Applied Mathematics, 2011, 2, 414-423
doi:10.4236/am.2011.24051 Published Online April 2011 (http://www.SciRP.org/journal/am)
RBFs Meshless Method of Lines for the Numerical
Solution of Time-Dependent Nonlinear Coupled
Partial Differential Equations
Sirajul Haq1, Arshad Hussain1, Marjan Uddin2,3*
1Faculty of Engineering Sciences, Ghulam Ishaq Khan Institute of Engineering Sciences and Technology,
Topi, Pakistan
2Department of Applied Mathematics, Illinois Institute of Technology, Chicago, USA
3Department of Mathematics, Karakoram International University, Gilgit, Pakistan
Received December 25, 2010; revised February 3, 2011; accepted February 6, 2011
Abstract
In this paper a meshless method of lines is proposed for the numerical solution of time-dependent nonlinear
coupled partial differential equations. Contrary to mesh oriented methods of lines using the finite-difference
and finite element methods to approximate spatial derivatives, this new technique does not require a mesh in
the problem domain, and a set of scattered nodes provided by initial data is required for the solution of the
problem using some radial basis functions. Accuracy of the method is assessed in terms of the error norms L2,
L∞ and the three invariants C1, C2, C3. Numerical experiments are performed to demonstrate the accuracy and
easy implementation of this method for the three classes of time-dependent nonlinear coupled partial diffe-
rential equations.
Keywords: RBFs, Meshless Method of Lines, Time-Dependent PDEs
1. Introduction
Nonlinear coupled partial differential equations have nu-
merous applications in the field of science and engineer-
ing, including solid state physics, fluid mechanics, che-
mical physics, plasma physics etc. (see [1-3] and the ref-
erences therein). In 1981 Hirota-Satsuma introduced the
coupled KdV equations, [4] which has many applications
in physical sciences. Coupled KdV equations describe an
interaction of the two long waves with different disper-
sion relation. The Burgers’ equations describe phenomena
such as a mathematical model of turbulence [5] and the
nonlinear hyperbolic system [6] represents interaction of
the two waves traveling in the opposite directions.
In the last decade many authors have studied the nu-
merical and approximate solution of time-dependent
nonlinear coupled partial differential equations by vari-
ous numerical methods. These include Adomian decom-
position method [7], the local discontinuous Galerkin
method [8], the variational iteration method [9], the
Chebyshev spectral collocation method [10] and the
radial basis functions method [6,11,12].
In the last decade, the theory of radial basis functions
(RBFs) has enjoyed a great success as scattered data in-
terpolating technique. A radial basis function,




jj
x
xxx   , is a continuous spline which de-
pends upon the separation distances of a subset of data
centers, n
X ,


,1,2,,
j
x
Xj N. Due to sphe-
rical symmetry about the centers
j
x
, the RBFs are called
radial. The distances,
j
x
x, are usually taken to be
the Euclidean metric.
Hardy [13] was the first to introduced a general scat-
tered data interpolation method, called radial basis func-
tions method for the approximation of two-dimensional
geographical surfaces. In 1982 Franke [14] in a review
paper made the comparison among all the interpolation
methods for scattered data sets available at that time, and
the radial basis functions outperformed all the other me-
thods regarding efficiency, stability and ease of imple-
mentations. Franke found that Hardy’s multiquadrics
(MQ) were ranked the best in accuracy, followed by thin
plate splines (TPS). Despite MQ’s excellent performance,
it contains a shape parameter c, and the accuracy of MQ
is greatly affected by the choice of shape parameter c
whose optimal value is still unknown. Franke [15] used
the formula

2
22
1.25cd where d is the mean dis-

S. HAQ ET AL. 415
tance from each data point to its nearest neighbor. Hick-
ernell and Hon [16] and Golberg et al. [17] had success-
fully used the technique of cross-validation to obtain an
optimal value of the shape parameter. Various research-
ers have contributed recently to this field (see [18-25]
ect.)
The method of lines (MOL) [26] is a general proce-
dure for the solution of time dependent partial differen-
tial equations (PDEs). The basic idea of the MOL is to
replace the spatial (boundary value) derivatives in the
PDEs with algebraic approximations. Once this is done,
the spatial derivatives are no longer stated explicitly in
terms of the spatial independent variables. Thus only the
initial value variable, typically time in a physical prob-
lem, remains. In other words, we have a system of ODEs
that approximate the original PDE. Now we can apply
any integration algorithm for initial value ODEs to com-
pute an approximate numerical solution to the PDE. Thus,
one of the salient features of the MOL is the use of ex-
isting, and generally well established, numerical methods
for ODEs. Very recently Quan Shen [27] use this ap-
proach for the numerical solution of KdV equation. In
this paper, we will use RBFs approximation method with
the method of lines (MOL) for the numerical solution of
time-dependent nonlinear partial differential equations
given as:
Class A: Coupled KdV equations
62,
txxx xx
uu uuvv

  
3,
txxx x
vv uv

  
Class B: Coupled Burgers’ equations

2,
txx x
x
u uuuuv   

2,
txx x
x
vv vvuv   
Class C: System of nonlinear hyperbolic equations
,
tx
uuuv

 
,
tx
vv uv


where ,,

 are positive parameters.
Rest of the paper is organized as follows: In Section 2,
The radial basis functions collocation method coupled
with MOL is presented. Section 3 is devoted to the nu-
merical tests of the method on the problems related to the
coupled KdV, the coupled Burgers’ equations and the
system of nonlinear hyperbolic equations. In Section 4,
the results are concluded.
2. RBFs Meshless Method of Lines
2.1. Coupled KdV Equations
Consider the nonlinear coupled KdV equations,
62,
3,
t xxxxx
t xxxx
uu uuvv
vv uv


 
  (1)
with boundary conditions



 
 
12
12
,, ,,
,,,,0,
uat ftubt ft
vatgtvbt gtt



(2)
and initial conditions


 
,0 ,
,0 ,,
uxfx
vxgx ax b


 (3)
where ,,

 are positive real constants.
For a given set of N collocation points

1
N
ii
x

in
the domain


,ab, the RBFs approximation for u and
v of (1) are given by
 
12
11
,,
NN
jj
jj
Uxr Vxr  



(4)
where


1
N
j
j


are the unknown constants to be deter-
mined, 


j
j
rxx   can be any well known ra-
dial basis function and
j
j
rxx is the Euclidean
norm between points
x
and .
j
x
Here we are using two
radial basis functions, the multiquadric 
22
rrc 


and cubic


3.rr
  Now for each node
, 1,2,3,,
i
x
iN

 in the domain


,ab , (4) can be
written as
12
,,

UAVA

 (5)
where
   
123
,,,, T
N
Ux UxUxUx





U

123
,,,,, 1,2.
T
ii iiNi
  
i




 
 
 
11 211 1
12 12 1
11
T
N
T
T
NN NN
N
x
xx x
xx x
xx x x
 

 

















 





 

A








 
123
,,,, ,
T
iiiiNi
xxxxx  

 



where 1,2, 3,,iN

.
Equation (4) can also be written as




 
1
1
,
T
T
Ux x
Vx x




A
UDxU
A
VDxV

 (6)
where






1
12
,,, .
T
N
x
DxD xDx






Dx A
Using the approximations

i
Ut and


i
Vt of the

S. HAQ ET AL. 416
solutions

,
i
uxt and

,
i
vxt given in (6), (1) at each
node , 1,2,3,,
i
x
iN can be written as
  
 
  
d6
d
2
d3
d
i
xxx iix i
ixi
i
xxx iix i
UDx UDx
t
VD x
VDx UDx
t



 

 
UU
V
VV

(7)
where
 


12
,,, ,
ixixiNxi
DxD xDx

 
x
Dx
 
,1,2,,,
jx ij i
DxDxjN
x



 


12
,,,
ixxx ixxx iNxxx i
DxDxDx

 
xxx
Dx
 
3
3,1,2,,.
jxxx iji
DxDxjN
x



In more compact form (7) can be written as
 

d62,
d
d3,
d
xxx xx
xxx x
t
t
 


 
U
D
UUDU VDV
VDVU DV
(8)
where the symbol * denotes component by component
multiplication of two vectors and
 
,1 ,1
,.
NN
xjxi xxxjxxxi
ij ij
DxD x

 

DD
For simplicity we write (8) as
 
12
dd
,, ,,
dd
FF
tt

UV
UVUV (9)
where
 


1,62, xxx xx
F   UVD UUDUVDV
 
2,3. xxx x
F   UVD VUDV
The corresponding boundary conditions are given by
 
 
11 2
11 2
,
,,
N
N
UftU ft
VgtVgt


(10)
and the initial conditions are as
   

  
012
012
,,,
,,,
N
N
tfxfx fx
tgxgxgx






U
V
(11)
Now we solve the system of ODEs (9)-(11) by using
the well known ODE solvers.
1) The classical forth order Runge-Kutta method (RK4)
given by

1
12 34
δ22 ,
6
nn
tKK KK
 UU

1
1234
22 ,
6
nn
t
J
JJJ

 
VV
where


1121 1
,, 2,,
nnn n
KFK FtKUVUV

 
31 241 3
δ2, ,, ,
nn nn
K FtKKFtK UV UV


12 221
,,, 2,
nn nn
JFJ FtJUVUV

 
322 423
,δ2,, δ,
nn nn
J
FtJJFtJ UVUV
2) Low-storage third-order (TVD-RK3) scheme given
by [28]


1
1
δ,,
nnn
tFUU UV
  

211
1
31 1
δ,,
44 4
nn
tF UUU UV
  

122
1
12 2
δ,,
333
nnn
tF
UUU UV


1
2
δ,,
nnn
tFVV UV
  

21 1
2
31 1
δ,,
44 4
nn
tF VVV UV
  

12 2
2
12 2
δ,.
33 3
nnn
tF
 VVV UV
2.2. Nonlinear Coupled Burgers’ Equations
Consider the nonlinear coupled Burgers’ equations


2,
2,
txx x
x
txx xx
u uuuuv
vv vvuv


 
  (12)
with the boundary conditions



 
 
12
12
,, ,,
,,,,0,
uat ftubt ft
vatgtvbt gtt



(13)
and initial conditions


 
,0 ,
,0 ,,
uxfx
vxgx ax b


 (14)
where ,,

 are positive parameters. The same proce-
dure as discussed in Section 2.1 can be used for the solu-
tion of (12)-(14).
2.3. Nonlinear Coupled Hyperbolics
Consider the nonlinear hyperbolic system
,,
tx tx
uuuv vvuv



  (15)
with boundary conditions

S. HAQ ET AL. 417
 

12
12
,,, ,
,,,,0,
uat ftubt ft
vatgtvbt gtt


(16)
and initial conditions
 
 
,0 ,
,0 ,,
uxfx
vxgxaxb


(17)
where

is a positive real constant. The same proce-
dure as discussed in Section 2.1 can be used for the solu-
tion of (15)-(17).
3. Numerical Examples
In this section, we apply the RBFs meshless method of
lines for the numerical solution of three classes of partial
differential equations, defined earlier. We use the 2 L
and L error norms to measure the difference between
the numerical and analytic solutions. The 2 L and L

error norms of the solution are defined by

12
2
221
δ,
max ,
N
j
j
LuU xuU
LuU uU



 


 
 (18)
We examine our results by calculating the following
three conservative laws. Hirota-Satsuma [4] proved that
the coupled KdV equations defined in (1) possesses three
conserved quantities for all values of

  and .


 



1
22
2
22
32
3
d,
2d
3
1
1d.
2
b
a
b
a
b
xx
a
Cux
Cu vx
Cauuuvvx





 





(19)
Later Hirota-Satsuma [29] showed that the system (7)
has infinitely many conserved quantities for the choice of
12

 and arbitrary values of

and .

In our in-
vestigation we consider the conserved quantities 1,C
2
C and 3 C only. In this section, we apply meshless
MOL using radial basis functions on the three classes of
partial differential equations defined earlier.
Problem 1 We consider the nonlinear coupled KdV
Equations (1) for v = 3 α = β and with exact solution [20]
 
 
2
2
1
,sec ,
2
1
,sec.
2
2
uxthx t
vxthx t


















(20)
where ,

 are arbitrary constants. The boundary con-
ditions


,,uat


,,ubt

,vat and

,vbt and the
initial conditions

 
,0 ,,0uxvx are extracted from the
exact solution (20). We solved the problem in the spatial
interval 55x

 by RBFs meshless method of lines
using RK4 and TVD-RK3 time integration schemes. In
our computations we used multiquadric (MQ) radial ba-
sis function. The results are presented in Tables 1-4, and
in Figure 1. It is observed that the two schemes RK4 and
TVD-RK3 show same order of accuracy, but TVD-RK3
scheme is more faster than RK4 scheme, and both re-
mained stable for small time step size δt It is also ob-
served that the three invariants 1,C 2
C and 3 C as
well as their normalized values,



 
11 11
00,NCC tCC



 
22 22
00,NCC tCC



 
33 33
00,NCC tCC
are absolutely conserved in time during the computations
which demonstrates the accuracy of the schemes. We also
noted that the value of MQ shape parameter for which the
solution converges belongs to the interval 0.1 0.6c


as shown in Table 3. The motion of solitary waves u
and v is shown in Figure 1, which are initially centered
at 0x

moving from left to right with the constant
speed ,

having the amplitudes

 and2


respectively.
Problem 2 Consider the nonlinear coupled Burgers’
Equations (12), whose exact solution [10] is given by
 
 
0
0
21
,2 tanh2,
41
21 21
,2tanh2.
2141
uxtaAAxAt
vxt aAAx At






  




 

 


 



(21)
where





 
0
12412 1,Aa  

 0,a ,


are arbitrary constants.
The boundary conditions

,,uat

,,ubt


,,vat


,vbt and the initial conditions

,0 ,ux


,0vx are
extracted from the exact solution (21). We solved the
problem in the domain 10 10x

 by using MOL
coupled with RBFs collocation method. The classical
RK4 and TVD-RK3 scheme are used in our computa-
tions. The results are listed in Table 5 and Figure 2, and
compare with earlier results [10]. It is observed that the
results are comparable with [10] and well agreed with the
exact solution.
Problem 3 Now we consider nonlinear coupled hyper-
bolic Equations (15). For the sake of comparison [6], we
take 0.5,a

 0.5,b

100

 and the initial condi-
tions.

S. HAQ ET AL. 418
(a) (b)
(c) (d)
Figure 1. Plots of Problem 1 corresponding to α = 0.1, β = 0.1, λ = 0.5, δx = 0.1, δt = 0.001, c = 0.58. Figures 1(a) and 1(b) show
the motion of the solitary waves u and v moving from left to right, initially centered at x = 0 when t = 0,1,2,3,4,5. Figures 1(c)
and 1(d) represent numerical solutions u and v over time [0,5].
 


0.51cos10,0.3,0.1;
,0
0, otherwise
xx
ux






(22)
 


0.51cos10,0.1,0.3 ;
,0
0, otherwise
xx
vx







(23)
and the boundary conditions



 
,,0,
,,0.
uat ubt
vat vbt


(24)
This problem is solved by RBFs meshless method of
lines using MQ with RK4 scheme. We take the initial so-
lutions v and

,0vx which are located at 0.2x


and 0.2,x respectively. When 0t the nonlinear
term, uv , causes these waves to move without change in
shape, u to the right and v to the left. The two waves
collide when 0.1t which results in change of shapes
of the waves. The two waves overlap each other near
0.25t

and they separate again at 0.3t approx-
imately. From this time onwards the linear term becomes
dominant and the pulses lose their symmetry and expe-
rience a decrease in the amplitude due to nonlinear inte-
raction as shown in the Figures 3 (a)-3(f). The numerical
results of the solutions u and v are presented graphi-
cally. Since the exact solution of this problem is not
known, we use cubic radial basis function 3 r to find the
numerical solution. These graphical results are agreed
well with the results obtained by quasi-linear interpola-
tion method [6].
4. Closure
We have applied the meshless method of lines using
radial basis functions for the numerical solutions of time-
dependent nonlinear coupled partial differential equa-

S. HAQ ET AL. 419
Table 1. Error norms and the three invariants for the solutions u, v using MQ when δt = 0.001, N = 100 c = 0.53. α = 0.01, β =
0.01 and λ = 0.01 corresponding to Problem 1.
t 
L
u


2
L
u


L
v



2
L
v 1 C 2 C 3 C
RK4
0.1 9.625E–05 3.676E–05 6.807E–06 2.601E–06 3.94906 2.69268 1.90965
1 3.368E–05 3.990E–05 2.373E–06 2.829E–06 3.94906 2.69268 1.90965
5 1.040E–04 1.471E–04 7.392E–06 1.040E–05 3.94896 2.69268 1.90966
10 2.514E–04 4.711E–04 1.791E–05 3.323E–05 3.94867 2.69267 1.90967
15 4.427E–04 1.003E–03 3.141E–05 7.109E–05 3.94819 2.69265 1.90968
20 6.962E–04 1.681E–03 4.916E–05 1.190E–04 3.94754 2.69261 1.90969
TVD-RK3
0.1 9.627E–05 3.676E–05 6.807E–06 2.601E–06 3.94906 2.69268 1.90965
1 3.353E–05 3.977E–05 2.373E–06 2.831E–06 3.94906 2.69268 1.90965
5 1.046E–04 1.459E–04 7.392E–06 1.041–05 3.94896 2.69267 1.90965
10 2.535E–04 4.675E–04 1.791E–05 3.326E–05 3.94867 2.69266 1.90965
15 4.446E–04 1.003E–03 3.141E–05 7.117E–05 3.94819 2.69263 1.90965
20 6.952E–04 1.679E–03 4.916E–05 1.191E–04 3.94754 2.69258 1.90965
Table 2. The three invariants and its normalized invariants for the solutions u, v using MQ, when, δt = 0.001, N = 100, c1 =
0.53, c2 = 0.53, α = 0.05, β = 0.05, λ = 0.05 corresponding to Problem 1.
t 1 C 2 C 3 C 1 NC 2 NC 3 NC

Amp u


Amp v
RK4
0.1 3.94906 2.79932 2.08037 1.248E–07 1.199E–08 4.186E–08 1.000 0.158
0.3 3.94905 2.79932 2.08037 1.570E–06 6.969E–07 2.310E–06 1.000 0.158
0.5 3.94903 2.79933 2.08038 6.670E–06 1.119E–06 3.999E–06 1.000 0.158
1 3.94896 2.79933 2.08039 2.409E–05 1.745E–06 8.336E–06 0.999 0.158
3 3.94819 2.79930 2.08043 2.195E–04 7.809E–06 2.828E–05 0.999 0.158
5 3.94671 2.79922 2.08046 5.954E–04 3.697E–05 4.491E–05 1.000 0.158
Table 3. Error norms and normalized invariants of the solutions u, v for different values of MQ shape parameter c when t =
1.0, δt = 0.001, N = 100, α = 0.1, β = 0.1, λ = 0.5 corresponding to Problem 1.
c


L
u



L
v
 1
NC 2 NC 3 NC
RK4
0.1 1.492E–01 3.101E–02 9.986E–05 3.112E–04 3.956E–03
0.2 8.632E–03 5.410E–04 2.656E–05 3.201E–04 7.502E–04
0.3 7.789E–03 4.934E–04 1.598E–05 3.202E–04 7.240E–04
0.4 7.795E–03 4.960E–04 9.505E–06 3.202E–04 7.239E–04
0.5 7.785E–03 4.944E–04 6.102E–06 3.202E–04 7.240E–04
0.6 7.816E–03 4.846E–04 7.137E–06 3.200E–004 7.236E–04

S. HAQ ET AL. 420
Table 4. Error norms and normalized invariants of the solutions u, v for different values of time step size δt when t = 1.0, c =
0.53, N = 100, α = 0.1, β = 0.1, λ = 0.5 corresponding to Problem 1.
δt


L
u



L
v
 1
NC 2 NC 3 NC
RK4
0.001 7.804E–03 4.932E–04 5.485E–06 3.203E–04 7.241E–04
0.0005 3.880E–03 2.458E–04 4.615E–06 9.281E–05 2.108E–04
0.0001 7.774E–04 6.294E–05 3.709E–06 8.735E–05 1.956E–04
0.00005 3.919E–04 6.257E–05 3.817E–06 1.098E–04 2.461E–04
0.00001 2.779E–04 6.230E–05 3.745E–06 1.277E–04 2.865E–04
Table 5. Error norms of solutions u and v v using MQ, corresponding to Problem 2.
L

t δt δ
x


a b 0 a c RK4 TVD-RK3 ChSC[10]
U
0.5 0.001 0.25 0.1 0.3 –10 10 0.05 0.58 4.169E–05 4.169E–05 4.16E–05
1.0 8.243E–05 8.243E–05 8.23E–05
0.5 0.001 0.25 0.3 0.03 –10 10 0.05 0.58 4.591E–05 4.591E–05 4.59E–05
1.0 9.183E–05 9.183E–05 9.16E–05
V
0.5 0.001 0.25 0.1 0.3 –10 10 0.05 0.58 2.157E–05 2.157E–05 2.19E–05
1.0 4.166E–05 4.166E–05 4.10E–05
0.5 0.001 0.25 0.3 0.03 –10 10 0.05 0.58 1.809E–04 1.809E–04 1.80E–04
1.0 3.617E–04 3.617E–04 3.59E–04
(a) (b)
Figure 2. Comparison of numerical solution using RK4 with MQ versus exact solutions u, v at time t = 1 w hen α = 1, β = 2, a0 =
0.1, c = 0.58 and δt =0.001, corresponding to Problem 2.
tions. Two time integration schemes RK4 and TVD-RK3
are used. The method is stable, efficient and very easy in
implementation. A large class of time-dependent nonli-
near partial differential equation can be solved by this

S. HAQ ET AL. 421
(a) (b)
(c) (d)
(e) (f)
Figure 3. Numerical solution using quintics. Figures 3(a)-3(f) show the motion and interaction of the waves u and v.

S. HAQ ET AL. 422
technique.
5. Acknowledgements
The work presented in this paper is supported by HEC
Pakistan through grants 063-281079-Ps3-169, IRSIP 14
Ps3.
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