Ijeart06501

International Journal of Engineering and Advanced Research Technology (IJEART) ISSN: 2454-9290, Volume-1, Issue-6, December 2015

The Change Effect for Boundary Conditions to Minimize Error Bound in Lacunary Interpolation

by Forth Spline Function Muhannad A. Mahmoud 

i , i  0,2,....,2m and n=2m+1 . We define 2m the class of Spline function S P ( 4,3, n ) as follows : any element S  ( x)S P (4,3, n) if the following xi 

Abstract—The researcher interested in spline function because it is necessary function in interpolation process, by putting a certain condition for spline function from forth degree which is as follows

Sn( x0 )  0, Sn( x2 m )  0 , then the aim of this research

condition satisfied :

is to change the previous boundary conditions which is as

Sn( x0 )  f ( x0 ), Sn( x2 m )  f ( x2 m )

  (ii) S  ( x) is polynomial of deg ree four in each [ x2i , x2i  2 ] (1)  i  0,1,..., m  1  (i ) S  ( x) C 3[0,1]

to get minimum

error for spline function from forth degree.

In this paper we prove the following : Theorem 1:

Index Terms— Spline function ; Boundary Conditions.

Given arbitrary number

f ( 2) ( x ), f ( 2) ( x2 m ) there S n ( x)  S p (4,3, n) such that

and

I. INTRODUCTION Spline functions appears in the late middle of twentieth century, According to Schoenbery [5] , the interest in Spline functions is due to the fact that , Spline function are good tool for the numerical approximation of functions and that they suggest new ,challenging and rewarding problems . For more information about a Spline function , one is referred to Alberg ,Milson and walsh [1] .Lacunary interpolation by Spline appears whenever observation gives scattered or irregular information about a function and its derivatives , but with out Hermite condition . Mathematically , in the problem of interpolation given data

a i, j

f ( x2 i ) , f (t 2 i )

by polynomial

p n (x )

S n ( x2 i )  f ( x2i )

i  0,1,2....., m

S n (t2i )  f (t2i )

i  0,1,2...., m  1

S

( 2) n

( x0 )  f

( 2)

( x0 ) , S

( 2) n

( x2 m )  f

( 2)

i=0,1,2…,n-1

exists a unique spline

    

( x2 m )

(2)

Theorem 2:

f C 4 [0,1] and S n  Sp(4,3, n) be

Let

a unique

spline

satisfying the condition of (theorem 1.1) , then 1 S ( x)  f ( r ) ( x)  5.6404257m r  4 w( f , )  m r  4 f ( 4) ; r  0,1,2,3 m

of

(r ) n

degree of most n satisfying :

where

p ( xi )  ai , j

, i  1,2,3,..., n , j  0,1,2 ,......., m

( j) n

w( f f

we have Hermit interpolation if for i , the order j of derivatives in form unbroken sequence . If some of the sequence are broken , we have lacunary interpolation . In another communications we shall give the applicationsof Spline functions obtaining the approximation solution of boundary value problem , for more about applications of Spline functions , see [2,4] . For description our problem , let

of

the

interval

[0,1]

,

II. TECHNICAL PRELIMINARIES: If p(x) is a polynomial of degree Four on [0,1] (because we want to construct a spline function of degree four ) then we have :

1 P( x)  P(0) A0 ( x)  P( ) A1 ( x)  P(1) A2 ( x)  P // (0) A3 ( x)  P // (1) A4 ( x) 3

 :0  xo  x2  x3  .....  x2 m  1 by a uniform partition

( 4)

1 )  max f ( 4 ) ( x)  f ( 4 ) ( y ) : x  y  h, x, y  [0,1] m  max  f ( 4 ) ( x) : 0  x  1

( 4)

with

……..(3) Where

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The Change Effect for Boundary Conditions to Minimize Error Bound in Lacunary Interpolation by Forth Spline Function

       ....( 4)       

1 [27 x 4  54x 3  38x  11] 11 1 A1 ( x)  [81x 4  162x 3  81x] 22 1 A2 ( x)  [27 x 4  54x 3  5 x] 22 1 A3 ( x)  [15x 4  41x 3  33x 2  7 x] 66 1 A4 ( x)  [12x 4  13x 3  x] 66 A0 ( x) 

f  C 4 [0,1] we have the following expansions: 4 2 f ( x2i  2 )  f ( x2i )  2hf / ( x2i )  2h 2 f // ( x2i )  h 3 f /// ( x2i )  h 4 f ( 4 ) (1, 2i ) 3 3 x2i  1, 2i  x2i  2 4 2 f ( x2i  2 )  f ( x2i )  2hf / ( x2i )  2h 2 f // ( x2i )  h 3 f /// ( x2i )  h 4 f ( 4 ) ( 2 , 2i ) 3 3 x2 i  2   2 , 2 i  x2 i 2 2 4 2 4 ( 4) f (t 2i )  f ( x2i )  hf / ( x2 i )  h 2 f // ( x2i )  h 3 f /// ( x2i )  h f ( 3 , 2 i ) 3 9 81 243 x2 i   3 , 2 i  t 2 i

In the subsequent section we need the following values for

4 8 32 32 4 ( 4) f (t2i  2 )  f ( x2i )  hf / ( x2i )  h 2 f // ( x2i )  h3 f /// ( x2i )  h f (4 , 2i ) 3 9 81 243 t2i  2  4 , 2i  x2i

2 // 2 4 hf ( x2i )  h 2 f /// ( x2i )  h 3 f ( 4 ) (5, 2i ) 3 9 81 x2 i   5 , 2 i  t 2 i 4 8 f // (t2i  2 )  f // ( x2i )  hf /// ( x2i )  h 2 f ( 4 ) (6, 2i ), t2i  2  6, 2i  x2i 3 9 2 2 f // (t2i )  f // ( x2i )  hf /// ( x2i )  h 2 f ( 4) (7 , 2i ), x2i  7 , 2i  t2i 3 9 // // /// f ( x2i  2 )  f ( x2i )  2hf ( x2i )  2h2 f ( 4) (8, 2i ) , x2i  2  8, 2i  x2i

f / (t 2i )  f / ( x2i ) 

f // ( x2i  2 )  f // ( x2i )  2hf /// ( x2i )  2h2 f ( 4) (9, 2i ), x2i  9, 2i  x2i  2 f /// (t2i )  f /// ( x2i ) 

2 ( 4) hf (10, 2i ) 3

 , x2i  10 , 2i  t2i  ....( 5)  III. PROOF OF THEOREM 1:

The proof depends on the following representation of

Sn ( x) for 2ih  x  (2i  2)h , i  0,1,...., m  1

we

have

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International Journal of Engineering and Advanced Research Technology (IJEART) ISSN: 2454-9290, Volume-1, Issue-6, December 2015

x  2ih x  2ih x  2ih )  f (t2i ) A1 ( )  f ( x2i  2 ) A2 ( ) 2h 2h 2h x  2ih x  2ih // //  4h 2 S n ( x2i ) A3 ( )  4h 2 S n ( x2i  2 ) A4 ( ) .......... (6) 2h h

S n ( x)  f ( x2i ) A0 (

On using (3.0) and the conditions

Sn (0)  f // (0) , Sn (1)  f // (1) //

//

we say that

S n (x )

(7)

as given by (6) satisfies (1) is fourth in

[ x2 i , x2 i  2 ] , i  0,1,2,...., m  1.we also need to show that

S // n ( x2i ) where i  1,2,...., m  1 uniquely .For this purpose we use the fact that S /// n ( x2i  2 )  S /// n ( x2i  ) , i  1,2,..., m  1 with the help of (6) and (7) reduce to

whether it is possible to determine

41 2 // 32 35 h S n ( x2i 1 )  h 2 S // n ( x2i )  h 2 S // n ( x2i  2 )  22 22 22 243 243 81 243 81 f ( x2i )  f (t2i )  f ( x2i  2 )  f (t2i  2 )  f ( x2i  2 ) 44 44 44 44 22 ........ (8)

but (7) is a strictly tri

diagonal dominant system has a unique solution[3] .

Thus

S // n ( x2i ) , i  1,2,..., m  1 can be obtained uniquely by the system (8)

This theorem is completed.

IV. ESITIMATES: In order to prove theorem 1.2 , we need the following: Lemma 4.0 : Let

E2i  S // n ( x2i )  f // ( x2i )

max E2i 

f C 4 [0,1] we have

then for

81 2 1 h w( f ( 4) ; ) , i  0,1,2,..., m  1 108 m

(9)

Proof: From (8) we have

32 2 //  41 2 // // h ( S h ( S n ( x2i )  f // ( x2i )) n ( x 2 i  2 )  f ( x 2 i  2 ))   22 22 

35 2 // 243 81   243 h ( S n ( x2i  2 )  f // ( x2i  2 ))   f ( x2i )  f (t 2i )  f ( x2i  2 )  22 44 44   44

243 81 41 32 35  f (t 2i 2 )  f ( x2i 2 )  h 2 f // ( x2i 2 ) h 2 f // ( x2i )  h 2 f // ( x2i  2 ) 44 22 22 22 22   2 4   h f  44 

( 4)

(  3, 2 i ) 

82 4 h f 22

( 4)

54 4 h f 44

( 8 , 2 i ) 

( 4)

(1, 2i ) 

70 4 h f 22

( 4)

32 4 h f 44

( 4)

( 4 , 2 i ) 

54 4 h f 22

  81 (9, 2i )    1 h 4 w( f   22

( 4)

,

( 4)

( 2 , 2 i )

1  ) , 1  1 m 

The result (9) follows on using the property of diagonal dominant [3].

Lemma 4.1: Let f  C [0,1] then 4

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The Change Effect for Boundary Conditions to Minimize Error Bound in Lacunary Interpolation by Forth Spline Function

135 1 hw( f ( 4 ) ; ) 44 m 378 1 (ii) S lll n ( x 2i  )  f lll ( x 2i )  hw( f ( 4) ; ) 88 m 33 1 (iii) S lll n (t 2i )  f lll (t 2i )  hw( f ( 4) ; ) 22 m 31 2 1 (iv) S ll n (t 2i )  f ll (t 2i )  h w( f ( 4) ; ) 44 m 2546 3 1 (v) S l n (t 2i )  f l (t 2i )  h w( f ( 4 ) ; ) 9801 m (i ) S lll n ( x 2i  )  f lll ( x 2i ) 

Proof: From (6) we have

h 3 S lll n ( x 2i  2 ) 

81 243 81 123 2 ll 39 f ( x 2i )  f (t 2i )  f ( x 2i  2 )  h S n ( x 2i )  h 2 S ll n ( x 2i  2 ) 22 44 44 66 66

Hence

81 243 81 123 2 ll f ( x 2i )  f (t 2i )  f ( x 2i  2 )  h ( S n ( x 2i )  f ll ( x 2i )) 22 44 44 66 39 123 2 ll 39  h 2 ( S ll n ( x 2i  2 )  f ll ( x 2i  2 ))  h 3 f lll ( x 2i )  h f ( x 2i )  h 2 f ll ( x 2i  2 ) 66 66 66 1 27 4 ( 4 ) 39 123 2 ll   h 4 f ( 4 ) (  3, 2 i )  h f (1, 2i )  h 4 f ( 4 ) ( 4, 2i )  h ( S n ( x 2i )  f ll ( x 2i )) 22 22 33 66 39  h 2 ( S ll n ( x 2i  2 )  f ll ( x 2i  2 )) 66 h 3 ( S lll n ( x 2i  )  f lll ( x 2i )) 



27 4 1 123 ll 39 h  2 w( f ( 4 ) ; )  ( S n ( x 2i )  f ll ( x 2i ))  h 2 ( S ll n ( x 2i  2 )  f ll ( x 2i  2 )) ;  2  1 22 m 66 66

By using (9) the lemma (4.1)(i) follows ,the proof of lemma (4.1)(ii-v) are similar ,we have only mention that

h 3 S lll n ( x 2i  )  

81 243 81 57 2 ll 105 2 ll f ( x 2i )  f (t 2i  2 )  f ( x 2i  2 )  h S n ( x 2i  2 )  h S n ( x 2i ) 44 44 22 66 66

h 3 S lll n (t 2i ) 

27 81 27 63 9 2 ll f ( x 2i )  f (t 2i )  f ( x 2i  2 )  h 2 S ll n ( x 2i )  h S n ( x 2i  2 ) 22 44 44 66 66

h 2 S ll n (t 2i ) 

18 27 9 2 5 f ( x 2i )  f (t 2i )  f ( x 2i  2 )  h 2 S ll n ( x 2i )  h 2 S ll n ( x 2i  2 ) 11 11 11 33 33

and hS l n (t 2i ) 

12 39 37 34 2 ll 14 2 ll f ( x 2i )  f (t 2i )  f ( x 2i  2 )  h S n ( x 2i )  h S n ( x 2i  2 ) 11 44 44 297 297

Proof of theorem (2): For 0  Z  1, we have

A0 ( Z )  A1 ( Z )  A2 ( Z )  1

....(10)

let x 2i  Z  x 2i  2 . On using (10) and (7) , we obtain

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International Journal of Engineering and Advanced Research Technology (IJEART) ISSN: 2454-9290, Volume-1, Issue-6, December 2015

x  2ih x  2ih )  ( S lll n ( x2i  2 )  f lll ( x)) A1 ( ) 2h 2h x  2ih  ( S lll n (t2i )  f lll ( x)) A2 ( )  L1  L2  L3 2h .......... .......(11)

S lll n ( x)  f lll ( x)  ( S lll n ( x2i )  f lll ( x)) A0 (

A0 ( x)  1, A1 ( x)  1 and A2 ( x)  1

from (4) it follows that Since

f lll ( x)  f lll ( x2i )  ( x  x2i ) f ( 4) ( ) , x2i    x

Therefore

x  2ih x  2ih )  ( S lll n ( x2i )  f lll ( x2i )  ( x  x2i ) f ( 4) ( )) A0 ( ) On using lemma 2h 2h  2h , we obtain

L1  ( S lll n ( x2i )  f lll ( x)) A0 (

x  x 2i

(4.1)(i) and

L1  S lll n ( x2i )  f lll ( x2i )  ( x  x2i ) f ( 4 ) ( ) A0 (

x  2ih ) 2h

Therefore

L1 

135 1 hw( f ( 4 ) ; )  2h f ( 4 ) 44 m

(12)

Similarly

378 1 hw( f ( 4 ) ; )  2h f ( 4) 88 m x  2ih L3  ( S lll n (t2i )  f lll ( x)) A2 ( )  ( S lll n (t2i )  f lll (t2i )  f lll (t2i )  f lll ( x2i ) 2h x  2ih  ( x  x2i ) f ( 4 ) ( )) A2 ( ) (13) 2h L2 

.

From (5) we have

2 ( 4) hf (10, 2i ) 3  t2i , but (3t2i  x2i )  2h

f lll (t 2i )  f lll ( x 2i )  where x2 i 

10 , 2i

L3  (S lll n (t2i )  f lll (t2i )  2h f ( 4) (0 )  f ( 4) ( ) , x2i   , 0  x

Hence

On using the above result with lemma (4.1)(iii), we obtain

L3 

77 1 hw( f ( 4 ) ; ) 22 m

......(14)

putting (12) ,(14) in (11) ,we obtain

956 1 hw( f ( 4) , )  4h f ( 4 ) 88 m

S lll n ( x)  f lll ( x) 

.......(15)

this prove (2) for r=3 to prove (2), when r=2 x

Since S

ll

n

( x)  f ( x)   ( S lll n (t )  f lll (t ))dt  ( S ll n (t 2i )  f ll (t 2i )) ll

t2 i

on using lemma (4.1)(iv) and (15) we obtain

S ll n ( x)  f ll ( x) 

987 2 1 h w( f ( 4 ) , )  8h 2 f ( 4 ) 44 m

.......(16)

which is proof (2) for r=2 the proof (2) for r=1 since x

S l n ( x)  f l ( x)   ( S ll n (t )  f ll (t ))dt  S l n (t 2i )  f l (t 2i ) t2 i

On using lemma (4.1)(v) and (16),we get

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The Change Effect for Boundary Conditions to Minimize Error Bound in Lacunary Interpolation by Forth Spline Function

S l n ( x)  f l ( x) 

884509 3 1 h w( f ( 4 ) , )  16h3 f ( 4 ) 19602 m

.......(17)

Which proof (2) for r=1, if r=0, we have x

S n ( x)  f ( x)   ( S l n (t )  f l (t ))dt  S n (t 2i )  f (t 2i ) t2 i

sin ce S n (t 2i )  f (t 2i )  0 Thus

884509 4 1 h w( f ( 4 ) , )  32h 4 f ( 4 ) 9801 m 1 Since 2mh=1 , then h  put it in above result this completes the proof of theorem (1.2) .

2m Sn ( x)  f ( x) 

V. CONCLUSIONS: Throughout this paper we mentioned that some time the changes of boundary conditions effect on minimizing error bound in the subject of lacunary interpolation by Spline function .

REFERENCES [1] Ahlberg,J.H., E.N.Nilson and J.L.walsh ,(1967) .The theory of Spline and their applications ,Academic press ,new york and London.

[2] Bokhary,M.A.,H.P. Dikshit and A.A. Sharma .(2000),Birkhoff Interpolation on some perturbed roots of unity. Numerical Algorithms.

Vol.25,No.(1-4),(2000),pp.(47-54).

[3] Kincaid, D and W. Cheney,(2002). Nuumerical analysis , Mathematics of scientific computing third edition, printed in USA. [4] Lenard , M., (1999), Modified (0,2) interpolation on the roots of Jacobi polynomial ,I(Explicit formulae), Acta Math.,Hungar ,Vol.84, No.(4), pp.(317-327).

[5] Schoenberg, I.J. , (1986),On Ahlberg – Nilson extention of Spline interpolation : the g-Spline and their optimal properties .J math. Anal .Appl.,Vol.4,No.(2),pp.(207-231).

Muhannad A. Mahmoud, Department of Mathematics, Science , University of Kirkuk, Iraq

College of

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