A Mathematical Modelfor Approximate SolutionofLine Integral

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ISSN 2319-8133 (Online)

A Mathematical Model for Approximate Solution of Line Integral

Saumya Ranjan Jena* and Arjun Singh Department of Mathematics,

School of Applied Sciences, KIIT Deemed to be University, INDIA.

email: [email protected],[email protected].

(Received on: May 11, 2019) ABSTRACT

In this paper, a new mathematical model has been designed for approximate solution of line integration over any closed region with the joint application of mixed quadrature rule and Green's Theorem. These rules have been tested over five test problems over various kind of closed domain to obtain numerical convergence to the exact approach and the error bound has been determined.

MSC2010: 65D30, 65D32.

Keywords: Error bound, Green's Theorem, Line integral, Maclaurin's Theorem, Mixed quadrature.

1. INTRODUCTION

There are wide range of methods available for numerical integration. Majid Rostami et al.¹ proposed numerical integration rules based on block-pulse functions and Chevyshev wavelet to approximate value of definite integrals. Siraj-ul-islam et al.² established wavelet and hybrid functions for numerical integrations of real definite integrals. A Chakrabarti et al.³ developed a straight method for numerical integration of Fredholm integral equation of second kind using least square method. S.R. Jena et al.⁴ applied mixed quadrature rule on Electromagnetic Field Problems. Tripathy et al.⁵ used mixed quadrature of Lobatto 4-point rule with Gaussian quadrature approximate evaluation of real definite integrals. H. T. Rathod et al.⁶ presented Gauss Legendre quadrature method for numerical integration over the standard triangular surface.

Our main objective in this paper is to highlight the fact that a mixed quadrature of same precision contributes greater approximation towards the exact result than that of a single quadrature rule that of same precision for numerical integration of line integration.

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Motivated by the excellent performance of above described methods we would apply the mixed quadrature technique through Green's theorem for the approximate solution of line integration. The organisation of this paper is as follows. In section-2 Green Theorem and a mixed quadrature rule has been constructed with degree of precision-5. Error analysis for proposed rule has been proposed in section-3. Numerical results with mixed quadrature rule and absolute errors with constituent rules like Gauss Legendre 2 point rule and Clenshaw Curtis -5 point rule have been reported in section-4. Some conclusions has drawn in section-5.

2. GREEN'S THEOREM AND MIXED QUADRATURE RULE

This section contains basic idea about Green's theorem and construction of mixed quadrature rule.

2.1 Green's Theorem

Green's Theorem⁸ establishes the relationship between line integral around simple closed curvecand a double integral over the region R bounded by c. In XY plane the mathematical notation of Green's theorem takes the form



cK(r).dr



^ ^



^_^^_ ^^_ ^_^

R c

y dxdy K x dy K

K dx

K ) (1)

( ₁ ₂ ² ¹ (1)

Where K is a vector function and integration in counter clockwise direction.

Taking transformation

2

) ( )

(b a b a

x   

, and

2

) ( )

(d c d c

y   

equation(1) takes the form

) 2 ( )

, ( ).

(

1

1 1

1

 



 

 f   Jdd dr

r

cK

(2)







 



 



 



 x y

y x y Jacobian x

J ( , )

) , (

2.2 Construction of mixed quadrature rule

This section contains mixed quadrature rule of higher precision which has been obtained by taking convex combination of two constituent rules with equal lower degree of precison for approximate evaluation of real definite integral

 

( , ) (3)

1

1 1

1

 

 

 f x y dxdy f

I (3)

(3)

The Simpson’s rd 3

1 rd rule is

 



( 1,1) 4 (0,1) (1,1)



⁽⁴⁾

) 0 , 1 ( ) 0 , 0 ( 4 ) 0 , 1 ( 4

) 1 , 1 ( ) 1 , 0 ( 4 ) 1 , 1 ( 9 ) 1 ( ) (

3 1









































f f

f

f f

f

f f

R f

I _s (4)

Gauss-Legendre – 2 point rule

) 5 ( 3

, 1 3 1 3

, 1 3 ) 1

( )

( ₂ 

 



 



 



 





 









 



 



R f f f f f

f

I _GL (5)

Where each rule of equation (4) and equation (5) is of precision 3 Hence

) 6 ( )

( ) ( ) (

3

3 1

1 f E f

R f I

s  s

 (6)

) 7 ( )

( )

(f R ₂ f E ₂ f

I  _GL  _GL (7)

Where ( )

3

1 f

ES and E_GL₂(f)are error in approximating the integrals I( f)by equation (4) and equation (5) respectively.

Assuming

f ( y x , )

to be sufficiently differentiable in 1x1,

 1  y  1

in equation (3)and using Maclaurin’s expansion

 

 

) 1 . 2 ( )

8 (

) 0 , 0 ( )

0 , 0 ( 6

) 0 , 0 ( 15

) 0 , 0 ( 20

) 0 , 0 ( 15

) 0 , 0 ( 6

) 0 , 0 (

! 6 1

) 0 , 0 ( )

0 , 0 ( 5

) 0 , 0 ( 10

) 0 , 0 ( 5

) 0 , 0 (

! 5 1

) 0 , 0 ( )

0 , 0 ( 4

) 0 , 0 ( 6

) 0 , 0 ( 4

) 0 , 0 (

! 4 1

) 0 , 0 ( )

0 , 0 ( 3

) 0 , 0 ( 3

) 0 , 0

! ( 3 1

) 0 , 0 ( )

0 , 0 ( 2 ) 0 , 0

! ( 2 1

) 0 , 0 ( )

0 , 0 ( )

0 , 0 (

1

1 1

1

6 , 0 6 5

, 1 5

4 , 2 4 2 3

, 3 3 3

2 , 4 2 4 1

, 5 5 0

, 6 6

5 , 0 5 4

, 1 4 3

, 2 3 2

2 , 3 2 3 1

, 4 4 0

, 5 5

4 , 0 4 3

, 1 3

2 , 2 2 2 1

, 3 3 0

, 4 4

3 , 0 3 2

, 1 2 1

, 2 2 0

, 3 3

2 , 0 2 1

, 1 0

, 2 2

1 , 0 0

, 1 0

, 0

 



















































 















 



 dxdy

f y f

xy

f y x f

y x

f y x f

y x f

x

f y f

xy f

y x

f y x f

y x f

x

f y f

xy

f y x f

y x f

x

f y f

xy f

y x f

x

f y f

xy f

x

f y f

x f

f

I (8)

(4)

   

  

⁽⁰^,⁰⁾ ⁽⁰^,⁰⁾



^... ⁽⁹⁾

! 7 ) 4 0 , 0 ( ) 0 , 0 180 ( ) 1 0 , 0 9 ( 1

) 0 , 0 ( ) 0 , 0 30 ( ) 1 0 , 0 ( ) 0 , 0 3 ( ) 2 0 , 0 ( 4 ) (

6 , 0 0

, 6 4

, 2 2

, 4 2

, 2

4 , 0 0

, 4 2

, 0 0

, 2 0

, 0





f f

f

f f

f f I

(9)

Using Maclaurin’s expansion for each term in equation (4) and substituting in it

   

  

(0,0) (0,0)

  

10

! 6 9 ) 11 0 , 0 ( ) 0 , 0 108 ( ) 1 0 , 0 9 ( 1

) 0 , 0 ( ) 0 , 0 18 ( ) 1 0 , 0 ( ) 0 , 0 3 ( ) 2 0 , 0 ( 4 ) (

6 , 0 0

, 6 4

, 2 2

, 4 2

, 2

4 , 0 0

, 4 2

, 0 0

, 2 0

, 0

3 1

f f

f

f f

f f R_s

 





(10) 2.3 Error in Simpson’s

3

1rd rule

Using equation (9)and equation (10) in equation (6) error associated with Simpson’s

3rd

1 rule is

) ( ) ( ) (

3 1 3

1 f I f R f

E_S   _S

   



⁽⁰^,⁰⁾ ⁽⁰^,⁰⁾



! 6 63

41

) 0 , 0 ( ) 0 , 0 270 ( ) 1 0 , 0 ( ) 0 , 0 45 ( ) 1 (

6 , 0 0

, 6

4 , 2 2

, 4 4

, 0 0

,

3 4

1

f f

f E_S

 













(11)

The degree of precision of the said formula is 3.

2.4 Gauss Legendre-2 point rule

Using Maclaurin’s series for each term in equation (5) and substituting in it.

   

  

⁽⁰^,⁰⁾ ⁽⁰^,⁰⁾



^... ⁽¹²⁾

! 6 27 ) 4 0 , 0 9 ( ) 1 0 , 0 ( ) 0 , 0

! ( 6 9

20

) 0 , 0 ( ) 0 , 0

! ( 3 9 ) 1 0 , 0 ( ) 0 , 0 3 ( ) 2 0 , 0 ( 4 ) (

6 , 0 0

, 6 2

, 2 4

, 2 2

, 4

4 , 0 0

, 4 2

, 0 0

, 2 2



 



 



 





f f

f

f f

f f R_GL

2.5 Error in Gauss-Legendre-2 point rule

Now the error associated with the Gauss- Legendre-2 point rule is obtained substituting equation (9) and (12) in equation (7)

 

  

⁽⁰^,⁰⁾ ⁽⁰^,⁰⁾



^.... ⁽¹³⁾

63 27 ) 1 0 , 0 ( ) 0 , 0 405 (

1

) 0 , 0 ( ) 0 , 0 135 ( ) 2 ( )

( ) (

6 , 0 0

, 6 4

, 2 2

, 4

4 , 0 0

, 4 2

2



 









f f

f R f I f

E_GL _GL

(5)

In this case also, the error contains at least fourth derivative of the integral function.

Thus the degree of the precision is 3.

2.6 Mixed Quadrature Rule.

Now multiplying 2 by equation (11) and 3 by equation (13) and adding them, we get ) 14 ( )

( 3 ) ( 5 2

) 1 ( 3 ) ( 5 2

) 1

( ₂

3 1 2

3

1 









 











 

 R f R f E f E f

f

I _GL

GL S

S (14)

) 15 ( )

( )

(

3 2 2 1

3

1 f E f

R f I

GL S GL

mix  S  (15)

 

2 ( ) 3 ( ) (16)

5 1

2 3

2 1

3

1 









 

 R f R f

f

R _GL

GL s

S (16)

Where ₂( )

3

1 f

R_S _GL and ₂( )

3

1 f

E_S _GL are mixed quadrature rule and its error obtained by Simpson’s rd

3

1 and Gauss-Legendre- 2 point rule respectively.

The truncation error generated by this approximation is given by

  ^

⁽⁰^,⁰⁾ ⁽⁰^,⁰⁾

^

⁽¹⁷⁾

! 5 189 ) 1 ( 3 ) ( 5 2 1

6 , 0 0

, 6

2 ₃¹ 2

3

1 E f R f f f

E_S _GL _s _GL 

 







The equation (14) is mixed quadrature rule of degree of precision 5 2.7 Clenshaw Curtis-5 point rule

We choose Clenshaw-Curtis -5 point rule in same vein^9,10

     

) 18 (

0 , 0 12 2 , 1 0 8 2 , 1 0 8 1 , 0 1 , 0 12

0 , 2 12 1 2 , 1 2 8 1 2 , 1 2 8 1 1 , 2 1 1

, 2 8 1

0 , 2 12 1 2 , 1 2 8 1 2 , 1 2 8 1 1 , 2 1 1

, 2 8 1

0 , 1 12 2 , 1 1 8 2 , 1 1 8 1 , 1 1

, 1

0 , 1 2 12

, 1 1 2 8

, 1 1 8 1 , 1 1 , 1

225 1



















 

 



 





 



 











 

 









 



 





 









 



 





 











 

 



 



 



 





 



 



 



 





 











  

 



 





 



















 

 



 





 



 







f f

f

f f

f

f f

f

f f

f

f f

f

 

f f

 

x y dxdy R

 

f

I _{CC 5}

1

1 1

1

, 



 

 

(6)

Using Maclaurin’s series for each term in equation (18) and substituting in it.

           ^{ } ^{ } 

             

          

⁰^,⁰

 

⁰^,⁰



^...

! 8 5 0 2 , 0 0

,

! 0 8 45 0 224 , 3600 0

1

0 , 0 0

,

! 0 6 15 0 8 , 0 0

, 180 0 0 1 , 9 0 1

0 , 0 0

, 30 0 0 1 , 0 0

, 3 0 0 2 , 0 4

8 , 0 0

, 8 6

, 2 2

, 6 4

, 4

6 , 0 0

, 6 4

, 2 2

, 4 2

, 2

4 , 0 0

, 4 2

, 0 0

, 2 0

, 0 5



 



 



 





f f

f

f f

f

f f

f f R_CC

(19)

The error associated with The error associated with Clenshaw -Curtis-5 Point rule is

     

   

       

   



⁰^,⁰ ⁰^,⁰



! 8 45

2

0 , 0 0

,

! 0 8 45 0 16 , 0 0

,

! 0 6 105

4

8 , 0 0

, 8

6 , 2 2

, 6 6

, 0 0

, 6

5 5

f f

f R f I f

E_CC _CC

 



 



 







 

20 3. ERROR ANALYSIS

In this section an error analysis has been developed by Theorem- 1 and Theorem- 2 Theorem 3.1

Let f(x,y)be sufficiently differentiable function in 1 x,y1. The truncational error

bound ₂( )

3

1 f

E_S _GL associated with the rule ₂( )

3

1 f

R_S _GL



⁽⁰^,⁰⁾ ⁽⁰^,⁰⁾



! 5 189 ) 1

( ₆_,₀ ₀_,₆

3 2

1 f f f

E_S _GL 

  Proof:

The proof obviously follows from the equation (17) Theorem 4.2

The bounds for the truncational error ₂ ₂ ₁ 225

) 2 (

3

1  M  

f E_S _GL Where1^,2

 

¹^,¹

Where M Max f₆_,₀(0,0) f₀_,₆(0,0), 1x1, 1 y1 Proof:

We have

) ( ) 45 (

) 1

( ₄_,₀ ₁ ₀_,₄ ₂

3

1 f f  f 

E_S  

) ( ) 135 (

) 2

( ₁ ₄_,₀ ₁ ₀_,₄ ₂

2 f  f  f 

E_GL  

(7)

Where

 

¹^,¹

, ₂

1   

 Hence

) ( 3 ) ( 52 ) 1

( ₂

3 2 1

3

1 f E f E f

E _GL

GL s

S  



⁽ ^,⁰⁾ ⁽⁰^, ⁾ ⁽ ^,⁰⁾ ⁽⁰^, ⁾



225 2

1 4 , 0 1 0 , 4 2 4 , 0 2 0 ,

4  f  f  f 

f   



 

 

^

 ²

1 2

1

) (*, ,*)

225 ( 2

5 , 0 0

, 5 n

n n

n

y f x f

225 ) 2

( ¹ ²

3 2

1



 

 M f

E_S _GL

WhereM max f₅_,₀(x,*) f₀_,₅(*,y) ; 1x1, 1 y1

Which, gives only the truncation error bound on₁,₂are closed to each other.

Corollary:

The error bound for the truncated error

225 ) 4

2(

3 1

f M ES GL 

When₁₂ 2 ⁷

4. NUMERICAL EXAMPLES

Here five numerical examples have been taken to illustrate the proposed rule.

Example-1

Fig- Example - 1

 

 

 











1

1 1

1

3 2

0 0 2 3

3

1 3  3( 1)   



d d Jdrd

r dy

x dx y I

c

r

(8)

Example-2

;

Example-3

 





 



 



 



 



 

































1

1 1

1

2 1 2

1 1

0 1

0 3

2 2

4 ) 1

(  





d d e

e dxdy

e e

dy xe d ye I

x y c

y x

Fig - Example - 3

Example-4

     

 

  

 

















1

1 1

1

2 3

5 1

0 2

0

1

0 1

0

2 3 5 3

4

] 1 4 1 1 32 [

1

) 2 32

( )

2 (







 d d

dudv u v u dydx

x xy I

x

Two transformations has been taken in Example-4

2 , 1

2 2 1

, 

 



  

v u

and uv

y u x

Example-5

         





 

















 



 



















 



 



 



  



 



  

1

1 1

1

2 2 1

0 1

0

2 2

1 1

5

2 256

8 2

256 1 64

8 64

1

sin 8 tan 8



 

 

 d d

dxdy y y x

x

y dy dx x

y I x

c

   

 

 

 



 











 







 











1

1 1

1 2 2

4 1 sin 4 1

cos 16 2

) sin cos 2 ( cos

2 sin





 

 

 d d

dxdy x y ydy

x xdx y I

c R

(9)

Table - 1 (Comparison between Mixed quadrature, Gauss Legendre - 2 point and Clenshaw Curtis-5 point rules)

5. CONCLUSION

Error analysis of these methods besides numerical examples provide a solid foundation to compare between Gauss Legendre 2 point rule, Clenshaw Curtis-5 point rule and mixed quadrature rule with same degree of precision for numerical evaluation of line integration. The numerical examples in Table - 1 and Graph of Example -3, Example - 4 ensures that mixed quadrature rule is far suitable to Gauss Legendre 2 point rule as well as Clenshaw Curtis-5 point rule for convergency towards exact results. Though Clenshaw Curtis-5 point rule is of same precision five, but the absolute error is more than mixed quadrature rule for evaluation of line integral. In the same manner it can be shown that the proposed mixed quadrature rule is superior to Gauss-Legendre -3 point rule with respect to same precision five.

REFERENCES

1. Majid Rostami, Ehlam Hashemizadeh and Mohammd Heidari: Mathematical Sciences: A

Integ -rals

) (

3 2

1 f

R

GL

S R_GL₂(f) R_CC₅(f) ^I⁽^f⁾^^Exact Absolute error

I1 75.3982236861550 35

75.39822368615502 1

75.39822368615550 49

75.3982236861550

35 ( ) 0

2 3

1 f 

ES GL

14 5(f)1.410^ E_CC

2

EGL =1.410^¹³ I2 1.57079528811837

3

1.568397113540485 1.570801674301462 1.57079632679489 7

-6 3 2

1 (f)1.010 E

GL S

-6 55.410 ECC

,E_GL₂ =2.410^³ I3 1.49364796906704

0

1.493189376565719 1.493653971584915 9

1.49364826562485 4

-7 3 2

1 (f)3.010 E

GL S

6 55.710^ ECC

4 24.510^ EGL

I4 0.64444444444444 4

0.555555555555555 55

0.641785113019776 0.66666666666666 6

-2 3 2

1 (f)2.210 E

GL S

2

EGL = 1.010^¹, 5

ECC =2.510^² I5 0.12954500567172

3

0.129544951242143 0.129545005552005 0.12954500576844 54

-11 3 2

1 (f)9.710 E

GL S

2

EGL = 5.510^⁸,

5

ECC =2.210^¹⁰

(10)

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International Journal of Computing Science and Mathematics International Journal of Computing Science and Mathematics, 6(4), 366-377 (2015).

6. H. T. Rathod et al. "Gauss-Legendre Quadrature over Triangles", J. Indian Inst. Sci, 84, 183-188 (2004).

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A Mathematical Modelfor Approximate SolutionofLine Integral