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S- . ., . .

(, )
S- . S- н - , : , . S- . 3- C 1 5- C 2 (.. ). , 3- C 1 . S-. S-. 1. S- 1.1. S- [a, b] {xk }k k
L {l }ll =0 , =

=K =0

,

xk = a + kh , h -- . [a, b] -

l = a + lH ,

H = mh, m Z .

85


5. Part 5. Mathematical Theories

y = ( y0 , y1 ,..., yK ) R
n

K +1

y0 R .
ajx
j

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


j =2

n

n a0 , a1 . :

l (u ) =


k =0

M

(u (l + kh) - y

ml + k

)2 .

P n , l :
0 a0 = y0 , a10 = y 0

(1.1)


l a0 = gl -1 (l - l -1 ) = gl -1 ( H ), a1l = g l -1( H ).

(1.2)
n m, M l +1

1. S- S

( x) ,
.

gl ( x) l x <

2. S- S, [a, b] . (1.1) :
0 a0 = g L -1

( H ), a10 = g L -1( H ).

(1.3)

L н , .
86


. ., . . -- -- 2006, . 2, . 85н104 Silaev D. A., Korotaev D. O. -- MCE -- 2006, v. 2, p. 85н104

1.2. l (u ) :
l l l S2 a0 + S3a1l h + S4 a2 h 2 + S5 a3h3 = 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

.

(1.5)

a i = ahi , i = 01, 2, 3. , i (1.2) (1.4) :
l l a l0-1 + ma1-1 + m 2 a l2-1 + m3 a 3-1 = a l0; 2 l -1 l -1 l -1 l a1 + 2ma 2 + 3m a 3 = a1,
l S 2 a l0 + S3 a1 + S4 a l2 + S5 a l3 = P l ; 1 l l l l l. S3 a 0+ S4 a1+ S5 a 2+ S6 a 3= P2

(1.6)

(1.7)

:

S2 A1 = S3
l 1 l 2

S3 S4 , A2 = S4 S5
l 0 l 1

S5 1m m 2 m3 ,B = ,B = . S6 1 0 1 2 2m 3m 2
l a2 , l = 01,..., L - 1. , l a3

,

Pl =

P X P

2l

=

a , X a

2 l +1

=

:
87


5. Part 5. Mathematical Theories


-E A1 B1 0 0 0

0 A2 B2 0 0 0

0 0 -E A1 B1 0

0 0 0 A2 B2 0

... B1 ... ... ... ... 0 0 0 0

B2 0 0 0 0 A

... A1

2



X

X X X X X

1 2 3 4 2 L -1 0

=P



0 P 0 0 P

0 1 L -1

.

(1.8)

-- 4 L x 4 L . E, , -- : E =

10 . 01

(1.1). 1:

1+ U=

12 1 m T35 - m3T34 A A m m2 2 T35 -3 T34 A A S4 S5

1 m+ m2T45 - A m 1+2 T45 -3 A

13 m T34 A , m2 T44 A

(1.9)



A = det( A2 ) = det

S5 = S 4 S6 - S52 > 0, S6

(1.10) (1.11)

Tij = Si S j - Si -1S j +1.

, (1.8) 2 L x 2 L :
U , , ,
88
1


. ., . . -- -- 2006, . 2, . 85н104 Silaev D. A., Korotaev D. O. -- MCE -- 2006, v. 2, p. 85н104



-E U 0 0

0 -E U 0

0 0 -E 0

U X 0 - B2 A2-1 P L -1 - 0 X 2 - B2 A2 1P0 X 4 = - B A-1P1 , (1.12) 0 22 - - E X 2 L - 2 - B2 A2 1 P L - 2

. (1.4). , U, , U = B1 - B2 A2 -1 A1 . 1.3. S- 1. m M S m, M [ y ]( x) . 2. m M , p U L . S m, M [ y ]( x) . 1.4. S- 3. f ( x) C4 [a, b] нн :

| f ( xk ) - f k | Ch

4 +

,

(1.13)

C h . , , U (1.9) . S m, M ( x)

l = a + lH (.. S

3 m, M

( x) C1[a, b] ),

x [a, b] :

89


5. Part 5. Mathematical Theories

f

( p)

( x) - S

( p) m, M

( x) C p h

4- p

, p = 01, 2, 3. ,

(1.14)

4. :

f ( x) C4 [a, b] , , | f (0) - f 0 | Ch
3+

| f ( xk ) - f k | Ch

4 +

,

(1.15)

C h . , , U (1.9) . S m, M ( x) l = a + lH
( p)




4- p

(..

S

3 m, M

( x) C1[a, b] )

x [a, b] :

f

( x) - S

( p) m, M

( x) C p h

, p = 01, 2, 3. ,

(1.16)

. [1], m M *, * 0.93, , U . 1.5. S- S- B j ( x) нн

y = ( y0 , y1 ,..., yK ) R

S-,
K +1

y0 R : { y i = ij, i = 0,..., K }.

,


j =0

K

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

{

S-, yi , i = 0,..., K } .



S- :

90


. ., . . -- -- 2006, . 2, . 85н104 Silaev D. A., Korotaev D. O. -- MCE -- 2006, v. 2, p. 85н104

0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2

. 1.

2. S - 2.1. - r - :

{i = ih1 , i = 0,1,..., K1},{ k = kH1 , k = 0,..., L1},
H1 = m1h1 , K1 = m1 L1 , 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] Ѕ [0, 2 ].
{ yij = f (i , rj ),

(2.1)

i = 0,..., K1 ,

j = 0,..., K 2 } нн

, . j = 1,..., K 2 S

S j ( ) [0, 2 ]

{ yij , i = 0,..., K1} .

91


5. Part 5. Mathematical Theories

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 нн , {z j } , f r (r , ) ,
r =0

.

z 0 =

1 h2

3 1 3( z1 - z0 ) - 2 ( z2 - z0 ) + 3 ( z3 - z0 )

(2.2)

нн . {z j } z 0 S (r ) нн S

[0,1] . , m2 < M 2



( = 0.93096) . , U . S (r )

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

92


. ., . . -- -- 2006, . 2, . 85н104 Silaev D. A., Korotaev D. O. -- MCE -- 2006, v. 2, p. 85н104

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

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

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

4. p- - r - r - ,

dp S j ( ), z j = d p

j = 1,..., K 2 , z0 = 0 ,

p = 1, 2, 3.

r , = k = k + 0 . 2.2. S- Ci ( ) , r 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

K1 -1

D j (r )


i =0

yij Ci ( ) =

K1 -1 K


i =0 j =0

2

yij Ci ( ) D j (r ) .

(2.4)

{z j = S j ( )} -

93


5. Part 5. Mathematical Theories

S j ( ) =

K1 -1


i =0

yij Ci ( ) .

.2.


0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04

. 2.

{ k = kH1 , k = 0,..., L1}, H1 = m1h1 {Rl = lH 2 , l = 0,..., L2 }, H 2 = m2 h2 . S- :

= kH1 + , r = lH 2 + r , | | H1 | r | H 2 .

(2.5)

BS-, S-, :

Ci ( ) =


p =0

3

c ipk p, D j (r ) =


q =0

3

cqjl r q.

94


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S ( , r ) , :

S ( , r ) =
=

K1 -1 K


i =0 j =0

2

3

yi

j

p =0

c ipk
i ij pk

p


q =0
j ql

3

j d ql r q =


p =0 q =0

3

3

r

K1 -1 K 2 p q i =0 j =0



yc d

=


p =0 q =0

3

3

a kl p r q. pq

(2.6)

(2.4) . 4.



p+q

p r

q

S (r , ), 0 p + q 3
K1 -1 K

:


i =0 j =0

2

dp dq yij Ci ( ) q D j (r ) , d p dr

r . 2.3. h = max(h1 , h2 ) . 5. m1 < M 1 , m2 < M 2 :

f C 4 [0,1] Ѕ [0, 2 ].
S ( , r ) :

p+q q

(2.7)

p r

S ( , r ) -



p+q q

p r

f ( , r ) < C pq h

4- p -q

, 0 p + q 3.

95


5. Part 5. Mathematical Theories

, , . 2.4. c :

1 r r

2 u 1 u = - p (r , ), (r , ) D, r + 2 2 r r u (r , ) D = f (r , ).

(2.8)

D н , . . 1) . 2) . 3) . .

S ( , r ) =

K1 -1 K i =0

u
j =1

2

ij

Ci ( ) D j (r ) ,

(2.9)

Ci ( ) D j (r ) н . r . Cl ( ) Dk (r ) , l , k l = 0,... , K1 - 1,

k = 1,... , K 2 ,

, (h2 k , h1l ) D (..

96


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D). D. :


D

2u u 1 2u + r 2 + r r 2 r

Cl ( ) Dk (r )rdrd =
2

=-


D

p(r , )Cl ( ) Dk (r )r drd

(2.10)

.

, r . . . r [ Rq , Rq +1 ], [ p , p +1 ], Rq = qH 2 ,

q = 0, ..., L2 , p = pH1 , p = 0, ..., L1 . . :


D

2u u 1 2u + r 2 + r r 2 r

Cl ( ) Dk (r )rdrd =

=


p =0 q =0 Rq

L1 -1 L2 -1



Rq

+1



p +1

p

2u u 1 2u r 2+ Cl ( ) Dk (r )rdrd . + 2 r r r

(2.9) :


i , j p ,q

uij

Rq

Rq



+1



r D(r ) Dk (r ) + rDj (r ) Dk (r )dr j
2





p +1

Ci ( )Cl ( )d +

p

97


5. Part 5. Mathematical Theories
Rq
+1



+

Rq



D j (r ) Dk (r )dr





p +1

p

Ci( )Cl ( )d =
Rq Rq
+1

=


i , j p ,q

uij r 2 D j (r ) Dk (r )


Rq

+

Rq



+1

rDj (r ) Dk (r ) -

-2rD j (r ) Dk (r )dr
Rq
+1





p +1

Ci ( )Cl ( )d +

p +1

p

+

Rq



D j (r ) Dk (r )dr Ci ( )Cl ( )

p +1

-

p





p

Ci ( )Cl ( )d =

=


i, j

2 uij Cl ( )Ci ( )d Dk (1) D j (1) + 0

(

(2.11)

1 - rDk (r ) D j (r ) - 2rDk (r ) D j (r )dr + 0

2 1 - Cl ( )Ci ( )d Dk (r ) Dj (r )dr = - 0 0


D

p(r, )Cl ( ) Dk (r )r 2 drd.

l k uij . , , :


i, j

uij C i ( ) D j (r )
D

= f ( , r ) .

:

98


. ., . . -- -- 2006, . 2, . 85н104 Silaev D. A., Korotaev D. O. -- MCE -- 2006, v. 2, p. 85н104


i, j

uij C i (l ) D j (1) = f ( ), l = 0,... , K1 - 1.

(2.12)

D , , ( , ). , (i, j ) , , . -, , , . -, , , , . (2.11) (2.12) uij , .. . 6. u (r , ) -- (2.8), S (r , ) нн , . , , U, r , . :

u (r , ) - S (r , ) < Ch 4 , h = max(h1 , h2 ).
. u S (r , ) =
K1 -1 K


2






i = 0 j =1

vij Ci ( ) D j (r ) . u C 4 . -

y = u - S . , 5, ,

99


5. Part 5. Mathematical Theories

y ( n ) (r , ) < C1h

4-n

, n = 0,1, 2, 3 . .. u --

(2.8), :


D

uCl ( ) Dk (r )r 2 drd =


D

p (r , )Cl ( ) Dk (r )r 2 drd .

, . u = y + S :
L1 -1 L2 -1 p =0 q =0


Rq




Rq

+1



p +1

p



2 1 2 ++ ( y + S ) Cl ( ) Dk (r )r 2 drd 2 2 r r r

=




=


D

r 2 p (r , )Cl ( ) Dk (r )drd .

y , , :


i, j

1 2 vij Cl ( )Ci ( )d Dk (1) Dj (1) + rDk (r ) Dj (r ) - 2rDk (r ) Dj (r )dr - 0 0

2 1 - Cl (r )Ci ( )d Dk (r ) D j (r )dr = 0 0

=-


D

p (r , )Cl ( ) Dk (r )r drd -
2


p , q Rq


Rq

+1



p +1

y (r , )Cl ( ) Dk (r )drd .

p

, (2.11) y . ij A = alk .

{}

z = S (r , ) - S (r , ) =


i, j

(uij - vij )Ci ( ) D j (r ) .

100


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(2.11), :


i, j

ij alk (uij - vij ) = ylk , Az = y ,

z = zij zij = uij - vij ,

{

}

y = ylk ylk =


p , q Rq

Rq

+1



p +1

p

y (r , )Cl ( ) Dk (r )drd .


p , q Rq Rq
+1

Rq

+1



p +1

y (r , )Cl ( ) Dk (r )drd
p +1

p




p ,q Rq



y (r , )Cl ( ) Dk (r )drd
Rq
+1

p

sup y


p , q Rq



p +1

Cl ( ) Dk (r )r 2 drd .

p



c

p ,l

=

[ p ,

sup

Cl ( ) , d

q ,k

= sup

p +1

]

r[ Rq , Rq +1 ]

Dk (r )r 2 .

5, y , :
sup y


p , q Rq


Rq

+1



p +1

Cl ( ) Dk (r )r 2 drd Ch

2


p ,q

c p , l d q , k H1 H 2

p

Cm1m2 h

4


p ,q

c p ,l d

q ,k

= Cm1m2 h 4


p

c p ,l


q

d

q ,k

.

101


5. Part 5. Mathematical Theories

, . 3 ( ), [1] [2]. , , , zn = Cl (nH1 ) -
l , m1n



n -t

zt ,

l , m1n

--

, = max( 1 , 2 ), 1 , 2 н U (1.9). r . , , , , , h1 h2 , . S- , , A , :

z = A-1 y
zij A
-1 l ,k

zA

-1

y
4




max ylk C2 h



S (r , ) - S (r , ) C2 h 4 ( 5).

u(r,) - S (r,) u(r,) - S (r,) + S (r,) - S (r,) < C1h4 + C2h4 = Ch4.

. 2.5. , , :

1 u 1 2u = 2, r (0,1), (0, 2 ); r + 2 2 r r r r u (r , ) r =1 = sin 2 ( ).
102


. ., . . -- -- 2006, . 2, . 85н104 Silaev D. A., Korotaev D. O. -- MCE -- 2006, v. 2, p. 85н104

. 1, 54 = 5, 0625 .
1

, , , 1,5 -

- r 2 4 6 9 14 21




-2

1, 99746 10 8, 608 10 1, 791 10 5, 46 10 1, 01 10 2,14 10

-3

2,32 4,804 3,28 5,45 4,66

-3

-4

-4 -5


1 0,9 0,8 0,7 0,6 u(r,phi) 0,5 0,4 0,3 0,2 0,1 0 1 4 7 10 13 16 19
r

19 10 22 1
phi

. 3.

103


5. Part 5. Mathematical Theories

1. .., .. S- . .: . . . .10. -- .: - , 1984, -- . 197. 2. .., .., .. . . N6, 1996 . , 175- . .1, . 22н25.

SOLVING OF BOUNDARY TASKS BY USING S-SPLINE Silaev D. A., Korotaev D. O. (Russia, Moscow)

This article is dedicated to use of S-spline theory for solving equations in partial derivatives. For example, we consider solving of Puasson equation. S-spline н is a piecewise-polynomial. Its koefficients are defined by two states. Its first part of koefficients are defined by smoothness of spline. The least koefficients are determined by least-squares method. According to order of considered polynomial and number of conditions of first and second type we get S-splines with different properties. At this moment we have investigated order 3 S-splines of class C 1 and order 5 S-splines of class C 2 (they meets conditions of smoothness of order 1 and 2 accordinally). We will consider how the order 3 S-splines of class C 1 can be applied for solving equation of Puasson on circle and on other areas.

104