Third Order Newton-Cotes Integration Rule for Solving Goursat
Partial Differential Equation
ROS FADILAH DERAMAN1, MOHD AGOS SALIM NASIR
1, SITI SALMAH YASIRAN
1, MOHD
IDRIS JAYES1
1Faculty of Computer and Mathematical Sciences
Universiti Teknologi MARA Malaysia
40450 Shah Alam
MALAYSIA
[email protected]; [email protected]; [email protected]
Abstract: - The Goursat partial differential equation is a second order hyperbolic partial differential equation
which arises in various fields of study. There are many approaches have been suggested to approximate the
solution of the Goursat partial differential equation such as finite difference method, Runge-Kutta method,
finite element method, differential transform method, and modified variational iteration method. All of the
suggested methods traditionally focused on numerical differentiation approaches include forward and central
difference in deriving the schemes. In this paper we have developed a new scheme to solve the Goursat partial
differential equation that has applied Adomian decomposition method (ADM) and associated with numerical
integration method based on Newton-cotes formula of order three to approximate the integration terms. Several
numerical tests to the new scheme indicate encouraging results.
Key-Words: - Goursat problem, Partial differential equation, Adomian decomposition method, Numerical
integration, Newton-Cotes formula, Accuracy.
1 Introduction ADM was introduced and developed by [1], is an
elegant method which has been proved its powerful,
effectiveness, efficiency and can easily handle a
wide class of linear or non linear, ordinary or partial
differential equation, and integral equation. The
method is successfully implemented to establish an
analytical solution and a reliable numerical
approximation to the Goursat partial differential
equation that appear in several physic models and
scientific applications.
Numerical integration is used to describe the
numerical solution of differential equation. [4]
explained the significance of numerical integration
in the reformulation of mathematical problems,
convert mathematical problems to ordinary or
partial differential equation into algebraic equation,
calculate the integral transform, fundamental
computation technique and applied statistical
computation. The Newton-Cotes formula is a group
of numerical integration methods based on the
evaluating integral at equal subinterval and efficient
in solving several numerical problems.
The Goursat problem is an initial value problem
involving a partial differential equation hyperbolic
type that consists of two independent variables
arises in several areas of physics and engineering.
There are various applications involving Goursat
problem such as in supersonic flows, reacting gas
flow, trajectory generation for the N-Trailer, sonic
barrier, steering of mobile robots, isotropic plate and
micro differential operator studied by the numerous
researchers such as [8], [3], [10], [7], [6], [5], and
[13] respectively. These models, usually relating
space and time derivatives, need to be solved in
order to gain a better insight into the underlying
physical problem. Many of these equations are such
that analytical methods cannot be utilized and
numerical approximations need to be used.
Recently a great deal of interest has been focused
on the numerical method for solving Goursat
problem include Runge-Kutta [9], Trapezoidal
Computational Methods in Science and Engineering
ISBN: 978-1-61804-174-6 62
formula [11], finite difference [15], two dimensional
differential transform [14] and variational iteration
method [17] have been used to investigate this
problem. However it is known that many of the
techniques focus on numerical differentiation
method in deriving the schemes. In this paper we
develop a new scheme by using ADM and
associated with a Newton-cotes integration formula
to solve (linear, derivative linear and non linear)
Goursat partial differential equations. Study its
implementation and compare the accuracy results
with the existing scheme in the literature.
2 The Goursat Problem and ADM The Goursat problem arises in linear and nonlinear
partial differential equation with mixed derivatives.
The standard form of Goursat problem [15]:
byax
hgu
yhyuxgxu
uuuyxfu yxxy
≤≤≤≤
==
==
=
0,0
)0()0()0,0(
)(),0(),()0,(
),,,,(
(1)
The established finite difference scheme is based on
arithmetic mean averaging of functional values and
is given by [15]:
)(
4
11,,1,1,1
2
1,,1,1,1
++++
++++
+++=
−−+
jijijiji
jijijiji
ffff
h
uuuu
…(2)
where h denotes the grid size.
If the ADM is used, we obtain [16]:
Let the linear operators as,
yL
xL yx ∂
∂=
∂
∂= , (3)
By the operators (3), left hand side of the standard
Goursat problem (1) becomes:
),,,,(),( yxyx uuuyxfyxuLL = (4)
Let the operators 1−xL and 1−
yL as a definite integrals
from 0 to x and from 0 to y respectively, i.e.:
( ) ( ) ( ) ( )∫ ∫== −−x y
yx dyLdxL
0 0
11 ..,.. (5)
which means that,
[ ] ),,,,(),( 11yxyyyx uuuyxfLyxuLLL −− =
(6)
Then, equation (6) becomes
[ ] ),,,,()0,(),(1
yxyx uuuyxfLxuyxuL−=−
(7)
where,
)0,(),(),(
1xuyxuyxuLL yy −=−
(8)
or equally
),,,,()0,(),(1
yxyxx uuuyxfLxuLyxuL−+=
(9)
Then, the multiplication with 1−xL of the both sides
(9) will yield:
),,,,()0,(
),(
111
1
yxyxxx
xx
uuuyxfLLxuLL
yxuLL
−−−
−
+=
…(10)
Now substitute
),0(),(),(1 yuyxuyxuLL xx −=− and
)0,0()0,()0,(1 uxuxuLL xx −=− (11)
into equation (10), then
),,,,()0,0()0,(
),0(),(
11yxyx uuuyxfLLuxu
yuyxu
−−+−=
−
…(12)
The ADM derivation of equation (1) given as
follows:
Computational Methods in Science and Engineering
ISBN: 978-1-61804-174-6 63
),,,,()0,0(),0()0,(
),(
11yxyx uuuyxfLLuyuxu
yxu
−−+−+=
…(13)
equation (13) can be write as:
∫ ∫+ +
+−
+++=++hx
x
hy
y
yx00
000000
0
0
0
0
)dydxu,uu,y,f(x,)y,u(x
h)y,u(x)yh,u(xh)yh,u(x
(14)
3 ADM Associated With Numerical
Integration Rule for the Goursat
Problem By indexing the independent variables, equation
(14) becomes:
∫ ∫+ +
++++ +−+=ni
i
nj
j
ji,ji,nji,j,ninj,ni djdifuuuu
(15)
where n is a number of order in the Newton-Cotes
formula.
The Newton-Cotes formula of order three with triple
segment is as follows [2]:
[ ]f(b)h)f(ah)f(af(a)hdxf(x)
b
a
+++++=∫ 2338
3
…(16)
By letting 3n = into equation (15), we get:
∫ ∫+ +
++++ +−+=3 3
3333
i
i
j
j
i,ji,ji,j,ji,ji djdi.fuuuu
…(17)
Utilize the rule (16) to approximate the double
integral in the scheme (17) to get,
[ ]
( )
( )
( )
( )
+++
++++
+
++
+++++=
+++
++++
++++
++++
=
+++=
+++++++
+++++++
++++++
++++
+++++++
+++++++
+++++++
+++
+++
+++
∫∫∫
2322212
13121112
333231
33212
3332313
2322212
1312111
321
321
333
33
33
64
27
33
33
64
9
338
3
338
9
338
9
338
3
8
3
338
3
,ji,ji,jii,j
,ji,ji,jii,j
,ji,ji,ji
i,j,ji,ji,jii,j
,ji,ji,jii,j
,ji,ji,jii,j
,ji,ji,jii,j
,ji,ji,jii,j
i,ji,ji,ji,j
i
i
j
j
i,j
i
i
ffff
ffffh
fff
fffffh
ffffk
ffffk
ffffk
ffffk
h
diffffh
didjf
...(18)
where kh =
Substitute approximation (18) into scheme (17).
Then, the new scheme can be written as,
++
+++++
+
+
++
++
++
+−+=
++++++
++++++++
++++
+++
++
+
++++
232221
213121112
3332
313
32
1
2
3333
33
33
64
27
3
3
3
3
64
9
,ji,ji,ji
i,j,ji,ji,jii,j
,ji,ji
,jii,j
,ji,ji
,jii,j
i,ji,j,ji,ji
fff
fffffh
ff
ff
ff
ff
huuuu
…(19)
4 Numerical Illustrations We consider the linear Goursat problems:
( )( )
210210
0
0
.y,.x
ey,u
ex,u
uu
y
x
xy
≤≤≤≤
=
=
=
(20)
The analytical solution is yxeyxu +=),( , given by
[16]. The MATLAB programs for the application of
the new scheme (19) and standard scheme (2) was
developed. The graphs and results presented below
are approximate solutions and relative errors at
several selected grid points respectively. For the
linear Goursat problem (20) we obtained:
Computational Methods in Science and Engineering
ISBN: 978-1-61804-174-6 64
Fig.1: Solution of problem (20) in graphical form
using scheme (19) at h = 0.05
Table 1: Relative errors for scheme (19) at 05.0=h
x
y 0.3 0.6 0.9 1.2
0.3 3.44918
42 x10-7
8.93437
03 x10-7
1.52546
42 x10-6
2.16744
14 x10-6
0.6 8.93437
03 x10-7
1.92833
67 x10-6
2.99575
31 x10-6
4.02854
46 x10-6
0.9 1.52546
42 x10-6
2.99575
31 x10-6
4.37485
35 x10-6
5.64212
06 x10-6
1.2 2.16744
14 x10-6
4.02854
46 x10-6
5.64212
06 x10-6
7.04996
31 x10-6
Average relative error for scheme (19) = 1.2147999
x10-5
Table 2: Relative errors for scheme (2) at 05.0=h
x
y 0.3 0.6 0.9 1.2
0.3 2.85671
86 x10-5
5.06442
01 x10-5
6.76965
13 x10-5
8.08610
35 x10-5
0.6 5.06442
01 x10-5
9.12579
88 x10-5
1.23770
98 x10-4
1.49756
41 x10-4
0.9 6.76965
13 x10-5
1.23770
98 x10-4
1.70070
01 x10-4
2.08185
74 x10-4
1.2 8.08610
35 x10-5
1.49756
41 x10-4
2.08185
74 x10-4
2.57532
28 x10-4
Average relative error for scheme (2) = 8.5630853
x10-5
Table 3: Average relative errors for grid size
010.0and020.0,025.0,040.0=h
0.040 0.025 0.020 0.010
Scheme
(19) 6.17559
52 x10-6
1.49110
05 x10-6
7.60550
17 x10-7
9.62674
01 x10-8
Scheme
(2)
5.40328
95 x10-5
2.06585
62 x10-5
1.31265
15 x10-5
3.32988
36 x10-6
We consider the derivative linear Goursat problem
below:
4.20,4.20
1),0(
1)0,(
1
≤≤≤≤
+−=
+−=
++−=
yx
eyu
exu
uyu
y
x
xxy
(21)
The analytical solution is yxexyyxu ++−−= 1),(
can be found in [12].
We developed MATLAB programs for the
application of standard scheme (19) and new
scheme (2) to derivative linear Goursat problem
(21). The graphs and results presented below are
approximate solutions and relative errors at several
selected grid points respectively.
Fig.2: Solution of problem (21) in graphical form
using scheme (19) at 040.0=h
00.25
0.50.75
00.5
11.5
0
5
10
15
u
00.8
1.62.4
2.8
0
0.8
1.6
2.4
3.2
0
20
40
60
80
100
120
xy
u
Computational Methods in Science and Engineering
ISBN: 978-1-61804-174-6 65
Table 4: Relative errors for scheme (19) at
040.0=h
x
y 0.6 1.2 1.8 2.4
0.6 2.07401
26 x10-6
2.64981
85 x10-6
2.79208
12 x10-6
2.80885
48 x10-6
1.2 2.64981
85 x10-6
3.68483
93 x10-6
4.06701
08 x10-6
4.17611
29 x10-6
1.8 2.79208
12 x10-6
4.06701
08 x10-6
4.55465
07 x10-6
4.76709
76 x10-6
2.4 2.80885
48 x10-6
4.17611
29 x10-6
4.76709
76 x10-6
5.00679
47 x10-6
Average relative error for scheme (19) = 7.5780008
x10-6
Table 5: Relative errors for scheme (2) at 040.0=h
x
y 0.6 1.2 1.8 2.4
0.6 6.11562
58 x10-5
7.81350
04 x10-5
8.23298
95 x10-5
8.28244
95 x10-5
1.2 7.81350
04 x10-5
1.43639
60 x10-4
1.58537
12 x10-4
1.62790
06 x10-4
1.8 8.23298
95 x10-5
1.58537
12 x10-4
2.26659
50 x10-4
2.37231
80 x10-4
2.4 8.28244
95 x10-5
1.62790
06 x10-4
2.37231
80 x10-4
3.08234
74 x10-4
Average relative error for scheme (2) = 1.2239002
x10-4
Table 6: Average relative errors for grid size
0100and016002000250 ..,.,.h =
0.025 0.020 0.016 0.010
Scheme
(19) 1.79459
53 x10-6
9.34414
39 x10-7
4.77086
71 x10-7
1.15984
97 x10-7
Scheme
(2)
4.57960
09 x10-5
3.01944
91 x10-5
1.92733
56 x10-5
7.49881
50 x10-6
We consider the non linear Goursat problem as
follows:
120120
1ln2
0
1ln2
0
2
.y,.x
)e(y
,y)u(
)e(x
)u(x,
eu
y
x
uxy
≤≤≤≤
+−=
+−=
=
(22)
The analytical solution is
)ln(2
),( yx eeyx
yxu +−+
= developed by [15]. The
MATLAB programs to implement the new scheme
(19) and standard scheme (2) for the non linear
Goursat problem (22) were developed. The graphs
and results presented below are approximate
solutions and relative errors at several selected grid
points respectively.
Fig.3: Solution of problem (22) in graphical form
using scheme (19) at 025.0=h
Table 7: Relative errors for scheme (19) at
025.0=h
x
y 1.2 1.5 1.8 2.1
1.2 3.78052
73 x10-7
5.002455
7 x10-7
5.924216
3 x10-7
6.45845
77 x10-7
1.5 5.00245
57 x10-7
6.959142
4 x10-7
8.655151
0 x10-7
9.85041
19 x10-7
1.8 5.92421
63 x10-7
8.655151
0 x10-7
1.134338
7 x10-6
1.35760
61 x10-6
2.1 6.45845
77 x10-7
9.850411
9 x10-7
1.357606
1 x10-6
1.71359
41 x10-6
Average relative error of scheme (19) = 7.5641119
x10-7
0
0.5
1
0
1
2
3-1.5
-1
-0.5
xy
u
Computational Methods in Science and Engineering
ISBN: 978-1-61804-174-6 66
Table 8: Relative errors for scheme (2) at 025.0=h
x
y 1.2 1.5 1.8 2.1
1.2 2.60634
65 x10-5
3.123954
7 x10-5
3.357548
9 x10-5
3.32392
67 x10-5
1.5 3.12395
47 x10-5
4.056010
5 x10-5
4.701967
1 x10-5
4.97556
07 x10-5
1.8 3.35754
89 x10-5
4.701967
1 x10-5
5.886354
3 x10-5
6.41707
33 x10-5
2.1 3.32392
67 x10-5
4.975560
7 x10-5
6.417073
3 x10-5
8.20265
85 x10-5
Average relative error of scheme (2) = 1.7052382
x10-5
Table 9: Average relative errors for grid size
0050and007001000200 ..,.,.h =
0.020 0.010 0.007 0.005
Scheme
(19) 3.86570
92 x10-7
4.81421
33 x10-8
1.66490
18 x10-8
6.04671
93 x10-9
Scheme
(2)
1.08650
54 x10-5
2.69211
33 x10-6
1.33220
14 x10-6
6.76055
47 x10-7
As can be seen from the results of the average
relative errors, for all Goursat problems with the
grid sizes investigated, the new scheme (19) is more
accurate than the standard scheme (2). Furthermore,
the graph shows the new scheme (19) successfully
perform the approximate solutions for problem (20),
(21) and (22).
5 Conclusions In this paper, we have developed a new scheme
based on ADM associated with the well known
Newton-Cotes integration formula for Goursat
partial differential equations (linear, derivative
linear and non linear). Our new scheme preserves
the linearity of Goursat problems (20) and (21). We
successfully applied the scheme (19) to find the
approximate solutions of the problems. The
numerical results obtained confirm the superiority of
the new scheme (19) related to the high accuracy
level over established scheme (2).
Acknowledgement
We acknowledge the support of the Research
Intensive Faculty Fund (600-RMI/DANA 5/3/RIF
(60/2012)) UiTM Malaysia.
References:
[1] G. Adomian, A Review of the Decomposition
Method in Applied Mathematics, Journal of
Mathematical Analysis and Application,
Vol.135, No.2, 1988, pp.501-544.
[2] M.A. A1-Alaoui, A Class of Numerical
Integration Rule With First Class Order
Derivatives, University Research Board of the
American University of Beirut, 1990, pp.25-44.
[3] Y.L. An and W.C. Hua, The Discontinuous Ivp
of A Reacting Gas Flow System, Transaction
of the American Mathematical Society, 1981.
[4] R.K. Arnold and W.U. Christoph, Numerical
Integration on Advanced Computer System,
Germany: Springer-Verlag Berlin Heidelberg,
1994, pp.5-23.
[5] S.A. Aseeri, Goursat Functions for A Problem
of An Isotropic Plate with A Curvilinear Hole,
International Journal for Open Problems
Computer Mathematic, Vol.1, 2008, pp.266-
285.
[6] L.G. Bushnell, D. Tilbury and S.S. Sastry,
Extended Goursat Normal Form With
Application to Nonholonomic Motion
Planning, Electronic Research Laboratory
University of California, 1994, pp.1-12.
[7] S.M. Cathleen, The Mathematical Approach to
the Sonic Barrier, Bulletin of the American
Mathematical Society, Vol.2, 1982, pp.127-
145.
[8] S. Chen and D. Li, Supersonic Flow Past A
Symmetrically Curved Cone, Indiana
University Mathematics Journal, 2000.
[9] J.T. Day, A Rungge-Kutta Method for the
Numerical Solution of the Goursat Problem in
Hyperbolic Partial Differential Equation,
Computer Journal, Vol.9, 1966, pp.81-83.
[10] T. Dawn, Trajectory Generation for the N-
Trailer Problem Using Goursat Normal Form,
IEEE Transactions on Automatic Control,
Vol.40, 1995, pp.802-819.
[11] D.J. Evans and B.B. Sanugi, Numerical
Solution of the Goursat Problem by A
Nonlinear Trapezoidal Formula, Applied
Mathematics and Letter, Vol.1, 1988, pp.221-
223.
[12] A.I. Ismail and M.A.S. Nasir, Numerical
Solution of the Goursat Problem, IASTED
Computational Methods in Science and Engineering
ISBN: 978-1-61804-174-6 67
Conference on Applied Simulation And
Modelling, 2004, pp.243-246.
[13] Y. Susumu, Goursat Problem for A
Microdifferential Operator of Fuchsian Type
and Its Application, RIMS Kyoto University,
Vol.33, 1997, pp.559-641.
[14] H. Taghvafard and H.G. Erjaee, Two-
Dimensional Differential Transform Method
for Solving Linear and Non-Linear Goursat
problem, International Journal for Engineering
and Mathematical Sciences, Vol.6, No.2, 2010,
pp.103-106.
[15] A.M. Wazwaz, On the Numerical Solution for
the Goursat Problem, Applied Mathematics and
Computational, Vol.59, 1993, pp.89-95.
[16] A.M. Wazwaz, The Decomposition Method for
Approximate Solution of Goursat Problem,
Applied Mathematics and Computational,
Vol.69, 1995, pp.229-311.
[17] A.M. Wazwaz, The Variational Iteration
Method for A Reliable Treatment of Linear
and Nonlinear Goursat Problem, Applied
Mathematics and Computational, Vol.193,
2007, pp.455-462.
Computational Methods in Science and Engineering
ISBN: 978-1-61804-174-6 68