FUZZY NUMERICAL SOLUTION OF POISSON EQUATION USING FUZZY DATA

(1)

FUZZY NUMERICAL SOLUTION OF

POISSON EQUATION

USING FUZZY DATA

RAPHEL KUMAR SAIKIA

Associate Professor, Department of Mathematics Jorhat Institute of Science and Technology

Jorhat-785010, Assam, India. Email: [email protected]

Abstract

In this paper, we have discussed fuzzy numerical solution of elliptic partial differential equation taking Poisson’s Equation (in two variable) into consideration. While solving this equation numerically at different grid points, on the other hand we want to observe findings using fuzzy intervals. Finite difference method is applied in solving the equation numerically.

Key Words: α-cut, fuzzy membership function(f.m.f.), interval of confidence, finite difference. 1. INTRODUCTION

In the field of numerical solution of partial differential equation , not much work have been done

using fuzzy interval. Fuzzy numerical solution of differential equation is the process of computing fuzzy membership functions at different grid points with the help of fuzzy intervals.

It has been discussed here taking an elliptic partial differential equation, namely Poisson Equation.

2. APPLICATION OF FINITE DIFFERENCE METHOD IN POISSON EQUATION.

The Poisson Equation in two variable is ( , ) y

x 2

2

2 2

y x u u _

_

    

=> uxx uyy 



(x,y)(1)

which has broad utility in electrostatics, mechanical engineering and theoretical physics is an elliptic partial differential equation since B2 - 4AC = -4 < 0. Replacing uxx and uyy by finite difference method [1] ,equation

(1) becomes

) , ( ) , ( ) , ( 2 ) , ( ) , ( ) , ( 2 ) , (

2

2 x y

k

k y x u y x u k y x u h

y h x u y x u y h x

u     _     _

_



( , ) ( , ) ( , ) ( , )



( , )

4 1 ) ,

(x y u x h y u x y h u x h y u x y h h2 x y

u         









(, )

4

1 2

, , ,

,

, u u u u i i j

uij  i h j i j h  i h j i j h 



 _ _ _ _

Here we will use two formulae called Standard Five Point Formula (SFPF) and Diagonal Five Point Formula (DFPF) represented respectively by diagrams Fig-1 and Fig-2

(2)

C

1

C

₂

C

₃

C

4

C

12

C

11

C

10

C

9

C

8

C

7

C

6

C

5

Fig-1

Fig-2

We will use Diagonal Five Point Formula (DFPF) wherever necessary as

,



1, 1 1, 1 1, 1 1, 1



4

1

      





_i _j _i _j _i _j _i _j

j

i

u

3. APPLICATION OF FUZZY INTERVAL IN POISSON EQUATION.

Boundary points have been taken from C

1 to C12 due to equation no. (1) with fuzzy intervals [3] as follows:

u1,u2,u3 and u4 are four interior points due to the square grids as shown in the fig-3.

Here

Fig-3

and

]

[

)

(

C

1



l

1,1;

l

1,2;

l

1,3

C

5

(



)

[

l

5,1;

l

5,2;

l

5,3

]

C

9

(



)

[

l

9,1;

l

9,2;

l

9,3

]

[

)

(

C

₂



l

₂_,₁_;

l

₂_,₂_;

l

₂_,₃

C

₆

(



)

[

l

₆_,₁_;

l

₆_,₂_;

l

₆_,₃

]

C

₁₀

(



)

[

l

₁₀_,₁_;

l

₁₀_,₂_;

l

₁₀_,₃

]

[

)

(

C

₃



l

₃_,₁_;

l

₃_,₂_;

l

₃_,₃

C

₇

(



)

[

l

₇_,₁_;

l

₇_,₂_;

l

₇_,₃

]

C

₁₁

(



)

[

l

₁₁_,₁_;

l

₁₁_,₂_;

l

₁₁_,₃

]

[

)

(

C

₄



l

₄_,₁_;

l

₄_,₂_;

l

₄_,₃

C

₈

(



)

[

l

₈_,₁_;

l

₈_,₂_;

l

₈_,₃

]

C

₁₂

(



)

[

l

₁₂_,₁_;

l

₁₂_,₂_;

l

₁₂_,₃

]

u

1

u

2

u

3

u

4

U

i,j

U

i,j

U

i-1,j

U

i+1,j

U

i,j+1

U

i,j-1

U

i-1,j

U

i,j-1

U

i+1,j

U

i,j+1

U

i-1,j-1

U

i+1,j-1

U

i-1,j+1

U

i+1,j+1



12 2 2 3 1



1

4 1

K U C U C

U     



1 5 3 4 2



2

4 1

K U C C U

U     



11 4 1 9 3



3

4 1

K C U U C

U     



3 6 2 8 4



4

4 1

K C U C U

(3)

where Ki’s are constants (i= 1,2,3,4 ) . Using Gauss-Seidel method, let





,0,





)

(

) 0 (

2  

U U₃(0)()







,0,





C2()



l2,1;l2,2;l2,3



C12()



l12,1;l12,2;l12,3



K1()



k1,1;k1,2;k1,3



with the supposition

ε , ηrestrictedly very small positive real number less than 1 ( Assumed less than .1)



; ;



( .1)

4 1 )

( 12,1 2,1 1,1 12,2 2,2 1,2 12,3 2,3 1,3

) 1 (

1 l l k l l k l l k A

U  



 



   



 



 

Now f.m.f. of U2(0) , U3(0),C2,C12and K1 and their respective interval of confidence are

  

   

 

  

   

 

  



   



Otherwise x x

x x

x

u

; 0

0 ;

) )( (

) 0 ( 2





;

α-cut for

u

2(0) is

 

( )



( 1), (1 )



)

( ) 0 (

2









 



u

  

   

 

  

   

 

  



   



Otherwise x x

x x

x

u

; 0

0 ;

) )( (

) 0 ( 3





;

α-cut for

u

₃(0)

is

 

3(0) ( )( )





(



1),



(1



)





 



u

   

   

 

   

   

 

  

 

  





Otherwise k x k k k

k x

k x k k k

k x

x

K

; 0

; ;

) )(

( ₁_,₂ ₁_,₃

2 , 1 3 , 1

3 , 1

2 , 1 1

, 1 1 , 1 2 , 1

1 , 1

1



α-cut

 

K₁ ()()



(k₁_,₂k₁_,₁)



k₁_,₁,(k₁_,₃k₁_,₂)



k₁_,₃



   

   

 

   

   

 

  

 

  





Otherwise l x l l l

l x

l x l l l

l x

x

C

; 0

; ;

) )(

( ₂_,₂ ₂_,₃

2 , 2 3 , 2

3 , 2

2 , 2 1

, 2 1 , 2 2 , 2

1 , 2

2



;

α-cut ( )



₂_,₂ ₂_,₁ ₂_,₁ ₂_,₃ ₂_,₂ ₂_,₃



2 ( )(l l ) l , (l l ) l

C   



  





   

   

 

   

   

 

  

 

  





Otherwise l x l l l

l x

l x l l l

l x

x

C

; 0

; ;

) )(

( ₁₂_,₂ ₁₂_,₃

2 , 12 3 , 12

3 , 12

2 , 12 1

, 12 1 , 12 2 , 12

1 , 12

12



;

α-cut



12,2 12,1 12,1 12,3 12,2 12,3



) (

12 ( )(l l ) l , (l l ) l

C  



  





(4)

 

                                       4 4 ) ( ) ( ) ( , 4 4 ) ( ) ( ) ( ) ( 3 , 12 3 , 2 3 , 1 2 , 12 3 , 12 2 , 2 3 , 2 2 , 1 3 , 1 1 , 12 1 , 2 1 , 1 1 , 12 2 , 12 1 , 2 2 , 2 1 , 1 2 , 1 ) ( ) 1 ( 1 l l k l l l l k k l l k l l l l k k U























To retain two roots with αε [0,1], let

4 4 ) ( ) ( )

( 1,2 1,1 2,2 2,1 12,2 12,1 1,1 2,1 12,1 1 l l k l l l l k k

x 







     











  

4 4 ) ( ) ( )

( 1,3 1,2 2,3 2,2 12,3 12,2 1,3 2,3 12,3 2 l l k l l l l k k

x 







     











  

Hence ) ( ) ( ) ( ) ( 4 1 , 12 2 , 12 1 , 2 2 , 2 1 , 1 2 , 1 1 , 12 1 , 2 1 , 1 1 l l l l k k l l k x              











and ) ( ) ( ) ( ) ( 4 2 , 12 3 , 12 2 , 2 3 , 2 2 , 1 3 , 1 3 , 12 3 , 2 3 , 1 2 l l l l k k l l k x              











Hence f.m.f. for U1(1)

is

) 1 . 1 . ( ; 0 4 4 ) ( ) ( ) ( ) ( 4 4 4 ) ( ) ( ) ( ) ( 4 ) )( ( 3 , 12 3 , 2 3 , 1 2 , 12 2 , 2 2 , 1 2 , 12 3 , 12 2 , 2 3 , 2 2 , 1 3 , 1 3 , 12 3 , 2 3 , 1 2 , 12 2 , 2 2 , 1 1 , 12 1 , 2 1 , 1 1 , 12 2 , 12 1 , 2 2 , 2 1 , 1 2 , 1 1 , 12 1 , 2 1 , 1 ) 1 ( 1 A Otherwise l l k x l l k where l l l l k k l l k x l l k x l l k where l l l l k k l l k x x

U 

                                                                           



























In a similar way we can find out all fuzzy membership functions for

4(1) ) 1 ( 3 ) 1 (

2 ,U andU

U .Next

successive approximations upto desired accuracy with their f.m.f. as required may be obtained from previous

approximations and given boundary conditions.

4. A numerical example

To solve the equation 2u10



x2 y210



(2)

over the square mesh with sides x=0, y=0, x=3, y=3 with u=0 on the boundary with mesh length 1,with boundary conditions u(0,y)=0 , u(3,y)=0, u(x,0)=0, u(x,3)=0.

Let u1, u2, u3, u4 be the values of u at the four mesh points A,B,C,D

(5)

Fig-4

At the point A, (i=1,j=2)

u

0,2



u

2,2



u

1,1



u

1,3



4 u

1,2





10 (

1 

4 

10 )



0 

u

2



u

3



0 

4 u

1





150

) 2 . 2 ( ) 150 (

4 1

3 2

1   

u u u

;

At B,

( 180) (2.3)

4 1

2  u u  

u

At C, ( 120) (2.4)

4 1

3 u u  

u

;

At D,

( 180) (2.5)

4 1

3 2

4  u u  

u

As

u

₁



u

₄

,

hence (2.3 ) & (2.4) are reduced to

( 90) (2.6) 2

1

2  u  

u

( 60) (2.7) 2

1

&u3  u1 

By Gauss-Seidel method, let

(

)



0 .

01 ,

0 ,

0 .

01 

,

3(0)

(

)



0 .

01 ,

0 ,

0 .

01 

) 0 (

2





u





u

and

k

1

(



)



149 .

99 ,

150 ,

150 .

01 





1



) 0 ( 3 ) 0 ( 2 )

1 ( 1

4 1 )

( u u k

u   

u₁(1)()



37.4925,37.5,37.5075



(2.2.1)

Now f.m.f. of

u

₂(0),

u

₃(0),and

k

₁ and their respective interval of confidence are

  

   

 

  

   

 

  



   



Otherwise x x

x x

x

u

; 0

01 . 0 0 ; 01 . 0

01 . 0

0 01 . 0 ; 01 . 0

01 . 0

) )( (

) 0 ( 2



;

α-cut is

 

u

₂(0) ()

(



)



0 .

01 



0 .

01 ,



0 .

01 



0 .

01 

  

   

 

  

   

 

  



   



Otherwise x x

x x

x

u

; 0

01 . 0 0 ; 01 . 0

01 . 0

0 01 . 0 ; 01 . 0

01 . 0

) )( (

) 0 ( 3



;

α-cut is

 

3(0) ( )

(



)



0 .

01 



0 .

01 ,



0 .

01 



0 .

01 



u

A(1,2)

U

1

B(2,2)

U

2

C(1,1)

U

3

D(2,1)

U

4

) 1 . 2 ( ) 10 (

10

4 , 2 2

1 , 1 , , 1 ,

1           

 u u u u i j

(6)

  

   

 

  

   

 

  

 

  





Otherwise x x

x x

x

k

; 0

01 . 150 0

15 ; 150 01 . 150

01 . 150

150 99

. 149 ; 99 . 149 150

99 . 149

) )( (

1



;

 

1 ( )

(



)



0 .

01 



149 .

99 ,



0 .

01 



150 .

01 



k

Hence from (2.2.1) interval of confidence for

u

1(1) is

 

1(1) ( )()



0.0075



37.4925,0.0075



37.5075



(2.2.2)



u

We are to retain two roots with





 

0 ,

1

where

0075 . 0

5075 . 37 0075

. 0

4925 .

37 2

1 _ 

 x and



x



Hence f.m.f. for

u

₁(1) is

) 3 . 2 . 2 (

; 0

5075 . 37 5

. 37 ; 5 . 37 5075 . 37

5075 . 37

5 . 37 4925

. 37 ; 4925 . 37 5 . 37

4925 . 37

) )( (

) 1 ( 1

                 

  

   

 

  

   

 

  

 

  





Otherwise x x

x x

x

u



In a similar way f.m.f. for

u

(₂1) is

) 4 . 2 . 2 (

; 0

75875 . 63 75

. 63 ; 75 . 63 75875 . 63

75875 . 63

75 . 63 74125

. 63 ; 74125 . 63 75 . 63

74125 . 63

) )( (

) 1 ( 2

                 

  

   

 

  

   

 

  

 

  





Otherwise x x

x x

x

u



And f.m.f. for

u

3(1) is

(2.2.5)

; 0

75875 . 48 75

. 48 ; 75 . 48 75875 . 48

75875 . 48

75 . 48 74125

. 48 ; 74125 . 48 75 . 48

74125 . 48

) )( (

) 1 ( 3

              

  

   

 

  

   

 

  

 

  





Otherwise x x

x x

x

u



Similarly for second iteration , f.m.f. for

u

₁(2),

u

₂(2)and

u

₃(2)

third iteration, f.m.f. for

u

1(3), ) 3 ( 2

u

and

u

3(3)

fourth iteration, f.m.f. for

u

₁(4),

u

₂(4)and

u

₃(4)

fifth iteration, f.m.f. for

u

1(5), ) 5 ( 2

u

and

u

3(5)

sixth iteration, f.m.f. for

u

₁(6),

u

₂(6)and

u

₃(6)

(7)

For seventh iteration, f.m.f. for

u

₁ ,

u

₂ and

u

₃ are respectively

  

   

 

  

   

 

  

 

  





Otherwise

x x

x

u

; 0

796875 9975115966 .

74 625

9908447265 .

74 ; 625 9908447265 .

74 796875 9975115966 .

74

796875 9975115966 .

74

625 9908447265 .

74 453125

9841778564 .

74 ; 453125 9841778564 .

74 625 9908447265 .

74

453125 9841778564 .

74

) )( (

) 7 ( 1



  

   

 

  

   

 

  

 

  





Otherwise

x x

x

u

; 0

3984375 5037557983

. 82 8125

4954223632 .

82 ; 8125 4954223632 .

82 3984375 5037557983

. 82

3984375 5037557983

. 82

8125 4954223632 .

82 2265625

4870889282 .

82 ; 2265625 4870889282

. 82 8125 4954223632 .

82

2265625 4870889282

. 82

) )( (

) 7 ( 2



  

   

 

  

   

 

  

 

  





Otherwise

x x

x

u

; 0

3984375 5037557983

. 67 8125

4954223632 .

67 ; 8125 4954223632 .

67 3984375 5037557983

. 67

3984375 5037557983

. 67

8125 4954223632 .

67 2265625

4870889282 .

67 ; 2265625 4870889282

. 67 8125 4954223632 .

67

2265625 4870889282

. 67

) )( (

) 7 ( 3



5.CONCLUSION

We calculated here up to seventh approximations while solving the numerical example. This may however be increased upto desired accuracy.

6. REFERENCES

[1] Grewal, B.S.(2010): Numerical methods in Engineering and sciences with Programs in C & C++_{, Khanna}

[2] Publishers, New Delhi-110002 pp. 343-358.

[3] Baruah, Hemanta K; (1999): Set Superimposition and its applications to the Theory of Fuzzy Sets, Journal

[4] of Assam Science Society, Vol. 40, Nos. 1 & 2, 25-31.

[5] Baruah, Hemanta K. (2010a): Construction of the Membership Function of a Fuzzy Number , ICIC Express [6] Letters.

[7] Baruah, Hemanta K. (2010b): The Mathematics of Fuzziness: Myths and Realities, Lambert Academic [8] Publishing, Saarbruken, Germany.

[9] Kaufmann A.; and Gupta M. M. (1984): Introduction to Fuzzy Arithmetic, Theory and Applications, Van [10] Nostrand Reinhold Co. Inc.,Wokingham, Berkshire.

[11] Zadeh L.A.: Probability Measure of Fuzzy Events, Journal of Mathematical Analysis and Applications, Vol. [12] 23 No. 2, August 1968 (pp 421-427).

[13] Sastry , S.S.(1995): Introductory Methods of Numerical Analysis, Prentice Hall, New Delhi-110001.