khan Institute of
Engineering Sciences and Technology
NUMERICAL SOLUTION OF
BOUNDARY-VALUE
AND
INITIAL-BOUNDARY-VALUE PROBLEMS
USING SPLINE FUNCTIONS
By
Fazal-i-Haq
Supervised by
Professor Dr. Syed Ikram Abbas Tirmizi
Co-Supervisor
Dr. Siraj-ul-Islam
This thesis is submitted in partial fulfillment of
The requirement of Degree of Doctor of Philosophy (PhD)
in Engineering Sciences
May 2009
Faculty of Engineering Sciences
GIK Institute of Engineering Sciences and Technology,
Topi, Pakistan.
ii
Dedicated to
My Late Mother,
Wife and Children
iii
iv
Acknowledgements
I would like to bow my head before Allah Almighty, the Most Gracious and the Most
Merciful, whose benediction bestowed upon me talented teachers, provided me
sufficient opportunity and enabled me to undertake and execute this research work.
Throughout the completion of this work, I have been supported and guided by
several people. I would like to take this opportunity to express my gratitude to all those
people. My deepest and sincere gratitude and appreciation goes to my supervisor
Professor Dr. S. I. A. Tirmizi for his guidance at each stage of this work. His patience,
encouragement and support have been very valuable in the completion of this thesis.
His advices were always stimulating and helpful when I was facing difficulties in my
research. His mission of producing high-quality work will always help me grow and
expand my thinking. Secondly, I wish to express a sincere acknowledgement to my co-
supervisor Dr. Siraj-ul-Islam for all the guidance, kind comments and constructive
criticism which have made the long hours of labor worthwhile. From the very
beginning, his dedication to the work, his concern for my well being and his confidence
in me and my work gave me the energy and inspiration to work through the difficulties.
I am also thankful to all my teachers in the Faculty of Engineering Sciences for their
encouragement during this research work.
I wish to record my deepest obligations to friends, parents, brothers and sisters for
their unfailing support and many sacrifices at every stage during these years. I would
like to thank my late mother, wife and children for their love, care, understanding,
encouragement and enduring support. I am also grateful to entire staff and research
students of faculty of Engineering Sciences, GIK Institute of Engineering Sciences and
Technology, Topi, for their help and support in one way or another during the course of
my research.
I would also like to gratefully acknowledge NWFP Agricultural University
Peshawar for study leave and to Higher Education Commission (HEC) Islamabad for
financial assistance during my Ph.D.
Fazal-i-Haq
May, 2009
v
Published and submitted papers
Certain aspects of this dissertation are based on the following published/submitted
papers:
Published Journal papers
[1] S. Islam, S. I. A. Tirmizi, F. Haq (2007): Quartic non-polynomial splines
approach to the solution of a system of second-order boundary-value problems.
Int J. High Performance Computing Applications 21 (1), 42-49.
[2] S. Islam, S. I. A. Tirmizi, F. Haq, M. A. Khan (2008): Non-Polynomial splines
approach to the solution of sixth-order boundary-value problems. Appl. Math.
Comput. 195, 270-284.
[3] S. Islam, S. I. A. Tirmizi, F. Haq, S. K. Taseer (2008): Family of numerical
methods based on non-polynomial splines for solution of contact problems.
Comm. Nonlinear Science and Numer. Simul. 13, 1448-1460.
[4] S. I. A. Tirmizi, F. Haq, S. Islam (2008): Non-polynomial spline solution of
We consider a model for nonlinear waves, the modified equal width (MEW) equation
2 0 t x xxtu u u u (4.1)
subject to the following physical boundary conditions
1 2, , , , u a t u b t (4.2a)
along with collocation boundary conditions necessary for unique quartic B-spline
solution
, , 0, , , 0, x x xx xxu a t u b t u a t u b t (4.2b)
and initial condition
,0 , .u x f x a x b (4.3)
The parameter is a positive constant and is an arbitrary constant, ( )f x is a
localized disturbance inside the interval [ , ]a b and 0 asU x . Here the
subscripts t and x denote differentiation with respect to t and x respectively. The
MEW equation was introduced by (Morrison et al. [91]) as a model for nonlinear
dispersive waves and is related to the modified regularized long wave (MRLW)
equation (Abdulloev et al. [3]) and the modified Korteweg-de Vries (MKdV) equation
(Gardner et al. [42]). Many authors have investigated numerical solution of the problem
(4.1)-(4.3). These include finite difference method given by (Esen and Kutluay [39]),
He's variational iteration method (Junfeng [68]), tanh and sine-cosine methods
(Wazwaz [159]), mesh free methods (Haq et al. [46]) and various forms of finite
element methods including collocation and Galerkin methods, (see (Esen [38], Evans
Chapter 4 MEW equation
78
and Raslan [40], Raslan [114], Saka [116], Zaki [162]) and the references therein).
In Refs. (Evans and Raslan [40], Raslan [114], Saka [116], Zaki [162]) the MEW
equation is solved numerically by collocation methods based on quadratic, cubic and
quintic B-splines. The present method solves Eqs. (4.1)-(4.3) by using quartic B-spline
collocation method. In Section 4.2, a new numerical method is developed. The stability
analysis of the method is established in section 4.3 and test problems are reported in
section 4.4 to validate performance of the method. Some conclusions are drawn in
section 4.5.
4.2 Quartic B-spline solution
In order to develop the numerical method for approximating solution of boundary value
problems like the one given in Eqs. (4.1)-(4.3), the interval [ , ]a b is partitioned into
1N uniformly spaced points mx such that 0 1 1... N Na x x x x b and
b a
hN
. The quartic B-splines , 2, 3,... 1mB x m N at the knots mx are defined
as (See (Prenter [102])):
4
1 2 2 1
4
2 1 1 1
4
3 2 14 4 4
3 2 1 2
4
3 2 3
, , ,
5 , , ,
10 , , ,1
5 , , ,
, , ,
0 otherwise,
m m m
m m m
m m mm
m m m m
m m m
d x x x x
d d x x x x
d d x x x xB x
h x x x x x x
x x x x
(4.4)
and the set 2 1 1, ,..., NB B B of quartic B-splines forms a basis over the interval ,a b .
The numerical solution ,U x t to ,u x t is given as:
1
2
, ,
N
m mm
U x t t B x (4.5)
where m t are time dependent parameters to be determined at each time level. The
nodal values , and m m mU U U at the knots mx are derived from Eqs. (4.4)-(4.5) in the
following form
Chapter 4 MEW equation
79
2 1 1
2 1 1
2 1 12
11 11 ,
43 3 ,
12.
m m m m m
m m m m m
m m m m m
U
Uh
Uh
(4.6)
where dashes represent differentiation with respect to space variable. Eq. (4.1) can be
rewritten as
2 0. xx xtu u u u (4.7)
We discretize the time derivative of Eq. (4.7) by a first order accurate forward
difference formula and apply the -weighted, (0 1) scheme to the space
derivative at two adjacent time levels to obtain the equation
1 1
12 21 0,
n n n nn nxx xx
x x
U U U UU U U U
t
(4.8)
where t is time step and the superscripts n and n+1 are successive time levels. In this
work we take 1/ 2 . Hence Eq. (4.8) takes the form
11 1 2 2
0,2
n nn n n nxx xx x xU U U U U U U U
t
(4.9)
The nonlinear term in Eq. (4.9) is approximated using the Taylor series:
2 1 1 2 1 1 2( ) ( ) 2 2( ) . n n n n n n n n nx x x xU U U U U U U U U (4.10)
At the nth time step, we denote , and m m mU U U at the knots mx by the following
expressions:
1 2 1 1
2 2 1 1
3 2 1 12
11 11 ,
43 3 ,
12.
n n n nm m m m m
n n n nm m m m m
n n n nm m m m m
L
Lh
Lh
(4.11)
Using the knots , 0,1,.., ,mx m N as collocation points, the following recurrence
relation at point mx is obtained using Eq. (4.9)-(4.11):
21 1 1 1 21 2 2 1 3 4 1 1 1 2 32 2n n n n
m m m m m m m m m m m mha a a a L tL L L (4.12)
where
Chapter 4 MEW equation
80
2 21 0 1 2 0 1
2 23 0 1 4 0 1
20 1 2
4 24 , 11 12 24 ,
11 12 24 , 4 24 ,
L 2 1 and 0,1,..., .
m m m m m m
m m m m m m
m m m
a L thL a L thL
a L thL a L thL
h L L m N
The Eq. (4.12) relates parameters at adjacent time levels and gives 1N equations in
4N unknowns , 2, 1,..., 1.i i N In order to get a unique solution; we eliminate
the parameters 2 1 1, , N . Using Eq. (4.6) and the boundary conditions (4.2), the
values of the parameters take the form
12 0 1
11 0 1
1 2 1 2
333 7,
4 4 87 1
,4 4 8
11 11 .N N N N
(4.13)
Elimination of the above parameters from Eq. (4.12) yields a 4-banded linear system of
(N+1) equations in (N+1) unknown parameters , 0,1,...,i i N . The linear system can
be solved by a four-diagonal solver successively for , 1, 2,..., ;ni n once we calculate
the initial parameters 0m . Finally the approximate solution ,U x t will be obtained
from Eq. (4.6).
Using the initial and boundary conditions, the values of the initial parameters 0m at
the initial time are determined with the help of the following expressions:
0 0 0 0
2 1 1
0 0 0 00 2 1 1 2
0 0 0 00 2 1 0 12
0 0 0 02 1 1 2
0 0 0 02 1 12
,0 ,0 11 11 , 0,1,..., ,
4,0 3 3 0,
12,0 0,
4,0 3 3 0,
12,0 0.
m m m m m m m
N N N N N
N N N N N
U x u x f x m N
U xh
U xh
U xh
U xh
(4.14)
Eq. (4.14) consists of a linear 4 4N N system which can also be solved by a
four-diagonal solver.
Chapter 4 MEW equation
81
4.3 Stability Analysis
In this section we apply the Von-Neumann stability method (Mitchell and Griffiths
[89]) for the stability of scheme developed in the previous section. Since this method is
applicable to linear schemes, the nonlinear term 2xU U is linearized by taking U as a
constant value k . The linearized form of proposed scheme takes the form
1 1 1 11 2 2 1 3 4 1 4 2 3 1 2 1 1,
n n n n n n n nm m m m m m m mpp p p p p p p (4.15)
where
2 21 2
2 22 24 4 , 22 24 12 ,p h h t k p h h t k
2 23 4
2 222 24 12 , 2 24 4 , 0,1,..., .p h h t k p h h t k m N
Substitution of exp , 1,n nm i mh i into Eq. (4.15) leads to
1 2 3 4
4 3 2 1
exp 2 exp exp
exp 2 exp exp .
p i h p i h p p i h
p i h p i h p p i h
(4.16)
Simplifying Eq. (4.16), we get
2 2 2
2 2
2 2
2 2
2 2 2
2 2
2 2
2
2
22
22
where
A= 12 24 24 8 cos
2 4 24 cos 2 ,
20 16 48 sin
2 4 24 sin 2 ,
= 12 24 24 8 cos
2 4 24 cos 2 ,
20 16
h
C h
A iB
C iD
dth k h dth k h
h dth k h
B h dth k h
h dth k h
dth k h dth k h
h dth k h
D h dth k
2 2
48 sin
2 4 24 sin 2 .
h
h dth k h
(4.17)
After simplification, we obtain same expressions for 2 2A B and 2 2C D in the
following form:
Chapter 4 MEW equation
82
2 2 2 2
4 2 2 2 4 2 2
4 2 2 2 4 2
4 2 2 2 2 4
4 2 2 2 4 2 2
122 40 240 2888 cos
143 12 2 144
8 22 288 24 10 cos 2
8 4 24 144 cos 3 ,
A B C D
h dt h k hh
h h dt k
h h dt k h
h dt h k h h
(4.18)
so that 2
1 and the linearized numerical scheme for the MEW equation is
unconditionally stable.
4.4 Test problems and discussion
In this section the numerical method outlined in the previous section is tested for a
single solitary wave and interactions of two solitary waves. Moreover, the Maxwellian
initial condition is also considered. The accuracy of the scheme is measured in terms of
the following discrete forms of 2L and L error norms:
2
21
, ,N
i i i ii
i
L Max u U L h u U
(4.19)
where u and U are exact and approximate solutions respectively. The exact solitary
wave solution of MEW equation is given in (Esen and Kutluay [39]):
0
2
where
( , ) sec ,
1 , ,
6
u x t A h k x x ct
Ac k
(4.20)
and A, c represent the amplitude and velocity of a single solitary wave initially centered
at 0x . The initial condition for the above problem is given by
0( ,0) sec ,u x A h k x x (4.21)
and the boundary conditions are taken from Eq. (4.2a) with 1 2 0 . We examine
the conservation properties of the MEW equation related to mass, momentum and
energy by calculating the following three invariants (See (Zaki [162])):
2 2 41 2 3, , .
b b b
x
a a a
C udx C u u dx C u dx (4.22)
Chapter 4 MEW equation
83
Integrals in Eq. (4.22) can be approximated by the trapezoidal rule. 4.4.1 A single solitary wave Problem 4.1 To compare our results with (Esen [38], Esen and Kutluay [39], Evans
and Raslan [40], Raslan [114]), we choose the following parameters:
00, 80, 3, 0.25, 0.1, 0.2,0.05, 1, 30.a b A h t x
In order to find error norms and the invariants 1 2 3, ,C C C at different times, the
computations are carried out for times upto 20.t We have compared present method
with earlier published papers (Esen [38], Esen and Kutluay [39], Evans and Raslan
[40], Raslan [114]) at t=20 and the results are reported in Table 4.1. At t=20 the error
norms of the present method are -4 -40.010451 10 , 0.009269 10 ,L -4
2 0.015789 10 ,L
-40.007867 10 for the time steps 0.2 and 0.05 respectively. It is clear from Table 4.1
that the errors in 1 2 3, ,C C C approach zero during the simulation, showing excellent
conservation properties of the new method. Hence the performance of the new method
is better than the above mentioned methods. Fig. 4.1 shows the graphs of the single
solitary wave solutions at 0 and 10t . Initially the centre of solitary wave of
amplitude 0.25 is located at 30x . At time 20t its magnitude is 0.249922 centered
at 30.6x . The absolute difference in amplitudes over the time interval 0, 20 is
observed to be 57.8 10 while it is 52 10 in the case of velocities. It can be
concluded that the solitary wave moves to the right with almost constant magnitude and
velocity. The error graph at time 20t is reported in Fig. 4.2. It can be observed from
the graph that the maximum errors occur around the central position of the solitary
wave.
The same problem is also considered for different values of the amplitude at time
step of 0.01. In Table 4.2 the error norms and invariants are summarized for
0.25,0.5,0.75,1.0A . It is observed that the errors are smaller and the invariants
remain constant during the simulation. The new method is compared with (Esen and
Kutluay [39]) and the comparison of error norms declares superiority of present
scheme. Fig. 3 shows the graphs of the solutions for 0.25,0.5,0.75,1.0A at 20t .
Chapter 4 MEW equation
84
Problem 4.2 In order to compare our method with earlier work (Esen [38], Saka [116],
Zaki [162]) we choose the parameters 0, 70, 3, 0.25,0.5,1.0, 0.1, 0.05,a b A h t
01, 30x . The simulation is performed upto time t=20. Error norms and the
invariants 1 2 3, ,C C C are recorded for different values of t and tabulated in Table 4.3. It
is observed that the accuracy of the scheme in terms of error norms increases for
decreasing values of A. For 1,0.25A and t=20 the error norms of the present method
are found as -3 -71.095929 10 ,9.27 10L and -3 -72 1.747622 10 ,7.878 10L . The
invariant quantities 1 2 3, ,C C C are almost constant during the simulation. In Table 4.3
we have also compared our results with lumped Galerkin method using quadratic B-
spline functions (Esen [38]), collocation and petro-Galerkin methods using quintic B-
splines (Saka [116], Zaki [162]) at t=20. In this problem it is observed that the accuracy
of different schemes also depends on amplitude A. Performance of the present method
is better than the methods given in (Esen [38], Zaki [162]) when A=0.25 and
comparable with (Zaki [162]) for 0.5A . It is observed that error norms are less in the
methods given in (Saka [116], Zaki [162]) for 1.A
4.4.2 Interaction of two solitary waves
Problem 4.3 To study the interaction of two solitary waves we use the following initial
condition:
2
1
( ,0) sec ,i iu x A h k x x (4.23)
where
6 1, .i
i
cA k
Parameters 1 2 1 23, 1, 0.5, 15, 30, 1, 0.1, 0.2, 0 80A A x x h t x are chosen
for the sake of comparison with the results of (Esen and Kutluay [39], Evans and
Raslan [40], Saka [116]). These parameters give two solitary waves having amplitudes
of ratio 2:1 and their peak positions are located at 15 and 30.x The analytical values
of the invariants 1 2 3, ,C C C for the above parameters are given in (Evans and Raslan
[40], Saka [116]) as:
Chapter 4 MEW equation
85
2 21 1 2 2 1 2
4 43 1 2
84.712389, 3.333333,
34
1.416667.3
C A A C A A
C A A
(4.24)
The calculations are performed from 0 to 80.t t Values of the invariant quantities
1 2 3, ,C C C are tabulated in Table 4.4 for the present method and are compared with
Refs. (Esen and Kutluay [39], Evans and Raslan [40]). It can be seen from the table that
the invariants remain satisfactorily constant throughout the simulation. The upper
bounds for absolute error in the invariants 1 2 3, ,C C C from t=0 to t=55 are less than
7 3 31.0 10 ,2.6 10 and 2.6 10 respectively. In Refs. (Esen and Kutluay [39], Evans
and Raslan [40]), the same errors are less than 6 3 32.0 10 ,3.1 10 ,2.6 10 and
3 3 21.8 10 ,8.7 10 ,1.8 10 . Fig. 4.4 shows the state of interaction and then separation
of solitary waves at times 30,35,40t and 55,80t in sequel. Initially the larger
wave of amplitude 1 is centered at 15x and the smaller one of amplitude 0.5 at
30x . Since the velocity of larger wave is 0.5 and that of the smaller 0.125, the larger
wave moves faster than the smaller and hence collides with later. At 80t the
amplitude of the larger wave is 0.9993 centered at 56.8x and that of smaller 0.4988
with peak position located at 37.7.x Hence during this interaction the amplitude is
almost unchanged. The absolute difference in amplitude for larger wave is 47.0 10
and that of smaller 31.2 10 , consequently the velocities of the waves are almost
maintained after the interaction. Thus the waves interact and then emerge from the
collision by preserving their shapes and velocities. We have solved the same problem
with 0.025t and the invariants are reported in Table 4.5 along with the invariants
reported in Ref. (Saka [116]). It is evident from the comparison of Tables 4.4-4.5 that
the conservation properties of the present method are excellent when time step is
reduced.
4.4.3 The Maxwellian initial condition
Problem 4.4 The birth of solitary waves is considered using the Maxwellian initial
condition
2,0 exp .u x x (4.25)
Chapter 4 MEW equation
86
The parameters 3, 1,0.5,0.1,0.05,0.02,0.005, 0.1, 0.01, 20 20h t x used in
Refs. (Saka [116], Zaki [162]) are chosen. In the case of Maxwellian condition the
behavior of the solution depends on the values of . The Maxwellian does not break up
into solutions for c where c is some critical number, and exhibits rapidly
oscillating wave packets. When ,c a mixed type of solution is obtained consisting
of a leading soliton with an oscillating tail (See (Zaki [162])). The Maxwellian breaks
up into a number of solitons according to the value of when .c Simulations
are performed upto time 12.t For 1,0.5 the Maxwellian shows an oscillatory
behavior and no clean waves are obtained as shown in Fig. 4.5.
For 0.1,0.05,0.02,0.005 the number of observed solitary waves is 1, 2, 3 and 7
respectively as shown in Fig. 4.5. The graphs are in good agreement with earlier work
(Saka [116], Zaki [162]). It is also clear from Fig. 4.5 that the peaks of solitary waves
lie on the straight line. For various values of the conservative quantities are tabulated
in Table 4.6 which remain almost constant during the simulation.
Chapter 4 MEW equation
87
Table 4.1
Invariants and error norms for single solitary wave, problem 4.1
t Time L 410 2L 410 1C 2C 3C
0.2 0 0.0 0.0 0.785398 0.166667 0.005208
5 0.002706 0.004075 0.785398 0.166667 0.005208
10 0.005377 0.008094 0.785398 0.166667 0.005208
15 0.007944 0.012009 0.785398 0.166667 0.005208
20 0.010451 0.015789 0.785398 0.166667 0.005208
20 (Esen and Kutluay [39]) 2.576377 2.701647 0.785398 0.166474 0.005208
20 (Evans and Raslan [40]) 1.569539 2.021476 0.785286 0.166582 0.005206
The generalized regularized long wave (GRLW) equation has the form
0,pt x x xxtU U U U U (5.1)
where parameters ,p and are positive constants. This equation has a major role in
the propagation of nonlinear dispersive waves and many authors have investigated its
numerical solution. These include finite difference method (Zhang [163]), Adomian
decomposition method (ADM) (Kaya [71], Kaya and El-Sayed [72]), mesh free
methods (Ali [12]) and Quasi linearization method (Ramos [107]). Exact solution was
obtained by using He’s variational iteration method (Soliman [135]) and Cosine-
function algorithm (Ali et al. [13]). A special case of Eq. (5.1) for 1p is the
regularized long wave (RLW) equation which is used to model a large number of
problems in various areas of science. The equation was originally introduced to explain
behavior of the undular bore development (Pergrine [101]). The analytical solution of
the RLW equation for some initial and boundary conditions are given in (Benjamin et
al. [20], Bona and Bryant [23]). It has been solved numerically by finite difference
methods (Eilbeck and McGuire [37], Pergrine [101]), Meshfree methods (Islam et al.
[53]), Fourier pseudospectral methods (Gou and Cao [45]) and various forms of finite
element methods including collocation, Galerkin and least square methods, (see (Dag
et al. [33], Dag et al. [34], Saka et al. [119]) for the details). We consider another
special case of Eq. (5.1) for 2p , the modified regularized long wave (MRLW)
equation given by
2 0t x x xxtu u u u u (5.2)
subject to the boundary conditions:
Chapter 5 MRLW equation
97
0 as .u x
We will consider the following periodic boundary conditions
, , 0u a t u b t (5.3)
and the collocation boundary conditions in order to get a unique B-spline solution,
given by
, , 0, , , 0.x x xx xxu a t u b t u a t u b t (5.4)
The initial condition is taken as
,0 , ,u x f x a x b (5.5)
where ( )f x is a localized disturbance inside the interval [ , ]a b . In Eq. (5.2) the
subscripts t and x denote differentiation with respect to t and x respectively. Various
methods have been used for the numerical solution of MRLW equation; for instance
cubic B-spline finite element method due to (Gardner et al. [43]), finite difference
method by (Khalifa et al. [74]), Adomian decomposition method in (Khalifa et al. [76])
and collocation method by (Khalifa et al. [75]).
We solve Eqs. (5.2)-(5.5) by quartic and quintic B-Spline collocation methods. The
outline of this chapter is as follows. In Section 5.2, the numerical methods are
presented. In section 5.3, stability is established and numerical results are reported in
section 5.4 to validate performance of the new methods. Conclusion is given in section
5.5.
5.2 The B-spline collocation methods
Let the partition of the space interval ,a b into uniformly spaced points mx be such
that 0 1 1... and N N
b aa x x x x b h
N
.
5.2.1 Quartic B-spline collocation method
We use the quartic B-splines given in Eq. (4.4). The approximate solution ,U x t of
MRLW equation to the exact solution ,u x t is given as:
1
2
, ,N
m mm
U x t t B x
(5.6)
Chapter 5 MRLW equation
98
where m t are time dependent parameters which will be determined for each time
level. Eq. (5.2) can be rewritten as
2 0.xx x xtu u u u u (5.7)
Discretizing the time derivative of Eq. (5.7) by a forward difference formula and
applying the -weighted scheme to the space derivative at two adjacent time levels, we
obtain the equation
1 111 2
21 0.
n n n nnnxx xx
x x
nn
x x
U U U UU U U
t
U U U
(5.8)
Substituting 1/ 2 in Eq. (5.8), we get
11 1 2 21
0.2 2
n nn n n n n n
xx xx x xx xU U U U U U U UU U
t
(5.9)
The nonlinear term in Eq. (5.9) is approximated by the Eq. (4.10). Using the knots
, 0,1,.., ,mx m N as collocation points, the following recurrence relation at point mx is
obtained using above equation and Eqs. (4.10)-(4.11):
2 2
1 2 1 3
1 1 1 1
1 2 2 1 3 4 1 2 1 2 ,m m m m
n n n n
m m m m m m m m h L tL L La a a a (5.10)
where
2 21 0 1 2 0 1
2 23 0 1 4 0 1
20 1 2
4 1 24 , 11 12 1 24 ,
11 12 1 24 , 4 1 24 ,
L 2 1 and 0,1,..., .
m m m m m m
m m m m m m
m m m
a L h t L a L h t L
a L h t L a L h t L
h L L m N
The Eq. (5.10) relates parameters at adjacent time levels and gives 1N equations in
4N unknowns , 2, 1,..., 1.i i N To assure a unique solution; we eliminate the
parameters 2 1 1, , N from Eq. (5.10) by making use of Eq. (4.13) with 1 2 0 .
After the elimination, Eq. (5.10) will give a 4-banded linear system of (N+1) equations
in (N+1) unknown parameters , 0,1,...,i i N . The linear system can be solved by a
four-diagonal solver successively for , 1, 2,...,ni n . Finally the approximate solution
,U x t will be obtained from Eq. (4.6). Before the commencement of the solution
process, we can find initial parameters 0m with the help of Eq. (4.14).
Chapter 5 MRLW equation
99
5.2.2 Quintic B-spline collocation method
Quintic B-spline functions , 2, 3,... 2mB x m N are defined at the knots mx by
(See (Prenter [102])):
5
1 3 3 2
5
2 1 2 2 1
5
3 2 1 1
5
4 3 15
5
5 4 1 1 2
5
6 5 2 2 3
, ,
6 , ,
15 , ,1
20 , ,
15 , ,
6 , ,
0 otherwise.
m m m
m m m
m m m
m m m m
m m m
m m m
e x x x x
e e x x x x
e e x x x x
B x e e x x x xh
e e x x x x
e e x x x x
(5.11)
The set of quintic B-splines 2 1 1 2, ,..., ,N NB B B B forms a basis over the interval
,a b . We take the approximate solution as
2
2
, ,N
m mm
U x t t B x
(5.12)
where m t are time dependent unknown real coefficients and mB x are five-degree
B-spline functions . Using Eqs. (5.11)-(5.12), the nodal values , and m m mU U U at the
knots mx can be written as:
2 1 1 2
2 1 1 2
2 1 1 22
26 66 26 ,
510 10 ,
202 6 2 .
m m m m m m
m m m m m
m m m m m m
U
Uh
Uh
(5.13)
At the nth time step, mU and its derivatives at the knots mx are denoted by the
following expressions
1 2 1 1 2
2 2 1 1 2
3 2 1 1 22
26 66 26 ,
510 10 ,
202 6 2 .
n n n n nm m m m m m
n n n nm m m m m
n n n n nm m m m m m
H
Hh
Hh
(5.14)
Using the knots , 0,1,.., ,mx m N as collocation points, we obtain the following
Chapter 5 MRLW equation
100
recurrence relation at point mx form Eq. (5.9) by making use of Eqs. (4.10) and (5.14):
2
1 2 3
1 1 1 1 1 2
1 2 2 1 3 4 1 5 2 12 1 2 ,m m m
n n n n n
m m m m m m m m m m mh H tH H Ha a a a a (5.15)
where
1 1 2 2 1 2
3 1 4 1 2
25 1 2 1 1 2
22 1
5 40 , 26 50 80 ,
66 240 , 26 50 80 ,
5 40 , 2 2 ,
1 and 0,1,..., .
m m m m m m
m m m m m
m m m m m m
m m
a K K a K K
a K a K K
a K K K h H H
K h t H m N
The Eq. (5.15) relates parameters at adjacent time levels and gives 1N equations in
5N unknowns , 2, 1,..., 2.i i N In order to have a closed form system; we
need to eliminate the parameters 2 1 1 2, , ,N N from the Eq. (5.15). Using Eq.
(5.13) and the boundary conditions, we arrive at the following values of the parameters:
2 0 1 2 1 0 1 2
1 2 1 2 2 1
165 65 9 33 9 1, ,
4 2 4 8 4 81 9 33 9 65 165
, .8 4 8 4 2 4N N N N N N N N
(5.16)
Elimination of the above parameters from Eq. (5.15) yields five-diagonal linear system
of (N+1) equations in (N+1) unknown parameters , 0,1,..., .i i N The linear system
can be solved by the penta-diagonal solver successively for , 1, 2,..., ;ni n once we
calculate the initial parameters 0m .Consequently the approximate solution ,U x t will
be obtained from Eq. (5.13).
The values of the initial parameters 0m at the initial time are determined with the
help of initial conditions and derivatives at boundaries in the following manner:
0 0 0 0 0
2 1 1 2
0 0 0 00 2 1 1 2
0 0 0 0 00 2 1 0 1 22
0 0 0 02 1 1 2
0 02 12
, 0 , 0 26 66 26 , 0,1,..., ,
5,0 10 10 0,
20,0 2 6 2 0,
5,0 10 10 0,
20,0 2
m m m m m m m m
N N N N N
N N N
U x u x f x m N
U xh
U xh
U xh
U xh
0 0 01 26 2 0.N N N
(5.17)
Eq. (5.17) consists of a five-diagonal linear system of 5N equations in
5N unknowns 0 , 2, 1,..., 2,m m N which can also be solved by penta-diagonal
Chapter 5 MRLW equation
101
algorithm.
5.3 Stability of the proposed scheme
Von-Neumann stability method (Mitchell and Griffiths [89]) is used for the stability of
scheme developed in the previous section. Being applicable to only linear schemes, the
nonlinear term 2xU U is linearized by taking U as a locally constant value k .
5.3.1 Stability of scheme based on quartic B-spline collocation method
The linearized form of proposed scheme using quartic B-spline collocation method
is given as
1 1 1 11 2 2 1 3 4 1 4 2 3 1 2 1 1,
n n n n n n n nm m m m m m m mpp p p p p p p (5.18)
where
2 2
2 21 2
2 23 4
2 2
1 1
1 12 24 4 , 22 24 12 ,
22 24 12 , 2 24 4 ,
0,1,..., .
k k
k kp h h t p h h t
p h h t p h h t
m N
Substitution of exp , 1,n nm i mh i into Eq. (5.18) leads to
1 2 3 4
4 3 2 1
exp 2 exp exp
exp 2 exp exp .
p i h p i h p p i h
p i h p i h p p i h
(5.19)
Simplifying Eq. (5.19), we get
,A iB
C iD
(5.20)
where
2 2 2
2 2
2 2
2 2
2 2 2 2
2 2
2
22 2A = 1 2 1 2 4 2 4 8 1 c o s
2 4 1 2 4 c o s 2 ,
2 0 1 6 1 4 8 s i n
2 4 1 2 4 s i n 2 ,
= 2 2 1 2 1 2 4 2 4 8 1 c o s
2 4 1 2 4 c o s 2 ,
2 0 1 6 1
h d t h k h d t h k h
h d t h k h
B h d t h k h
h d t h k h
C h d t h k h d t h k h
h d t h k h
D h d t h
2
2 2
4 8 s i n
2 4 1 2 4 s i n 2 .
k h
h d t h k h
After algebraic manipulation, we obtain similar expressions for 2 2A B and 2 2C D in
Chapter 5 MRLW equation
102
the following form:
2 2 2 2
4 2 2 2 2 2 2 2 4 2 2 2
4 2 2 2 2 2 2 2 4 2 2 2
4 2 2 2 2 2 2 2 4 2 2 2
4 2 2 2 2 2 2
16 61 20 40 20 120 144
8 143 12 24 12 24 144 cos
16 11 12 24 12 120 144 cos 2
8 4 8 4
A B C D
h dt h dt h k dt h k h
h dt h dt h k dt h k h h
h dt h dt h k dt h k h h
h dt h dt h k dt
2 4 2 2 224 144 cos 3 ,h k h h
so that 2
1, which proves unconditional stability of the linearized numerical
scheme for the MRLW equation.
5.3.2 Stability of scheme based on quintic B-spline collocation method
The linearized form of proposed scheme using quintic B-spline collocation method
is given by
1 1 1 1 11 2 2 1 3 4 1 5 2
5 2 4 1 3 2 1 1 2 ,
n n n n nm m m m m
n n n n nm m m m mp
p p p p p
p p p p
(5.21)
where
2 2 21 2 3
2 24 5
2 2
2 2
2 40 5 (1 ), 52 80 50 (1 ), 132 240 ,
52 80 50 (1 ), 2 40 5 (1 ), 0,1,..., .
p h h t k p h h t k p h
p h h t k p h h t k m N
Substitution of exp , 1,n nm i mh i into Eq. (5.21) leads to
1 2 3 4 5
5 4 3 2 1
exp 2 exp exp exp 2
exp 2 exp exp exp 2
m m m m m
m m m m m
p i h p i h p p i h p i h
p i h p i h p p i h p i h
(5.22)
Simplification of Eq. (5.22) yields
A iB
A iB
(5.23)
where
2 2 2
2 2
A= 4 80 cos 2 104 160 cos 132 240 ,
10 (1 )sin 2 100 (1 )sin ,
h h h h h
B h t k h h t k h
so that 1 and the linearized numerical scheme for the MRLW equation is
unconditionally stable.
Chapter 5 MRLW equation
103
5.4 Numerical Tests and Results
The numerical methods outlined in the previous section are tested for single solitary
wave and interactions of multiple solitary waves. The Maxwellian initial condition
which generates a train of solitary waves is also considered. The accuracy is checked
using the error norms L and 2L . The analytical solution of MRLW equation is given as
(See (Khalifa et al. [74])):
0
6( , ) sec 1 ,
1
c cu x t h x c t x
c
(5.24)
where 0x is an arbitrary constant. The initial condition is given by
0
6( ,0) sec ,
1
c cu x h x x
c
(5.25)
and the boundary conditions are extracted from the exact equation. The conservation
properties of the MRLW equation related to mass, momentum and energy are
determined by finding the following three invariants (Gardner et al. [43], Olver [98]):
2 2 4 21 2 3
6, , ,
b b b
x x
a a a
C udx C u u dx C u u dx
(5.26)
5.4.1 Single solitary wave
Problem 5.1 We take 01, 0.03, 0.2, 0.025, 1, 0, 40 60c h t x x to
compare our results with (Khalifa et al. [74]). In order to find error norms and the
invariants 1 2 3, ,C C C at different times, the computations are carried out for times upto
10t . The results are given in Table 5.1. At 10t the error norms are
-4 0.033611 10L and -42 0.048674 10L . The error in invariant 3C approaches
zero during the simulation and maximum absolute errors in 1 2,C C remain less than
3 63.55 10 and 2.0 10 throughout the simulation. In Table 5.1 the performance of
the new method is compared with finite difference method due to (Khalifa et al. [74]) at
10t . It is observed that errors of the method in (Khalifa et al. [74]) are considerably
larger than those obtained with method 5.2.1. Fig. 5.1(a) shows the solutions at
0,2,4,6,8,10t and illustrates the motion of solitary wave to the right along the
Chapter 5 MRLW equation
104
interval 40 60x when 0 10t . Initially the centre of the solitary wave of
amplitude 0.424264 is located at 0x . At time 10t its amplitude is 0.424202
centered at 10.2x . The absolute difference in amplitudes over the time interval
0,10 is found to be 56.2 10 . It can be inferred that the solitary wave moves to the
right with constant amplitude and velocity. Error graph is shown in Fig. 5.1(b).
Problem 5.2 For the sake of comparison with earlier work of authors (Gardner et al.
[43], Khalifa et al. [75]), we choose the following parameters for the solution of
MRLW equation using method 5.2.2:
00, 100, 6, 1, 0.2, 0.025, 1, 40.a b c h t x In order to find error norms
and the invariants 1 2 3, ,C C C at different times, the computations are carried out for
times upto 10.t The amplitude of the solitary wave is 0.9999644 located at 60.x
The absolute difference between numerical and exact peak values is 53.56 10 . The
results are summarized in Table 5.2. At 10t the error norms are -32.0248 10L
and -32 3.9314 10L . It is clear from Table 5.2 that 1C is constant and 2 3,C C remain
almost constant with the maximum error of 55.0 10 . In Table 5.2 the new method is
compared with (Gardner et al. [43], Khalifa et al. [75]) which shows superiority of our
scheme. Fig. 5.2(a) shows the graphs of the solutions at 0, 2,5,10t and depicts the
motion of solitary wave from left to the right along the interval 0 100.x The error
graph is reported in Fig. 5.2(b).
Problem 5.3 The parameters 06, 0.3, 0.1, 0.01, 1, 40,0 100c h t x x are
chosen to enable comparison with collocation method based on cubic splines (Khalifa
et al. [75]). The results are given in Table 5.3. At 20t the error norms are
-5 2.222848 10L and -52 5.089274 10L . The absolute error in invariants
1 2 3, ,C C C approach zero during the simulation. In Table 5.3 the new method is
compared with collocation method based on cubic splines given in (Khalifa et al. [75]).
Again errors are less in the present method, however both methods 5.2.1 and (Khalifa
et al. [75]) produce excellent conservation properties. Fig. 5.3(a) shows the solutions at
0,5,10,15,20t and illustrates the motion of solitary wave to the right along the
interval 0 100x when 0 20t . Initially the amplitude of solitary wave is
0.547723 and its peak position is located at 40x . At 20t its amplitude is recorded
Chapter 5 MRLW equation
105
as 0.547721 with center 66x . Hence the absolute difference in amplitudes over the
time interval 0, 20 is observed as 62.0 10 Error graph is displayed in Fig. 5.3(b).
The same problem is also solved by method 5.2.2 and the results are reported in
Table 5.4. Fig. 5.4(a) shows the solution at 0,6,12, 20t and the error graph is shown
in Fig. 5.4(b).
5.4.2 Two solitary waves
Problem 5.4 We consider interaction of two solitary waves using the following initial
condition:
2
1
6( ,0) sec .
1i i
ii i
c cu x h x x
c
(5.27)
In our numerical experiment, we choose 1 2 1 26, 4, 1, 25, 55, 1, 0.2,c c x x h
0.025,t 0 250.x The parameters give solitary waves of different amplitudes of
ratio 2:1 having centers at 25 and 55x x to make the interaction possible. For the
case of interaction of two solitary waves the analytical values of invariants are given in
(Khalifa et al. [75]) as 1 2 311.467698, 14.629243, 22.880466C C C . Computations
for both methods 5.2.1 and 5.2.2 are done up to time 20t and same values for
invariant quantities are obtained. The values of invariants 1 2 3, ,C C C are tabulated in
Table 5.5 and compared with (Khalifa et al. [75]) at 20t . The invariants are almost
constant , however it is observed that collocation method using cubic splines by
(Khalifa et al. [75]) has better conservative properties in this case. In Fig. 5.5 the
interaction of solitary waves is shown at different times 0,4,8,10,14,20t
respectively. It is clear from the figure that the two solitary waves interact at times
4,8,10t and then separate at 14,20t preserving their original shapes.
We have also taken 1 2 1 26, 0.03, 0.01, 18, 58, 40 180c c x x x to
compare our results with ADM method in (Khalifa et al. [76]). The results are tabulated
in Table 5.6. It is clear from the table that the conservation properties of present method
are better than ADM when smaller amplitudes and times are considered. Moreover, no
interaction of solitary waves is observed over the time interval 0, 2 .
Chapter 5 MRLW equation
106
5.4.3 Three solitary waves
Problem 5.5 Interaction of three solitary waves is considered using the initial
condition
3
1
6( ,0) sec ,
1i i
ii i
c cu x h x x
c
(5.28)
and the following parameters for the purpose of comparison.
1 2 3 1 2 36, 4, 1, 0.25, 15, 45, 60, 1, 0.2, 0.025, 1,0 250.c c c x x x h t x These parameters give solitary waves of different amplitudes 4, 2 and 1 having their
peaks located at 15,45 and 60x moving along the same direction. The analytical
values of invariants for the interaction of three waves are given in (Khalifa et al. [75])
as 1 2 314.9801, 15.8218, 22.9923C C C . Computations for both methods 5.2.1 and
5.2.2 are done from 0 to 45t t and same values for invariant quantities are
obtained. The values of the invariant quantities during the simulation are given in Table
5.7. It is evident from this table that the values produced by our method for 1 2,C C are
better when compared with the exact values whereas the performance of the method in
(Khalifa et al. [75]) is better in case of 3C . Fig. 5.6 shows the interaction of solitary
waves at times 0,5,8,15,20,40t respectively. The three solitary waves interact at
times 5,8,15t and then separate at 20,40t emerging unchanged.
The parameters 1 2 2 1 2 36, 0.03, 0.02, 0.01, 18, 48, 88, 40 180c c c x x x x are
chosen to compare our results with ADM of (Khalifa et al. [76]). The results are
tabulated in Table 5.8 along with those given in (Khalifa et al. [76]). It is again
observed that the conservation properties of present method are very good as compared
to ADM when smaller amplitudes and times are considered.
5.4.4 The Maxwellian initial condition
Problem 5.6 As the last problem, the Maxwellian initial condition
2,0 exp 40u x x , (5.29)
is used with boundary conditions given by Eq. (5.3) when 0a and 100b . The
numerical solutions of the MRLW equation are carried out for generation of a train of
solitary waves for various values of . We consider for the set of values
Chapter 5 MRLW equation
107
0.1,0.04,0.015,0.01, which is used in the earlier work(See (Gardner et al. [43], Khalifa
et al. [74])) and the simulations are performed up to time 15t . For 0.1 we
observe generation of a single solitary wave along with an oscillating tail. However
when is reduced, Maxwellian initial condition breaks up into a number of solitary
waves. For 0.04,0.015,0.01 the number of observed solitons is 2,3,4 respectively.
An oscillating wave behind the train of solitary waves is observed in each case. The
various cases for are shown in Fig. 5.7 and the graphs are in agreement with earlier
work (Gardner et al. [43], Khalifa et al. [74]). It is also clear from the Fig. 5.7 that the
peaks of solitary waves lie on the straight line. The conservative quantities are given in
Table 5.9 which remain almost constant during the simulation
Chapter 5 MRLW equation
108
Table 5.1 Accuracy test for problem 5.1 method 5.2.1 Time L
410 2L 410 1C 2C 3C
0 0.0 0.0 7.804400 2.129885 0.130251
1 0.001589 0.004743 7.805221 2.129885 0.130251
2 0.003456 0.009481 7.805897 2.129886 0.130251
3 0.005569 0.014214 7.806451 2.129886 0.130251
4 0.007960 0.018951 7.806898 2.129887 0.130251
5 0.010731 0.023719 7.807254 2.129887 0.130251
6 0.013944 0.028526 7.807528 2.129887 0.130251
7 0.017671 0.033387 7.807729 2.129887 0.130251
8 0.022125 0.038349 7.807864 2.129887 0.130251
9 0.027374 0.043428 7.807936 2.129887 0.130251
10 0.033611 0.048674 7.807948 2.129887 0.130251
10 (Khalifa et al. [74]) 1.99524 6.98280 7.80932 2.12988 0.130315
Invariants and error norms for single solitary wave, 1, 0.03, 0.2, 0.025,c h t 40 60x
Chapter 5 MRLW equation
109
Table 5.2 Accuracy test for problem 5.2 method 5.2.2 Time L
310 2L 310 1C 2C 3C
0 0.0 0.0 4.44288 3.29983 1.41421
1 0.2209 0.4036 4.44288 3.29983 1.41421
2 0.4283 0.8041 4.44288 3.29982 1.41420
3 0.6289 1.2018 4.44288 3.29982 1.41420
4 0.8277 1.5953 4.44288 3.29981 1.41419
5 1.0261 1.9859 4.44288 3.29980 1.41419
6 1.2248 2.3749 4.44288 3.29980 1.41418
7 1.4239 2.7633 4.44288 3.29980 1.41418
8 1.6236 3.1520 4.44288 3.29979 1.41417
9 1.8239 3.5413 4.44288 3.29979 1.41417
10 2.0248 3.9314 4.44288 3.29978 1.41416
10 (Khalifa et al. [75]) 5.43718 9.30196 4.44288 3.29983 1.41420 10 (Gardner et al. [43]) 9.24 16.39 4.442 3.299 1.413
Invariants and error norms for single solitary wave, 6, 1, 0.2, 0.025, 1, 40,0 100oc h t xx
Chapter 5 MRLW equation
110
Table 5.3 Accuracy test for problem 5.3 method 5.2.1 Time L
510 2L 510 1C 2C 3C
0 0.0 0.0 3.581967 1.345076 0.153723
2 0.267928 0.556873 3.581967 1.345076 0.153723
4 0.520815 1.103371 3.581967 1.345076 0.153723
6 0.768640 1.637097 3.581967 1.345076 0.153723
8 0.994829 2.156048 3.581967 1.345076 0.153723
10 1.209189 2.662302 3.581967 1.345076 0.153723
12 1.417087 3.158828 3.581967 1.345076 0.153723
14 1.621213 3.648135 3.581967 1.345076 0.153723
16 1.823033 4.132113 3.581967 1.345076 0.153723
18 2.023404 4.612153 3.581967 1.345076 0.153723
20 2.222848 5.089274 3.581967 1.345076 0.153723
20 (Khalifa et al.
[75])
29.6650 60.6885 3.58197 1.34508 0.153723
Invariants and error norms for single solitary wave, 6, 0.3, 0.1, 0.01, 40oc h t x
Chapter 5 MRLW equation
111
Table 5.4
Accuracy test for problem 5.2 method 5.2.2
Time L 510 2L 510 1C 2C 3C
0 0.0 0.0 3.58197 1.34508 0.15372
2 0.2683 0.5669 3.58197 1.34508 0.15372
4 0.5403 1.1236 3.58197 1.34508 0.15372
6 0.7902 1.6654 3.58197 1.34508 0.15372
8 1.0187 2.1912 3.58197 1.34508 0.15372
10 1.2354 2.7038 3.58197 1.34508 0.15372
12 1.4457 3.2064 3.58197 1.34508 0.15372
14 1.6523 3.7016 3.58197 1.34508 0.15372
16 1.8565 4.1914 3.58197 1.34508 0.15372
18 2.0593 4.6773 3.58197 1.34508 0.15372
20 2.2612 5.1602 3.58197 1.34508 0.15372
Invariants and error norms for single solitary wave, 6, 0.3, 0.1, 0.01, 1,c h t
40,0 100o xx
Table 5.5 Accuracy test for problem 5.4 Time
1C 2C 3C
0 (Analytical) 11.467698 14.629277 22.880432
2 11.467698 14.624259 22.860365
4 11.467698 14.619226 22.840279
6 11.467699 14.614169 22.820069
8 11.467700 14.606821 22.787857
10 11.467700 14.603687 22.771773
12 11.467699 14.603056 22.775766
14 11.467699 14.598059 22.756029
16 11.467700 14.593048 22.736127
18 11.467700 14.588061 22.716289
20 11.467701 14.583089 22.696510
20 (Khalifa et al. [75]) 11.4677 14.6292 22.8809
Invariants for two solitary waves, 1 2 1 2 0 2506, 4, 1, 25, 55, 0.2, 0.025, xc c x x h t
Chapter 5 MRLW equation
112
Table 5.6 Accuracy test for problem 5.4 Present method ADM (Khalifa et al. [76])
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