Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mce.biophys.msu.ru/archive/doc15531/doc.pdf Дата изменения: Wed Oct 17 18:32:04 2007 Дата индексирования: Mon Oct 1 20:24:51 2012 Кодировка:

S- . ., . .
(, )
S-. . (.. ). , . , ( ) . S- , .

1. S- [a, b] {xk }k k
{l }ll
y = ( y0 , y1 ,..., yK ) R
K +1

=K =0

,

xk = a + kh , h -- . [a, b] К
=L =0

, l = a + lH , H = mh, m Z .

y 0 R .



SP u : u ( x) = a0 + a1 +

n


j =2

n

aj x
M

j



n

a0 , a1 . : l (u ) =


k =0

(u (l + kh) - y

ml + k

)2 .

SP n , l :


6. Part 6. Mathematical modelling researches and methods
0 a0 = y0 , a10 = y 0,

(1.1)
l 1

a = gl -1 (l - l -1 ) = gl -1 ( H ), a = g l -1( H ).
l 0 l 0 l 1

(1.2)

a = gl (0) a = g l (0) , (1.2) . n 1. S- Sm, M ( x) , gl ( x) l x < l +1 . 2. S- S, [a, b] . (1.1) : 0 a0 = g L -1 ( H ), a10 = g L -1( H ) (1.3) L ­ , . :

l l l S2 a0 + S3 a1l h + S4 a2 h 2 + S5 a3 h3 = P l 1 l l l S3a0 + S4a1l h+ S5a2h2 + S6a3h3 = P2l

(1.4)



Sj =


k =0

M

k j , Pjl =


k =0

M

y

ml + k

k

j +1

.
i

(1.5)

a i = ah , i = 01, 2, 3. , % i (1.2) (1.4) : a l0-1 + m a1-1 + m 2 a l2-1 + m3 a 3-1 = a l0 % %l % %l % (1.6) 2 l -1 l -1 l -1 l % % % % a1 + 2m a 2 + 3m a 3 = a1


S2 a l0 + S3 a1 + S4 a l2 + S5 a 3 = P l % %l % %l 1 S3 a l0+ S4 a1+ S5 a l2+ S6 a 3= P2l % %l % %l

(1.7)

(1.1). . , 2 L x 2 L ,
594


. ., . . -- -- 2005, . 2, . 593 ­ 595 Silaev D. A., Korotaev D. O. -- MCE -- 2005, vol. 2, p. 593 ­ 595

. (1.4). [3]. [2] [3] S-. m < M ( 093) S- . .

1. f ( x) C4 [a, b] -

| f ( xk ) - f k | Ch
4+

,



C h. , . S m, M ( x) c
3 m, M


1



l = a + lH







(..

S ( x) C [a, b] ), x [a, b] : () f ( p ) ( x) - SmpM ( x) C p h 4 - p , p = 01, 2, 3 , (1.8) ,

: | f (0) - f 0 | Ch3+ 2. 2.1. - r - :
{ i = ih1, i = 0,1,..., K1},{ k = kH1, k = 0,..., L1}, H1 = m1h1, K1 = m1L1, K1h1 = 2 ,

{rj = jh2 , j = 0,1,..., K 2 },{Rl = lH 2 , l = 0,..., L2 }, H 2 = m2 h2 , K 2 = m2 L2 , K 2 h2 = 1.

f ( , r ) , f 4
r



,





f C 4 [0,1] в [0, 2 ] .



{ yij = f (i , rj ), i = 0,..., K1 , j = 0,..., K 2 } - , . j = 1,..., K 2 S- S j ( ) [0, 2 ] { yij , i = 0,..., K1} .

595


6. Part 6. Mathematical modelling researches and methods

f ( , rj ) rj ,

S

( p) j

( ) -

p f ( , rj )
p

< Ch14- p , [0, 2 ], p = 0,1, 2, 3.

% , [0, 2 ] . % {z j = S j ( ), j = 1,..., K 2 , z0 = y00 } . z 0 - ,

{z j } ,

% f r (r , )

r =0









. (2.1)

, 1 z 0 = 3( z1 - z0 ) - h2 . {z j } z

3 1 ( z2 - z0 ) + ( z3 - z0 2 3
0

)




S% (r ) - S-

[0,1] . , m2 < M 2 . , % S% (r )

% f ( , r ) r [0,1] . 3. - r {z j = S j ( ), j = 1,..., K 2 , z0 = y00 } ,
S ( , r ) = S (r ) . :

S ( , r ) , r : z 0 S (r ) ,

S ( , r ) = {S (r ) | {z j = S j ( ), j = 1,..., K 2 , z0 = y00 }} .

, r 3 , , r R j .
r = R j :

596


. ., . . -- -- 2005, . 2, . 593 ­ 597 Silaev D. A., Korotaev D. O. -- MCE -- 2005, vol. 2, p. 593 ­ 597

p p S ( , r ) = p S ( , r + 0), p = 0,1, 2, 3. r p r 4. p-

-r-

( p = 1, 2, 3 )
dp S j ( ), j = 1,..., K 2 , z0 = z j = d p

- r -,
0 , p = 1, 2, 3. r ,



p p

S ( , r ) ,



p - S ( , r ) , 3, 4. 2.2. - r -.
h = max(h1 , h2 ) .

= k = k + 0 . , : 5. p+q S ( , r ), 0 p + q 3 q - r p r q

2. m1 < M 1 , m2 < M 2 f C 4 [0,1] в [0, 2 ]


- r - S ( , r ) :

p+q q



p+q q

p r

S ( , r ) -

p r

f ( , r ) < C pq h

4- p - q

, 0 p + q 3,

p, q 0.

597


6. Part 6. Mathematical modelling researches and methods

2.3.

f1 ( , r ) = r 2 sin 2 e

-r

2

,

m1 = m2 = 3 M 1 = M 2 = 4 . k ( ) .
L1 = L2

f1
k
8



f1
k

r

f1
k
-3

2 r

f1



K
1.222 в 10
-2
-3 -4

12 24 48 96 192 384

6.297 в 10
3952 в 10 .

-4

1.031в 10

-3
-5 -6

7.652 в 10
15.259

-5 -6

15.933 6.759 в 10 16.084 4.648 в 10 16.012 3.617 в 10 16.003 3.330 в 10

9.406 в 10

-4

8.135 1.502 в 10 7.854 1.912 в 10

8.140 7.855 7.963 7.991 7.997

2.457 в 10 1.535 в 10

14.541 1.198 в 10 12.851 1.504 в 10 10.860 1.882 в 10 8.066

-4

-7 -9

-7 -8 -9

-5 -6 -7

7.963 2401 в 10 .

-5 -6 -7

9.590 в 10

7.991 3.004 в 10 7.998 3.757 в 10

5.993 в 10

-10

16.001 4.129 в 10

2.353 в 10

3. S- 3.1. S- S- B j ( x) -
S-, % : { y i = ij , i = 0,..., K } . ,


j =0

K

y j B j ( x) = S ( x)

S-, { yi , i = 0,..., K } . {i = ih1 , i = 0,1,..., K1},

{ k = kH1 , k = 0,..., L1}, H1 = m1h1 , K1 = m1 L1 , K1h1 = 2 ,

598


. ., . . -- -- 2005, . 2, . 593 ­ 599 Silaev D. A., Korotaev D. O. -- MCE -- 2005, vol. 2, p. 593 ­ 599

{rj = jh2 , j = 0,1,..., K 2 }, {Rl = lH 2 , l = 0,..., L2 }, H 2 = m2 h2 , K 2 = m2 L2 , K 2 h2 = 1.

0, 12 0, 1 0,08 0,06 0,04 0, 02 0 -0, 02 1 -0,04 7 13 52 35 19 18 25 31 37 43 49 55 1 0, 1-0,12 0, 08-0,1 0, 06-0,08 0, 04-0,06 0, 02-0,04 0-0,02 -0,02-0 -0,04--0,02

S-.


r

Ci ( ) ,
D j (r ) .

S ( , r ) = {S (r ) | {z j = S j ( ), j = 1,..., K 2 , z0 = y00 }} = S (r ) =

=


j =0

K

2

z j D j (r ) =


j =0

K

2

D j (r )


i =0

K1

yij Ci ( ) =


i =0 j =0

K1

K

2

yij Ci ( ) D j (r )

{z j = S j ( )}
S j ( ) =



i =0

K1

yij Ci ( ) .

{ k = kH1 , k = 0,..., L1},



H1 = mh1 1



{Rl = lH 2 , l = 0,..., L2 }, H 2 = m2 h2 . S- : % % % % = kH1 + , r = lH 2 + r , | | H1 | r | H 2 (3.2)
599


6. Part 6. Mathematical modelling researches and methods

BS- S- :

Ci ( ) =


p =0

3

%p cipk , D j (r ) =


q =0

3

cqjl r %

q

S ( , r ) , :
S ( , r ) =

y
i = 0 j =0 ij p =0

K1

K

2

3

% c ipk

p


q =0

3

j d ql r q = %


p =0 q =0

3

3

%% r

K1 K 2 p q i =0 j =0



yij c ipk d

j ql

=


p = 0 q =0

3

3

p a kl r pq % %

q

(3.3)

3.2. S B-. 2- : (3.4) y - y = sin(2 x), x (0,1) (3.5) y (0 ) = y (1), y ( 0 ) = y (1) y (0) = 1, y (1) = 0 : y ( x) = yi B i ( x) , , B j ( x) :
yi


k =0
k

L -1

k +1

Bi ( x) B j ( x) - B i ( x) B j ( x)dx = sin(2 x) B j ( x)dx
0

1

(3.6)

, :
yi B i ( x) B j ( x) + B i ( x) B j ( x) dx + sin(2 x) B j ( x) dx = yi
0 0 1 1


k =0

L -1

B i ( x ) B j ( x )



k +1

= yi B i ( x) B j ( x)

1 0

k

yi B i (0) = 1, yi B i (1) = 0 , ,

(3.6) : (3.7)
600


. ., . . -- -- 2005, . 2, . 593 ­ 601 Silaev D. A., Korotaev D. O. -- MCE -- 2005, vol. 2, p. 593 ­ 601

. :
k L ­ k 6 24.8 27.6 15.8 15.9 2*6=12 2*12=24 2*24=48 2*48=96 2.53x10 2.35x10 1.57x10 6.24x10
-5 -6 -7

L - 6 2*6=12 2*12=24 2*24=48 2*48=96

1.07x10 4.32x10 1.56x10 9.92x10 6.23x10

-4 -6 -7

10.8 14.9 15.7 15.93

-9
-10

1.001x10

-8

-10

3.3. S . . u = p( x, y ), x ( A1 , B1 ), y ( A2 , B2 ), u (0, 0) = u0 (3.8)
p( x, y ) = 1 (cos 2 x - 1) sin( y ) , A1 = A2 = 0 , 2

B1 = B2 = 2 . u u ( x, y ) = yij Bij ( x, y )
(3.8): 2 2 (3.9) uij 2 + 2 B ij ( x, y ) = p ( x, y ) y x .3.2, (3.9) B kl , , :
uij
B1 B2

A1 A2



2 B ij ( x, y ) kl 2 B ij ( x, y ) kl B ( x, y ) + B ( x, y )dxdy x 2 y 2

=

B1 B2

A1 A2



p( x, y ) B kl dxdy

(3.10)

601


6. Part 6. Mathematical modelling researches and methods
uij
B1 B2 B2

A2



B1 B ij ( x, y ) kl B ij ( x, y ) kl B ( x, y )dy + B ( x, y )dx - A x y A1 1 A2 =
B1 B2

B1

B2

(3.11)
kl

A1 A2



ij B ( x, y ) B kl ( x, y ) + B ij ( x, y ) B kl ( x, y ) dxdy x x y y

A1 A2



p( x, y ) B dxdy

, : uij B ij (0, 0) = u0 {uij } . : u = 2 + sin x sin y
154 = 50625 . .
L - 4 1.5*4=6 1.5*6=9 1.5*9=14 8.6x10 1.8x10
-2
-3

24 = 16



k

L - 2


-2



k

1.997x10 8.6x10
-3

4.8 3.28
-4

2*2=4 2*4=8 2*8=16

2.32 9.93 13.78

5.46x10

-4

8.67x10 6.27x10

-4
-5

1.0001x10

5.45

3.4.

B


A A

B

S ( x)dx =


A k =0 3

BK

yk Bk ( x)dx =
i +1

y
k =0 k

K

B

Bk ( x)dx =


k =0

K

yk ck ,

A



ck = Bk ( x)dx =


n=0 i =0

L -1

ain, k H i +1

-

. ain, k - i - n - k - (.. { yi = ik , i = 0,..., K } ). 4- .

602


. ., . . -- -- 2005, . 2, . 593 ­ 603 Silaev D. A., Korotaev D. O. -- MCE -- 2005, vol. 2, p. 593 ­ 603

3.5. - r (3.1) :


00

1 2

S ( , r )drd =
1

y
k =0 l =0 kl 0

K1

K

2

1

Ck ( ) Dl (r )drd =
0

2


k =0 l =0

K1

K

2

ck c l ykl , %



% c l = Bl (r )dr =
0

L2 -1 3


n=0 i =0

% ai i +1

n ,l

H

i +1 2

,

H2 =

1 L2

,



% a

n ,l i

-

i -

n - l - [0,1] (.. { y j = jl , j = 0,..., K 2 } ). ck =
2


0

Dk (r )dr =


n=0 i =0

L1 -1 3

ain ,k i +1

H

i +1 1

, H1 =

2 L1



ain, k - i - n - k - [0, 2 ] (..
{ y j = jk , j = 0,..., K1} ). sin rP6 ( ) 2 r sin 2
L - 12 2*12=24 2*24=48 2*48=96 2*96=192
-8 -9 -11



k

L - 12


-7 -8



k

9.15x10 2.82x10

3.11x10 3.66x10 2.8x10

32.45 2*12=24 32.34 2*24=48 32.19 2*48=96 31.41 2*96=192

8.49 12.8 14.56 15.3

8.711x10 2.71x10

-9 -10 -11

-12 -14

1.96x10 1.27x10

8.615x10

3.6. . M=80, m=40.

603


6. Part 6. Mathematical modelling researches and methods

: 1. .., .. S- . ". . ", , 2003 . .157. 2. .., .. S- . .: . . . .10. .: - , 1984, .197-206. 3. .., .., .. . . N6, 1996 . , 175 . .1, .22-25.

604


. ., . . -- -- 2005, . 2, . 593 ­ 605 Silaev D. A., Korotaev D. O. -- MCE -- 2005, vol. 2, p. 593 ­ 605

S-SPLINE APPLICATION Silaev D. A., Korotaev D. O.
(Russia, Moscow)

This paper considers application of special S-splines. These spline's construction is based on condition of 1st derivative's smoothness and on the method of least squares. Distinction of these splines is their semi-locality (i.e. every polynomial knows all the preceding function's values and doesn't know the following ones). Basic S-splines are constructed using these splines that give us an ability to consider any function as linear combination of basic splines. Basic splines are applied for function approximation, solving of differential equations (using method of Galerkin) and for getting a quadrature formulas. Sspline on circle is constructed in the same way that allows us to solve the same tasks for circle and for more complex areas.

605