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British Journal of Mathematics & Computer Science 5(3): 310-332, 2015, Article no.BJMCS.2015.021
ISSN: 2231-0851
SCIENCEDOMAIN international www.sciencedomain.org
______________________________________________________________________________________________________________________
_____________________________________
*Corresponding author: heba_salem28@yahoo.com, mam_el_kady@yahoo.com
Numerical Solutions of Coupled Nonlinear Evolution
Equations via El-gendi Nodal Galerkin Method
M. El-Kady
1*, Salah M. El-Sayed
2 and Heba. E. Salem
3
1Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt.
2Department of Scientific Computing, Faculty of Computers and Informatics, Benha University,
Egypt. 3Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt.
Article Information
DOI: 10.9734/BJMCS/2015/8245
Editor(s):
(1) Raducanu Razvan, Department of Applied Mathematics, Al. I. Cuza University, Romania.
(2) Chin-Chen Chang, Department of Information Engineering and Computer Science, Feng Chia University, Taiwan. (3) Sheng Zhang, Department of Mathematics, Bohai University, Jinzhou, China.
(4) Qiankun Song, Department of Mathematics, Chongqing Jiaotong University, China.
(5) Kai-Long Hsiao, Taiwan Shoufu University, Taiwan.
(6) Paul Bracken, Department of Mathematics, University of Texas-Pan American Edinburg, TX 78539, USA.
Reviewers:
(1) Anonymous, University, Elazg, Turkey. (2) Anonymous, HITEC University Taxila Cantt, Pakistan.
(3) Anonymous, Namık Kemal University, Turkey.
(4) Anonymous, King Mongkut's University of Technology Thonburi, Thailand.
(5) Anonymous, Taif University, Taif, Saudi Arabia.
Complete Peer review History: http://www.sciencedomain.org/review-history.php?iid=727&id=6&aid=6753
Received: 06 December 2013
Accepted: 19 February 2014
Published: 04 November 2014
_______________________________________________________________________
Abstract
In this research the solution of coupled modified Korteweg-de Vries equation (mKdV) and the
generalized Hirota–Satsuma coupled KdV equation by using El-gendi nodal Galerkin (EGG)
approaches are presented. El-gendi nodal Galerkin (EGG) (EGG) approaches consist of two
approaches, the first is El-gendi Chebyshev nodal Galerkin (ECG) and the second approach is
called El-gendi Legendre nodal Galerkin (ELG). In these new approaches spaces of the solution
and the weak form to the system are presented. The resulted systems of ODES are solved by the
fourth order Runge-Kutta solver. The convergence and the stability of these new methods are
analyzed numerically. Numerical results are presented and compared with the results obtained
by pseudo-spectral method.
Original Research Article
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
311
Keywords: Coupled mKdV equation; generalized Hirota-Satsuma Coupled KdV equation;
El-gendi nodal Galerkin method; Legendre and Chebyshev cardinal functions.
1 Introduction
The effort in finding exact solution to a nonlinear equation is important for understanding of the
most nonlinear physical phenomena. For instances, the nonlinear wave phenomena observed in
fluid dynamics, plasma and optical fibers are often modeled by the bell shaped sech solutions and
the kink shaped tanh solutions.
In this paper, we consider coupled mKdV and a generalized Hirota-Satsuma coupled KdV
equations. In [1] the authors introduced a 44× matrix spectral problem with three potentials and
they proposed a corresponding hierarchy of nonlinear equations; two typical equations in
hierarchy are coupled mKdV equation which is given by:
,3)(32
33
2
1 2
xxxxxxxxt uuvvuuuu λ−++−=
,3333 2
xxxxxxxxt vvuvuvvvv λ++−−−= ],0[, Ttbxa ∈<< , (1)
where λ is a real constant with the initial conditions:
),()0,( 1 xfxu = ),()0,( 2 xfxv = (2)
and the boundary conditions are given in the following form
],0[),(),(),(),( 21 Tttqtbutqtau ∈== ,
],0[),(),(),(),( 21 Tttgtbvtgtav ∈== . (3)
and the generalized Hirota-Satsuma coupled KdV equation is given as follows
,)(332
1xxxxxt vzuuuu +−=
,3 xxxxt uvvv +−=
,3 xxxxt uzzz +−= ],0[, Ttbxa ∈<< , (4)
with the initial conditions:
),()0,( 1 xfxu = ),()0,( 2 xfxv = ),()0,( 3 xfxz = (5)
and the boundary conditions are given in the following form
],0[),(),(),(),( 21 Tttqtbutqtau ∈== ,
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
312
],0[),(),(),(),( 21 Tttgtbvtgtav ∈== ,
],0[),(),(),(),( 21 Ttthtbzthtaz ∈== . (6)
Equations (1-3) become a generalized KdV equation for u = 0 and the mKdV equation for 0=v ,
respectively. The soliton solutions of these problems describe various phenomena in nature, such
as vibrations, solutions and propagation with a finite speed. Particularly, coupled KdV system,
describes interactions of two long waves with different dispersion relations [2].
Solitary solutions for various nonlinear wave equations have been investigated using different
methods which can only solve special kind of nonlinear problems due to the limitations or
shortcomings in the methods. Many studies of generalized Hirota-Satsuma coupled KdV and
coupled mKdV equations have been done by many authors via different approaches. In [3] the
author used the extended tanh-function method and symbolic computation to obtain respectively
four kinds of soliton solutions for a new coupled mKdV and new generalized Hirota–Satsuma
coupled KdV equations, which were introduced recently by Wu et al. [1].
In [4] the author presented the decomposition method to obtain the Soliton solution for
generalized Hirota–Satsuma coupled KdV equations and coupled mKdV equation. In this paper
the algorithm is illustrated by studying an initial value problem. The obtained results are
presented, and only few terms are required to obtain an approximate solution that is found to be
accurate and efficient.
In [5] the authors applied the variational iteration method to obtain approximate analytic solutions
of coupled mKdV and generalized Hirota–Satsuma coupled KdV equations. This method provides
a sequence of functions and is based on the use of the Lagrange multiplier for the identification of
optimal values of parameters in a functional.
Also, in [6] the authors demonstrated the feasibility and validity of the differential transform
method, namely DTM. Therefore, the method has been applied to solve coupled mKdV and
generalized Hirota–Satsuma coupled KdV equations with initial conditions of two types. The
DTM method is based on the Taylor series, but the series coefficients are calculated in an iterative
manner by the help of T-transform.
In [7] the authors presented the numerical solution of systems of Hirota-Satsuma coupled KdV
and coupled mKdV equations by means of Homotopy Analysis Method (HAM). The HAM can
extremely minimize the volume of computations with respect to traditional techniques and yields
the analytical solution of the desired problem in the form of a rapidly convergent series with easily
computable components.
In [8] the Homotopy Analysis Method (HAM) is presented for obtaining the approximate solution
of new coupled modified Korteweg-de Vries (mKdV) system. The approximate analytical solution
is obtained by using this method in the form of a convergent power series with components that
are easily computable.
In this work, we aim to introduce two reliable techniques in order to solve coupled mKdV and
generalized Hirota–Satsuma coupled KdV equations. The new techniques depend on El-gendi-
nodal Galerkin (EGG) methods. Nodal Galerkin methods start from a weak form of the equations,
but replace hard to evaluate integrals by quadrature. Gauss quadrature is always used in the week
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
313
form. In this research, we use El-gendi quadrature formula [9]. This formula has a symmetric
property and this property leads us to reduce the number of operations to 50% and hence reducing
the rounding error. In addition, this formula is an alternating series which converges as the number
of grid points tends to infinity.
Although, we use the differentiation matrix in these methods as the pseudo-spectral method which
is a very well-known method, we find that (EGG) methods are more accurate for a long time and
converge faster than the pseudo-spectral method.
Also, we aim to discuss numerically in this case the stability and convergence of the new
approaches and make a comparison with the pseudo-spectral error where the question of stability
of the spectral approximations tends to be critical for solving hyperbolic conservation laws. The
most important reason is that the nonlinear mixing of the Gibbs oscillations with the approximate
solution will eventually cause the scheme to become unstable [10]. In other words, Trefethen and
Trummer determined the relation between the eigenvalues of the differentiation matrix and the
allowable time step in explicit time integration. On a grid of N points per space dimension,
machine rounding leads to errors in the eigenvalues of size )( 2NO . This phenomenon may lead
to inconsistency between predicted and observed time step restriction [11, 12,13]. Contrary, when
we try to solve the model problem )(uLut = where dxdL /= by the new techniques the
resulted system was insensitive to round-off errors and the condition number scaled sub-linearly
with N .
This paper is organized as follows: In section 2, we employ El-gendi Chebyshev nodal Galerkin
(ECG) method for solving (1) and (4) and we present the approximate solutions at the extrema
points of the Chebyshev polynomial. In section 3, we employ El-gendi Legendre Nodal Galerkin
(ECG) method for solving (1). In section 4, we discuss the eigenvalues and the time step for
(EGG) and pseudo-spectral methods. In section 5, we present the numerical solutions with
graphics and we compare the results with pseudo-spectral solutions at different times and for
enough large grid points.
2. EL-gendi Nodal Galerkin Method for Coupled Mkdv
In this section we present the numerical solution of the coupled mKdV by using El-gendi
Chebyshev nodal Galerkin (ECG) method and by using El-gendi Legendre nodal Galerkin (ELG)
method as follow:
2.1 El-gendi chebyshev nodal galerkin method for coupled mkdv
In this section we will explain El-gendi Chebyshev nodal Galerkin method and illustrate how it
used to solve equations (1-3) in case 1−=a and 1=b . In this method, the trail and the test
spaces are identical, so that we define for 0≥m the space )1,1(−mH to be a vector space of
functions )1,1(2 −∈ Lv such that all distributional derivatives of v is of order up to m and it can
be represented by functions in )1,1(2 −L . Since the functions of )1,1(1 −H are continuous up to
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
314
the boundary by Sobolev imbedding theorem, it is meaningful to introduce the following solution
subspace of )1,1(1 −H :
}0),1(),1(:)1,1({)1,1( 11
0 ==−−∈=− tutuHuH .
The weak form of equation (1) is given by multiplying both sides in equation (1) by a test
function, and then we have:
>−∈∀+
+−−−=
>−∈∀−
++−=
−∈−∈
∫
∫∫∫∫∫
∫
∫∫∫∫∫
−
−−−−−
−
−−−−−
.0),1,1(,3
33)(2
3
,0),1,1(,3
)(32
3)(
2
1
,s.t.)1,1(),1,1(find
1
0
1
1
1
1
21
1
1
1
21
1
1
1
1
0
1
1
1
1
1
1
1
1
31
1
1
1
1
0
1
0
tHGdxGv
dxGvudxGvudxGvdxGvdxGv
tHYdxYu
dxYuvdxYvdxYudxYudxYu
HvHu
x
xxxxxxxt
x
xxxxxxxt
N
λ
λ
By using the integration by parts we have:
>−∈∀−
+−+=
>−∈∀+
−−+−=
−∈−∈
∫
∫∫∫∫∫
∫
∫∫∫∫∫
−
−−−−−
−
−−−−−
.0),1,1(,3
33)(2
3
,0),1,1(,3
)(32
3)(
2
1
,s.t.)1,1(),1,1(find
1
0
1
1
1
1
21
1
1
1
21
1
1
1
1
0
1
1
1
1
1
1
1
1
31
1
1
1
1
0
1
0
tHGdxGv
dxGvudxGvudxGvdxGvdxGv
tHYdxYu
dxYuvdxYvdxYudxYudxYu
HvHu
x
xxxxxxxt
x
xxxxxxxt
N
λ
λ
(7)
Let us denote the finite dimensional subspace of )1,1(1
0 −H to be given as follows:
}0),1(),1(:P{ ==−∈= tftffX N
N,
where NP is the space of polynomials with degree N . Let kT be the Chebyshev polynomial of
degree k , then the cardinal function which is based on the Chebyshev polynomial is defined as
follows:
NixTxTN
xN
k
kikki
i ...,,1,0,)()(2
)(0
== ∑=
θθ
ϕ , (8)
where we have for all 1=kθ , except 2/10 == Nθθ and . The grid points l
x are
the so-called Chebyshev Gauss–Lobatto points,
ll ii x δϕ =)(
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
315
)/cos( Nkxk π= , Nk ...,,1,0= ,
which are the extrema points of the Chebyshev polynomial ( )N
T x . Consequently, we have the
following spaces [14]:
,
)}(),...,(),({ 121 xxxspanX N
N
v −= ψψψ ,
where )(xiψ has the form (8). Now, we give the approximate solution in the form
∑=
=N
j
jj
NxtUu
0
)()( ϕ , ∑=
=N
j
jj
NxtVv
0
)()( ψ , (9)
to ensure the approximations satisfy the boundary conditions, we set 00 == NUU and
00 == NVV . Also, we can set test functions )(xY and )(xG as a function of thN order
polynomials so we can write these polynomials in an equivalent cardinal form
∑=
=N
i
ii xxY0
)()( ϕα , ∑=
=N
i
ii xxG0
)()( ψβ , (10)
where the nodal values ii βα , are arbitrary except that 00 == Nαα and 00 == Nββ , to
ensure that )(xY and )(xG satisfy the boundary conditions. Now, the nodal Galerkin
approximation to equations (7) is
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( )
( )
>∈∀−
+−+=
>∈∀+
−−+−=
∈∈
,0,,,3
,)(3,3,)(2
3,,
,0,,,3
,3,2
3,)(,
2
1,
,s.t.,find
22
3
tXGGv
GvuGvuGvGvGv
tXYYu
YvuYvYuYuYu
XvXu
N
vNx
N
N
N
x
N
N
N
x
N
xNx
N
Nx
N
xxN
N
t
N
uNx
N
Nx
NN
Nx
N
xNx
N
Nx
N
xxN
N
t
N
v
NN
u
N
λ
λ
(11)
where
( ) ∑=
=N
j
jjNjNxgxfbgf
0
)()(, , (12)
with Nkb are given by:[9]
∑= −
=2/
02
2cos
14
4 N
j
sNk
N
jk
jNb
πθ, 1...,,2,1 −= Nk ,
)}(),...,(),({ 121 xxxspanX N
N
u −= ϕϕϕ
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
316
1
120
−==
Nbb NNN
. (13)
Now, we evaluate the terms in (11) as follows:
( ) =N
N
t vu , ∑=
N
j
jjNjUb0
α& , (14)
and the second term is
( ) =Nx
N
xx Yu , ∑∑==
′′′N
k
kjkNk
N
j
j xub00
)(ϕα , (15)
where the first derivative of the cardinal functions )(xjϕ at the points lx have the entries of the
differentiation matrix: [15]
NjxTxTc
k
Nx
N
k
lnjk
n
kk
oddknn
j
lj ...,,1,0,)()(4
)(1
1
)(0
==′ ∑ ∑=
−
+=
θθϕ , (16)
the second derivative is:
NjxTxTc
nkk
Nx
N
k
lnjk
n
kk
evenknn
j
lj ...,,1,0,)()()(4
)(2
222
)(0
=−
=′′ ∑ ∑=
−
+=
θθϕ . (17)
and the rest terms are evaluated by the same fashion. Then (11) can be written in the form
====
−=′−′+
′′−′+′′′=
′+′′−
′−′+′′′−=
∑ ∑∑
∑ ∑∑∑∑
∑ ∑∑
∑ ∑∑∑∑
= ==
= ====
= ==
= ====
.0,0
,1...,,2,1,)(3)(3
)()(3)(2
3)()(
,)(3)()(2
3
)(3)()()(2
1
00
0 00
2
0 00
2
00
0 00
0 00
3
00
NN
N N
N
N
m
mNmm
j
N N
m
jmNjm
N
jN
N
k
kjkNK
N
jNj
N N
jN
N
k
kjkNk
N N
m
mjNmm
N
jN
N
k
kjkNK
N
jNj
VVUU
NjxbVxbUV
xxbUVxbVxxbVVb
xbUxxbV
xbVUxbUxxbUUb
l l
lllll
l
l
l
l
llll
l
l
l l
lllll
l
l
l
llll
l
l
&
&
ψλψ
ψϕψψψ
ϕλϕψ
ϕϕϕϕ
(18)
Now, we put ][ VUW = where
)()( tUtW ii = , )()( tVtW iNi =+ , Ni ...,,1,0= . (19)
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
317
with the boundary,
00 == NWW , 021 ==+ NN WW . (20)
Now, we can rewrite (18) by using (19-20) then the resulted system of ODEs has been solved by
using fourth order Runge-Kutta solver.
2.2 El-gendi Chebyshev Nodal Galerkin Method for Generalized Hirota-
Stasuma KdV
In this section, we have the following space
)}(),...,(),({ 121 xxxspanX N
N
z −= χχχ ,
where )(xiχ has the form (8). Now, we give the approximate solution of )(xz in the following
form
∑=
=N
j
jj
N xtZz0
)()( χ , (21)
also, we set 00 == NZZ to ensure the approximations satisfy the boundary conditions. Let
)(xV be a test function of thN order polynomials and we can write it in the equivalent cardinal
form
∑=
=N
i
ii xxV0
)()( χη , (22)
where the nodal values iη are arbitrary except that 00 == Nηη to ensure that )(xV satisfy the
boundary conditions. Now, the nodal Galerkin approximations to equations (4-6) are
======
−=′+′′′=
′+′′′=
′−′+′′′−=
∑ ∑∑∑
∑ ∑∑∑
∑ ∑∑∑∑
= ===
= ===
= ====
.0,0,0
,1...,,2,1),(3)()(
),(3)()(
,)(3)(2
3)()(
2
1
000
0 000
0 000
0 00
2
00
NNN
j
N N
m
Nmm
N
k
kjkNK
N
jNj
j
N N
m
Nmm
N
k
kjkNK
N
jNj
N N
m
mjNmm
N
jN
N
k
kjkNK
N
jNj
ZZVVUU
NjxbUZxxbZZb
xbUVxxbVVb
xbVUxbUxxbUUb
l
l
ll
l
l
l
l
ll
l
l
l
l
l
llll
l
l
&
&
&
χχχ
ψψψ
ϕϕϕϕ
(23)
Now, we put ][ ZVUW = where
)()( tUtW ii = , )()( tVtW iNi =+ , Ni ...,,1,0= . (24)
with the boundary conditions
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
318
00 == NWW , 021 ==+ NN WW . (25)
Now, we can rewrite (23) by using (24-25) then the resulted system of ODEs has been solved by
using fourth order Runge-Kutta solver.
3. EL-gendi Legendre Nodal Galerkin Method for Coupled
Mkdv In this section we consider the Legendre cardinal functions based on Chebyshev Gauss-Lobatto
(CGL) nodes and we present the approximate solution as a linear combination of these functions
which have the following form:
∑=
+=N
k
kiki xLxLkN
x0
)()()12(2
)(~ πϕ , Ni ...,,1,0= , (26)
where jx are the Chebyshev Gauss-Lobatto points and ll ii x δϕ =)(~
. Also, the spaces of the
solutions are constructed as before. Now, we give the approximate solution in the following form
∑=
=N
j
jj
N xtUu0
)(~)(~
ϕ , ∑=
=N
j
jj
N xtVv0
)(~)(~
ψ , (27)
Also, we set 0~~
0 == NUU and 0~~
0 == NVV . The test functions )(~
xY and )(~
xG are thN order
polynomials so we can write these polynomials in the equivalent cardinal form
∑=
=N
i
ii xxY0
)(~~)(~
ϕα , ∑=
=N
i
ii xxG0
)(~~)(
~ψβ , (28)
where the nodal values ii βα~
,~are arbitrary except that 0~~
0 == Nαα and 0~~
0 == Nββ . Now,
the nodal Galerkin approximation to equations (6-7):
====
−=′−′+
′′−′+′′′=
′+′′−
′−′+′′′−=
∑ ∑∑
∑ ∑∑∑∑
∑ ∑∑
∑ ∑∑∑∑
= ==
= ====
= ==
= ====
.0~~
,0~~
,1...,,2,1,)(~~3)(~~~
3
)(~)(~~~3)(~~
2
3)(~)(~~~
,)(~~3)(~)(~~
2
3
)(~~~3)(~~
)(~)(~~
2
1~
00
0 00
2
0 00
2
00
0 00
0 00
3
00
NN
N N
N
N
m
mNmm
j
N N
m
jmNmm
N
jN
N
k
kjkNk
N
jNj
N N
jN
N
k
kjkNk
N N
m
mjNmm
N
jN
N
k
kjkNK
N
jNj
VVUU
NjxbVxbUV
xxbUVxbVxxbVVb
xbUyxbV
xbVUxbUxxbUUb
l l
lllll
l
l
l
l
llll
l
l
l l
lllll
l
l
l
llll
l
l
&
&
ψλψ
ψϕψψψ
ϕλϕψ
ϕϕϕϕ
(29)
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
319
where
( ) ∑=
=N
j
jjNjNxgxfbgf
0
)()(, . (30)
and ( )lxxj =
′ψ~ is the first order differentiation matrix that depends on Legendre polynomial at the
CGL nodes and have the entries given by: [16]
( )lxxj =
′ψ ∑ ∑=
−
=
−−+=
N
mj
j
k
kj
l
j
kij xaxLN
j]2/)1[(
0
12)(
1,)(2
)12(π, (31)
where
!)!2()!(2
)!2()!22()1()(
1,kkjkj
kjkja
j
kj
k−−
−−−= ,
Also, ( )lxxj =
′′ψ has the following formula:
( )lxxj =
′′ψ ∑ ∑=
−
=
−−+=
N
mj
j
k
kj
l
j
kij xaxLN
j]2/)2[(
0
22)(
2,)(2
)12(π, (32)
where
!)!2()!(2
)!12()!2()!22()1()(
2,kkjkj
kjkjkja
j
kj
k−−
−−−−−= .
Also, as above we present the vector solution of the system as follows
]~~
[~
VUW = ,
where
)(~
)(~
tUtW ii = , )(~
)(~
tVtW iNi =+ , Ni ...,,1,0= . (33)
with
0~~
0 == NWW , 0~~
21 ==+ NN WW . (34)
Now, we can rewrite (29) by using (33-34) then the resulted system of ODEs has been solved by
using forth order Runge-Kutta solver.
4. Eigenvalues and Time Step
The aim of this section is to show where such stability restrictions come from so we consider the
first-order hyperbolic initial boundary value problem
xt uu = , )1,1(−∈x , 0>t , (35)
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
320
with the initial and boundary conditions
)()0,( xfxu = , 0),1( =tu , 0>t . (36)
which can be viewed as a model of more general hyperbolic system of equations with
appropriately specified boundary conditions. Now, El-gendi Galerkin approximation to this model
is given as follows:
)1()1(),(),( −−+−= ρρρ N
Nx
N
N
N
t uuu , NP∈∀ρ , (37)
where )(xiϕρ = in Chebyshev case or )(~ yiϕρ = in Legendre case. The associated
)1()1( +×+ NN matrix that represents the left hand side of (37) is
)0...,,0,1(diagMDB T
CC +−= ,
)0...,,0,1(diagMDB T
LL +−= ,
where )( jiC xD ϕ′= is the differentiation matrix in Chebyshev case, )(~jiL xD ϕ′= is the
differentiation matrix in Legendre case and TD is the transpose of the matrix,
)...,,,( 10 NNNN bbbdiagM = is the diagonal mass matrix and the resulted pseudo-spectral
matrix of (35-36) is ):1,:1(~
NNDD = after imposing the boundary condition.
Our concern is with the eigenvalues of D~
and B in the Chebyshev and Legendre cases. Since we
will use an explicit fourth order Runge-Kutta method to integrate in time, there will be a limit on
the size of the time step that depends on the size of the maximum eigenvalue of the D~
and B
matrices i.e. The approximation to be stable in time, the maximum eigenvaluemaxλ must fall
within the region of absolute stability of the time integration method. Analytic representations for
the eigenvalues of D~
and B matrices are not known, so we will find the eigenvalues
numerically. Now, for each time step t∆ , a stability region in the complex plane, defined as the
set of all C∈λ for which it reduces to a stable recurrence relation when applied to the model
problem uut λ= [12]. So the solutions of (35-36) will be bounded as ∞→t for a fixed time
step if and only if the eigenvalues of D~
and B lie in this stability region. Also, the stability region
expands in proportion to1)( −∆ t . Therefore if the eigenvalues of the matrix are of size )(NO , the
result is that the stability restriction )( 1−=∆ NOt , while if they are of size )( 2NO , the
restriction becomes )( 2−=∆ NOt .
In Fig. 1 we illustrate the stability region of the fourth order Rung-Kutta method and it is clear
from Figs. 2, 3 that the eigenvalues have large imaginary parts and some negative real parts. The
presence of the real parts indicates that the approximations are dissipative. Hence, some energy is
lost as the computations proceed. Dissipation is important for the stability of variable coefficient
and nonlinear problems. Moreover, it is clear that the eigenvalues are produced by round-off error
effects and the structure of the eigenvalues differs from the Chebyshev and Legendre
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
321
approximations. The largest eigenvalues of the Legendre approximation are very near the
imaginary axis, while the corresponding eigenvalues of the Chebyshev approximation have
significantly larger real parts. The time step will be limited by the location in the complex plane of
the largest eigenvalue. In Figs. 4 and 5 the eigenvalues of CB , LB are in the region of stability
and scaled sub-linearly with N and relatively insensitive to round-off errors where the algorithm
is said to be stable if its outcome is relatively insensitive to round-off error.
Fig. 1. The stability region of the fourth-order Runge-Kutta method in the complex plane.
Fig. 2. Distribution of the eigenvalues of CD~
for 100,32=N .
It is clear from Fig. 6 (a) that the condition numbers in the 2-norm of differentiation matrices scale
)(2
NO whereas (EGG) matrices scale sub-linearly with N . The maximum, minimum eigenvalues
and condition numbers of D~
and B matrices are indistinguishable graphically in the two cases
(Chebyshev and Legendre). We see from the Fig. 6 (b) that the largest eigenvalue of D~
grows
asymptotically as )( 2NO whereas the maximum eigenvalue of (EGG) matrices scale sub-linearly
with N .
-4 -3 -2 -1 0 1-3
-2
-1
0
1
2
3
R e a l
I m
a g
i n
a r
y
2.780
2.927i
-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2-100
-80
-60
-40
-20
0
20
40
60
80
100
i m
a g
i n
a r
y
r e a l
N=32
-250 -200 -150 -100 -50 0-1000
-800
-600
-400
-200
0
200
400
600
800
1000
i m
a g
i n
a r
y
r e a l
N=100
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
322
Fig. 3. Distribution of the eigenvalues of LD~
for 100,32=N .
Fig. 4. Distribution of the eigenvalues of CB for 100,32=N .
Fig. 5. Distribution of the eigenvalues of LB for 100,32=N .
-14 -12 -10 -8 -6 -4 -2 0-100
-80
-60
-40
-20
0
20
40
60
80
100i
m a
g i
n a
r y
r e a l
N=32
-160 -140 -120 -100 -80 -60 -40 -20 0-800
-600
-400
-200
0
200
400
600
800
i m
a g
i n
a r
y
r e a l
N=100
-3 -2 -1 0 1 2 3 4
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
R e a l
I m
a g
i n
a r
y
N=32
-3 -2 -1 0 1 2 3 4-3
-2
-1
0
1
2
3
R e a l
I m
a g
i n
a r
y
N=100
-3 -2 -1 0 1 2 3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
R e a l
I m
a g
i n
a r
y
N=32
-4 -3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
R e a l
I m
a g
i n
a r
y
N=100
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
323
(a) (b)
Fig. 6. (a) Spectral condition numbers of D~
and B , (b) maximum and minimum moduli of
eigenvalues of D~
and B .
When we solved the following problem
xxt uu = , )1,1(−∈x , 0>t , (38)
with the initial and boundary conditions
0),1(),1( ==− tutu , 0>t . (39)
El-gendi Galerkin approximation to this model is given as follows:
Nx
N
xN
N
t uu ),(),( ρρ −= , N
P∈∀ρ ,(40)
The associated )1()1( −×− NN matrix that represents the left hand side of (35) is
C
T
CC MDDB −=2, L
T
LL MDDB −=2, (41)
and the resulted pseudo-spectral matrix after imposing the boundary conditions is
):2,:2(2~ 2
NNDD = . In [12], experiments show that the second-order spectral differentiation,
with zero boundary conditions at both endpoints have real and negative eigenvalues and its
maximum magnitude is of )( 4NO for both Chebyshev and Legendre. Moreover, it is noted that
the maximum eigenvalue of LD2~
approximately one half that of CD2~
.Although, their
eigenvalues are less sensitive to perturbations than in the first-order case.
Fig. 7 (b) illustrates the size of the maximum and minimum eigenvalues of the two matrices ( 2D
and 2B ) for different numbers of grid points N . It is clear that the maximum eigenvalues of 2B
101
102
101
102
103
104
105
N
cond (D)
cond (B)
101
102
10-2
10-1
100
101
102
103
104
105
106
N
D-min
D-max
B-min
B=max
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
324
which is a symmetric positive definite matrix grow as )( 2NO , whereas the minimum eigenvalues
decay as )(1−
NO but the maximum eigenvalues of 2D grow as )(4
NO for both the Chebyshev
and the Legendre approximations and in computations we see that the maximum eigenvalues of
the Legendre methods is typically only half as large as that of the corresponding Chebyshev
method. On the other hand, the condition number of 2B grows like )( 3NO which is cleared from
Fig. 7 (a). Thus, the time step in the EGG will be of order )( 2−NO and the time step in pseudo-
spectral method will be of order )( 4−NO , so the EGG methods will be faster than the pseudo-
spectral method.
(a) (b)
Fig .7. (a) Spectral condition numbers of 2D and 2B and (b) maximum and minimum
moduli of eigenvalues of 2D and 2B .
5. Numerical Experiments
In this section we give two examples and we use MATLAB 7.0 softwaretoobtain the numerical
results.
Example 1. Consider the coupled mKdV equations are given in the following form:
,3)(32
33
2
1 2
xxxxxxxxt uuvvuuuu λ−++−=
,3333 2
xxxxxxxxt vvuvuvvvv λ++−−−= ],0[, Ttbxa ∈<< ,
with the solitary wave solutions,
)tanh(),( ξkktxu = ,
)(tanh4)4(2
1),( 222 ξλ kkktxv −+= , ],0[, Ttbxa ∈<< ,
101
102
10-4
10-2
100
102
104
106
N
Cond (D2)
Cond (B2)
101
102
10-4
10-2
100
102
104
106
N
D2-min
D2-max
B2-min
B2-max
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
325
where tkx )32(2
1 2 λξ −−+= , k and λ are arbitrary constants.
The following error notations are defined:
Absolute error )()( iappiex xuxu −= , 1,...,1 −= Ni ,
and
Maximum error )()(max iappiexi
xuxu −= , 1,...,1 −= Ni .
where ( )ex i
u x and ( )app i
u x are the exact and approximate solutions, respectively.
In Table (1), the problem solved in the domain ],[ ba and since Gauss-Lobatto points are in
interval[ 1,1]− , therefore, the interval ]1,1[− is mapped to ],[ ba by a linear mapping defined
by:
++
−=
22
ababX ii η , Ni ,...,0= ,
whereiη are the Gauss-Lobatto nodes. Now, we introduce the u -ELG, v -ELG, u -ECG, v -
ECG, u -pseudo and v -pseudo (i.e. the absolute error of the u -solution in the interior points by
ELG, ECG and pseudo-spectral methods respectively) in Table 1.
Table 1. The solution of mKdV at T=0.5 and 1.0=∆t , 1.0=k , 1.0=λ , 10=N ,
100−=a and 100=b .
X u-ELG u-ECG u-pseudo v-ELG v-ECG v-pseudo
X(2)
X(3)
X(4)
X(5)
X(6)
X(7)
X(8)
X(9)
5.4101E-12
1.7438E-10
3.5669E-08
2.2957E-05
2.3747E-05
3.6976E-08
1.8077E-10
5.6084E-12
8.0966E-12
3.0086E-10
5.5464E-08
2.6850E-05
2.7763E-05
5.7497E-08
3.1189E-10
8.3934E-12
1.0003E-04
1.0156E-04
1.4105E-04
2.4272E-04
2.4255E-04
1.4043E-04
1.0121E-04
9.9921E-05
2.1640E-12
6.9752E-11
1.4266E-08
8.6397E-06
8.9170E-06
1.4789E-08
7.2309E-11
2.2434E-12
3.2386E-12
1.2034E-10
2.2182E-08
9.9930E-06
1.0306E-05
2.2995E-08
1.2476E-10
3.3574E-12
1.2566E-06
1.3906E-06
3.4664E-06
4.0934E-05
4.0431E-05
3.4915E-06
1.3984E-06
1.2527E-06
Figs. 8 and 9 present the numerical solutions of coupled mKdV by pseudo-spectral and (EGG)
methods respectively 1.0=∆t , 1.0=k , 1=λ , 40−=a , 40=b and for 30=N . It is clear
that the v -solution by the pseudo-spectral method for long time is unstable in the boundary but
the v -solution by (EGG) methods is stable at the boundary for a long time.
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
326
Fig. 8. The approximate solution of the coupled mKdV by the pseudo-spectral method.
Fig. 9. The approximate solution of the coupled mKdV by (EGG) methods.
Fig. 10 displays the infinity norm of the pseudo-spectral method. The maximum absolute error
increased linearly from 0≈t to reach its maximum at 09.0≈t and then the error increases
linearly as a function of t .
Fig. 10. The infinity norm error of the pseudo-spectral method as a function of time for
10=N .
-40 -30 -20 -10 0 10 20 30 400
0.5
1
-0.1
-0.05
0
0.05
0.1
X
t
u
-40-20
020
40 0
0.2
0.4
0.6
0.8
1
0.5
0.505
0.51
0.515
t
X
v
-40 -30 -20 -10 0 10 20 30 400
0.5
1
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
x
t
u
-40 -30 -20 -10 0 10 20 30 400
0.5
1
0.502
0.504
0.506
0.508
0.51
0.512
0.514
0.516
0.518
t
x
v
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7x 10
-3
t
m a
x e
r r
U (
t )
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7x 10
-3
t
m a
x e
r r
V (
t )
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
327
Fig. 11 displays the infinity norm error of (EGG) methods. The infinity error oscillates
periodically with decreasing amplitude from 0≈t to reach their minimum at 3.0≈t and then
increases oscillatory in the same fashion to reach the same type of periodicity at 6.0≈t after that
amplitude from 67.0≈t will be fixed until 1≈t but the maximum error will increase.
Fig. 11. The infinity norm error of El-gendi Galerkin method as a function of time for
10=N .
Figs. 12 and 13 illustrate that the point-wise absolute error at each point in the grid when the
number of grid points is 30=N . We observe that (EGG) methods are better than the pseudo-
spectral method around the boundary but in case u -solution the absolute error will be maximum
around and at 0=x . In contrast, in case v -solution the absolute error is very small at 0=x and
very large around 0=x .
We observe from Fig. 14 that (EGG) methods outperform the pseudo-spectral method in terms of
accuracy. It is clear from Fig. 15 that for sufficiently large N (100 to 200) that the logarithm base
as a function of grid points N the logarithm decreases while N increases. So (EGG) methods are
converging faster and more accurate than the pseudo-spectral method.
Fig. 12. Absolute error of u and v by using the pseudo-spectral method for N=30 and
1.0=∆t , 5.0=t .
0 0.2 0.4 0.6 0.8 10
1
2
x 10-4
t
m a
x e
r r
U (
t )
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9x 10
-5
t
m a
x e
r r
V (
t )
-100 -50 0 50 1000
1
2
3
4
5
6
7
8x 10
-4
X
a b
s e
r r
U
-100 -50 0 50 1000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
-3
X
a b
s e
r r
V
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
328
Fig. 13. Absolute error of u and v by using El-gendi Galerkin method for N=30 and
1.0=∆t , 5.0=t .
Fig. 14. Logarithm of the maximum error at 5.0=t .
Fig. 15. Logarithm of the maximum error at 1=t and for large N .
-100 -50 0 50 1000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6x 10
-4
x
a b
s e
r r
U
-100 -50 0 50 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
-4
X
a b
s e
r r
V
0 20 40 60 80 100-12
-10
-8
-6
-4
-2
0
2
4
6
8
N
L o
g 1
0 (
m a
x -
n o
r m
) U
p s e u d o
EGG
0 20 40 60 80 100-15
-10
-5
0
5
10
15
N
L o
g 1
0 (
m a
x -
n o
r m
) V
p s e u d o
EGG
100 110 120 130 140 150 160 170 180 190 200-4.048
-4.046
-4.044
-4.042
-4.04
-4.038
-4.036
-4.034
-4.032
N
L o
g 1
0 (
m a
x -
n o
r m
) V
E G G V
F i t
100 110 120 130 140 150 160 170 180 190 200-3.232
-3.231
-3.23
-3.229
-3.228
-3.227
-3.226
-3.225
-3.224
-3.223
N
L o
g 1
0 (
m a
x -
n o
r m
) U
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
329
A practical way to verify that a wave solution is stable is to check if the maximum absolute error
remains, for long times, less than2)( tO ∆ [17]. So, if the errorincreases over this value,
oscillations will soon grow and become unbounded after relatively short times, not only because
of the numerical scheme, but also due to the nonlinear nature of the equations. This is clear from
Fig. 13 that the error of (EGG) methods are still less than 2)( tO ∆ in case N=100 and time step
)(10 1−=∆ Nt . In contrast, the pseudo-spectral method the error becomes very large and the
method fails to satisfy good accuracy. In other words, for a long time 10=t , N=40 and
)(0025.0 1−=∆ Nt we find the maximum error of the El-gendi Galerkin in the u-solution to be
(1.356445e-002) and the maximum error of the pseudo-spectral method is (5.888195e+005) which
is a very big error and refers to the instability of the method.
Example 2. Consider the generalized Hirota-Satsuma coupled KdV equation [18]
,)(332
1xxxxxt uzuuuu +−=
,3 xxxxt uvvv +−=
,3 xxxxt uzzz +−= ],0[, Ttbxa ∈<< ,
with the following exact solution
,3/3/8)(tanh4),( 2
22
2
2 cqkqktxu −−= ξ
,3/43/2)(tanh2),( 02
22
2
2 cqkqktxv −−−= ξ
,2)(tanh2),( 02
22
2
2 cqkqktxz +−= ξ
where )(2 ctxkq −=ξ . In Table (2) we solve this example for 30−=a , 30=b , 1.0=c ,
1.00 =c , 1.02 =q , 1.0=k , 01.0=∆t and 10=N .
Table 2. Comparison between the Maximum errors (Mer) of the pseudo-spectral methodand
ECG method for solving the generalized Hirota-Satsuma coupled KdV equation
N T Method 0.5 1.0 2.0
Mer-pseudo Mer-ECG Mer-pseudo Mer-ECG Mer-pseudo Mer-ECG
10 8.6790E-05 4.8267E-06 1.7359E-04 9.655 e-06 3.4720e-04 1.9318e-05 20 8.7593E-05 4.8664E-06 1.7519E-04 9.733 e-06 3.5040e-04 1.9468e-05 30 8.7482E-05 4.8604E-06 1.7497E-04 9.721 e-06 3.4996e-04 1.9446e-05 40 8.7638E-05 4.8686E-06 1.7528E-04 9.737 e-06 3.5056e-04 1.9474e-05 50 8.7562E-05 4.8647E-06 8.2176E-04 9.730 e-06 6.2498e+11 1.9462e-05 60 6.5429E-03 4.8686E-06 8.3951E+12 9.737 e-06 7.6754e+11 1.9474e-05 70 1.2814E+12 4.8676E-06 1.1556E+12 9.735 e-06 1.1302e+12 1.9472e-05
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
330
It is clear from Table (2) that the error which resulted from pseudo-spectral method is unbounded
when the number of grid points increases. The error of pseudo-spectral method becomes
unbounded for 70=N at a time 5.0=t and when the time increases the unbounded error will
appear for small number of grids. In El-gendi Chebyshev Galerkin (ECG) method the errors are
bounded for large number of grid points N and have the same order. On the other hand, the figures
16, 17 and 18 display the numerical solutions of the proposed problem where the left column
presents the behavior of the solution by the pseudo-spectral method for 45=N and the right
column illustrate the behavior of the numerical solution by using ECG method for the same
number of nodes.
(a) (b)
Fig. 16. (a) Presents the pseudo-spectral of u solution for 45=N and (b) presents the
nodal Galerkin of u solution for 45=N .
(a) (b)
Fig. 17. (a) Presents the pseudo-spectral of v solution for 45=N and (b) presents the
nodal Galerkin of v solution for 45=N .
(a) (b)
Fig. 18. (a) Presents the pseudo-spectral of z solution for 45=N and (b) presents the
nodal Galerkin of z solution for 45=N .
-30-20
-100
1020 30 0
0.5
1
1.5
2
-0.036
-0.0355
-0.035
-0.0345
-0.034
t
x
u p
se
ud
o
-30-20
-100
1020
30 0
0.5
1
1.5
2
-0.036
-0.0355
-0.035
-0.0345
-0.034
t
x
u G
al
-30 -20 -10 0 10 20 30 0
0.5
1
1.5
2
-1.434
-1.433
-1.432
-1.431
t
x
v p
se
ud
o
-30-20
-100
1020
30 0
0.5
1
1.5
2
-1.434
-1.4335
-1.433
t
x
v a
p p
-30-20
-100
1020
30 0
0.5
1
1.5
2
0.098
0.0982
0.0984
0.0986
0.0988
0.099
t
x
z a
p p
-30-20
-100
1020
30 0
0.5
1
1.5
2
0.098
0.0982
0.0984
0.0986
0.0988
0.099
t
x
z G
al
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
331
6. Conclusion
In this paper, two efficient methods depend on nodal Galerkin method are employed to solve the
coupled mKdV and the generalized Hirota-Satsuma coupled KdV equations. A study of the
stability of these methods and the pseudo-spectral method is presented. Numerical results are
given for long times and for a large number of grid points. EGG methods have smaller errors than
pseudo-spectral method and lead to stable approximations. Stability guarantees that the solution
remains bounded as N approaches infinity.
Acknowledgements
The authors are very grateful to the referees for carefully reading the paper and for their comments
and suggestions which have improved the paper.
Competing Interests The authors declare that no competing interests exist.
References
[1] Wu YT, Geng XG, Hu XB, Zhu SM. A generalized Hirota–Satsuma coupled Korteweg–de
Vries equation and Miura transformations, Phys. Lett. A. 1999;255:259-264.
[2] Zayed EM, Nofal TA, Gepreel KA. On using the homotopy perturbation method for finding
the travelling wave solutions of generalized nonlinear Hirota- Satsuma Coupled KdV
equations. International Journal of Nonlinear Science. 2009;7:159-166.
[3] Raslan KR. The decomposition method for a Hirota-Satsuma coupled KdV equation and
acoupled MKdV equation, Intern. J. Compu. Math. 2004;81:1497-1505.
[4] Fan EG. Soliton solutions for a generalized Hirota Satsuma coupled KdV equation and a
coupled MKdV equation, Physics Letters A. 2001;282:18-22.
[5] Guo-Zhong Z, Xi-Jun Y, Yun X, Jiang Z, Di W. Approximate analytic solutions for
ageneralized Hirota Satsuma coupled KdV equation and a coupled mKdV equation, Chin.
Phys. B. 2010;19.
[6] Kangalgil F, Ayaz F. Solitary wave solutions for Hirota-Satsuma coupled KdV equation
and coupled mKdV equation by differential transformation method. The Arabian Journal
for Science and Engineering. 2010;35:203-213.
[7] Arife AS, Vanani SK, Yildirim A. Numerical solution of Hirota-satsuma couple Kdv and a
coupled MKDV equation by means of homotopy analysis method. World Applied Sciences
Journal. 2011;13: 2271-2276.
El-Kady et al.; BJMCS, 5(3): 310-332, 2015; Article no.BJMCS.2015.021
332
[8] Ghoreishi M, Ismail A, Rashid A. The solution of coupled modified KDV system by
thehomotopy analysis method, TWMS Jour. Pure Appl. Math. 2012;3:122-134.
[9] El-gendi SE. Chebyshev solutions of Differential, Integral, and Integro-Differential
Equations. Computer Journal. 1969;12:282-287.
[10] Gottlieb D, Hesthaven JS. Spectral methods for hyperbolic problems. Journal of
Computational and Applied Mathematics. 2002;128:186-221.
[11] Canuto C, Hussaini M, Quarteroni A, Zang T. Spectral Methods Fundamentals in Single
Domains, Springer Verlag; 2006.
[12] Trefethen LN, Trummer MR. An instability phenomenon in spectral methods, SIAM J.
Numer. Anal. 1987;24:1008-1023.
[13] Trefethen LN. Spectral Methods in MATLAB, SIAM, Philadelphia; 2000.
[14] Chen F, Shen J. Efficient spectral-Galerkin methods for systems of coupled second-order
equations and their applications. Journal of Computational Physics. 2012;231:5016-5028.
[15] Elbarbary ME, El-Sayed M. Higher order pseudospectral differentiation matrices. Applied
Numerical Mathematics. 2005;55:425-438.
[16] El-Kady M, Biomy M. Interactive Chebyshev – Legendre Approach for Linearquadratic
optimal regulator systems. IJWMIP. 2011;9:459-483.
[17] Tzirtzilakis EE, Skokos CD, Bountis TC. Numerical solution of the Boussinesq equation
using spectral methods and stability of solitary wave propagation. 1st International
Conference “From Scientific Computing to Computational Engineering”, (III). 2005;928-
933.
[18] Khater AH, Temsah RS, Callebaut DK. Numerical solutions for some coupled nonlinear
evolution equations by using spectral collocation method. Mathematical and Computer
Modelling. 2008;48:1237-1253.
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