Third Order Newton-Cotes Integration Rule for Solving

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]

;[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:


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:


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


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


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.


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


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


x10-6


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


x10-4


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.


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


ISBN: 978-1-61804-174-6 66


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


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.

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ISBN: 978-1-61804-174-6 68

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Third Order Newton-Cotes Integration Rule for Solving