ChM_teory2

Ɋɨɡɝɥɹɧɟɦɨ ɧɚɣɩɪɨɫɬɿɲɭ ɬɚɤɭ ɿɧɬɟɪɩɨɥɹɰɿɸ – ɥɿɧɿɣɧɭ. ɇɟɯɚɣ ɡɚɞɚɧɚ

f x ɡɧɚɱɟɧɧɹɦɢ f xi

i

0,n. ɉɨɛɭɞɭɽɦɨ ɮɭɧɤɰɿɸ

1

x

– ɥɿɧɿɣɧɭ ɧɚ

x >xi 1, xi @, ɳɨ ɿɧɬɟɪɩɨɥɸɽ ɰɿ ɡɧɚɱɟɧɧɹ:

i

x

xi

1

xi

x

)1(x)

L1

(x)

f (xi 1)

f (xi

)

, x >xi 1, xi

@ (2)

x

x

i 1

x

i

x

ɉɪɟɞɫɬɚɜɢɦɨ ʀʀ ɭ ɜɢɝɥɹɞɿ

i

i 1

)1(x)

n

¦ f (xi )Mi (x).

(3)

Ɍɨɞɿ

d

max

max

z(x)

1

h2 M2 ,

(6)

ɞɟ M2

f cc(x)

i 1,n x [xi 1,xi ]

8

max

, h maxhi , ɳɨ ɞɨɜɨɞɢɬɶ (4)

[a,b]

i

Ɂɚɞɚɱɚ 16 Ⱦɨɜɟɫɬɢ ɨɰɿɧɤɭ

f c(x)

)1c (x)

d

h

M2 .

Ɉɬɠɟ ɦɚɽɦɨ ɡɛɿɠɧɿɫɬɶ ɩɪɨɰɟɫɭ ɿɧɬɟɪɩɨɥɸɜɚɧɧɹ ɡɚ ɞɨɩɨɦɨɝɨɸ ɤɭɫɤɨɜɨ-

ɥɿɧɿɣɧɨʀ ɮɭɧɤɰɿʀ

f (x)

Ɏ1(x)

C[a,b]

ho0(nof)o0

Ɏ1(x)` f (x).

Ɋɨɡɝɥɹɧɟɦɨ ɞɟɹɤɿ ɩɪɨɫɬɨɪɢ:

1. H0

L2

(a,b) - ɝɿɥɶɛɟɪɬɿɜ ɩɪɨɫɬɿɪ, ɜ ɹɤɨɦɭ ɫɤɚɥɹɪɧɢɣ ɞɨɛɭɬɨɤ ɜɢɡɧɚɱɚɽɬɶɫɹ

ɬɚɤ: (u,v)

³b u(x)v(x)dx, ɚ ɧɨɪɦɚ u 0

(u,u) .

a

k

b

2. Hk

W2k (a,b). Ɍɟɩɟɪ

ɫɤɚɥɹɪɧɢɣ

ɞɨɛɭɬɨɤ

(u,v)k

¦

³u(m) (x)v(m) (x)dx, ɚ

u 02 ... u(k) 2 .

m 0a

ɧɨɪɦɚ u k

Ɍɟɨɪɟɦɚ 2 ɇɟɯɚɣ f (x) H2 W22 (a,b). Ɍɨɞɿ

)1

d

2 k

f (k)

(k)

0

h

f

2 , k 1,2.

6.10. Ʉɭɫɤɨɜɨ-ɤɭɛɿɱɧɚ ɟɪɦɿɬɨɜɚ ɿɧɬɟɪɩɨɥɹɰɿɹ

ɇɟɯɚɣ ɞɟɹɤɚ ɮɭɧɤɰɿɹ

f

x ɡɚɞɚɧɚ ɜ ɬɨɱɤɚɯ xi ɫɜɨʀɦɢ ɡɧɚɱɟɧɧɹɦɢ ɬɚ

ɡɧɚɱɟɧɧɹɦɢ ɩɨɯɿɞɧɨʀ:

f xi

,

xi ,

. ɉɨɡɧɚɱɢɦɨ ɱɟɪɟɡ 3

x

yi

yi

f

i

0,n

ɮɭɧɤɰɿɸ, ɹɤɚ ɛɭɞɟ ɿɧɬɟɪɩɨɥɸɜɚɬɢ ɡɚɞɚɧɭ. Ɍɨɞɿ

)3

x

H3i

x ,

x

[xi 1, xi ]

(1)

ɇɟɜɚɠɤɨ ɧɚɩɢɫɚɬɢ ɹɜɧɢɣ ɜɢɝɥɹɞ ɰɶɨɝɨ ɛɚɝɚɬɨɱɥɟɧɚ H3i x ɧɚ ɩɪɨɦɿɠɤɭ :

xi

yi

yic

yi

1,i

yic

yic

yic 1

xi

yi

yic

hi

2 yi 1,i

yi

,

1,i

yi

xi

1

yi

1 yic

1

1,i

hi2

1

hi

x

i

1

y

i

1

i

yi 1,i

y

2

yi

2yi 1,i

yi 1

2

H3 x yi yic x xi

x xi

x xi

x xi 1

.

h

h2

i

i

Ɇɨɠɧɚ ɩɪɟɞɫɬɚɜɢɬɢ ɤɭɫɤɨɜɨ-ɤɭɛɿɱɧɭ ɮɭɧɤɰɿɸ ɿ ɜ ɬɚɤɨɦɭ ɜɢɝɥɹɞɿ:

x

n

yi

i0

x

yic

i1

x

@

.

(2)

)

3

¦

M

M

i

0

ɍɦɨɜɢ ɿɧɬɟɪɩɨɥɸɜɚɧɧɹ: )3

xi

3 xi

i

. əɤɳɨ

ɰɿ

ɭɦɨɜɢ

yi ,

yi ,

0,n

ɩɿɞɫɬɚɜɢɬɢ ɜ (2), ɬɨ ɨɬɪɢɦɚɽɦɨ ɭɦɨɜɢ ɧɚ ɛɚɡɢɫɧɿ ɮɭɧɤɰɿʀ:

Mi0 xj

Gij ,

Mi0

xj

0, i, j 0,n,

Mi1 xj

0, Mi1

xj

Gij .

ɐɿ ɮɭɧɤɰɿʀ ɤɭɫɤɨɜɨ-ɤɭɛɿɱɧɿ,

ɬɨɛɬɨ

Mik

x

3,

x

[xi 1,xi 1], k

0,1

( 3 –

ɦɧɨɠɢɧɚ ɛɚɝɚɬɨɱɥɟɧɿɜ ɬɪɟɬɶɨɝɨ ɫɬɟɩɟɧɹ), ɧɚ ɜɫɿɯ ɿɧɲɢɯ ɩɪɨɦɿɠɤɚɯ ɜɨɧɢ

ɧɭɥɶɨɜɿ. ɇɟɯɚɣ hi

h, ɿ ɩɨɡɧɚɱɢɦɨ s

x

xi

,

x [xi 1, xi ] s > 1, 0@.

1). ȼɜɟɞɟɦɨ M10 s

Mi0 x ,

h

0,1@. ɉɨɛɭɞɭɽɦɨ ɰɸ ɮɭɧɤɰɿɸ.

x [xi , xi 1],

s

ȼɨɧɚ ɡɚɞɨɜɨɥɶɧɹɽ ɭɦɨɜɚɦ :

M10 1 0,

M10

0

M10 1 0.

M10 0

1,

0

1

0

H3 s 1 0 s 1 s2

2s2 s 1

3s2 1

M10 s

0

1

1

2s3

1

0

1

1

2

0

1

0

>

1,0@;

ɧɚɥɨɝɿɱɧɨ

2).

M

20

s

2s3

3s2

1

, Mi0

x

M10 s , x

[xi

1, xi ], s

3).

M

11

s

s s

1 2 , h

M10

s

Mi0 x

, x [xi , xi 1], s 0,1@;

4).

M

21

s

s s

1 2 , h

M10

s

Mi0

x

, x

[xi

1, xi ], s

1,0@.

ɬɟɩɟɪ

ɛɭɞɭɽɦɨ

ɹɜɧɢɣ

ɜɢɝɥɹɞ

ɮɭɧɤɰɿɣ Mik x

ɞɥɹ ɞɨɜɿɥɶɧɨɝɨ

ɩɪɨɦɿɠɤɭ

x [xi 1 , xi 1 ]:­ 2s3

3s2 1, xi d x d xi 1

­hs s 1 2 , xi d x d xi 1

0

°

2s

3

3s

2

1, xi 1 d x d xi

1

x

°

2

Mi x

®

Mi

®hs s 1 , xi 1 d x d xi

°

0, xi 1

x x xi 1

° 0, xi 1

x x xi 1

x

xi

¯

¯

ɞɟ s

(ɹɤɳɨ ɫɿɬɤɚ ɧɟɪɿɜɧɨɦɿɪɧɚ, ɬɨ ɜ ɮɨɪɦɭɥɚɯ ɡɚɦɿɫɬɶ h, ɛɭɞɟ h ɚɛɨ

h

i

hi 1

ɧɚ ɜɿɞɩɨɜɿɞɧɢɯ ɿɧɬɟɪɜɚɥɚɯ).

Ɉɰɿɧɢɦɨ

f

x

)

3

x

C[a,b] . Ɋɨɡɝɥɹɧɟɦɨ ɞɥɹ x

[xi 1, xi

]:

f

x

)3

x

f x

H3i x

f 4 [

x xi 1

2

x xi 2 .

4!

Ɂɪɚɡɭ ɩɨɬɪɿɛɧɨ ɡɪɨɛɢɬɢ ɩɪɢɩɭɳɟɧɧɹ, ɳɨ

f x C4 a,b@. Ɂ ɬɢɯ ɠɟ ɦɿɪɤɭɜɚɧɶ,

ɳɨ

ɿ

ɞɥɹ

ɤɭɫɤɨɜɨ-ɥɿɧɿɣɧɨʀ

ɮɭɧɤɰɿʀ,

ɦɚɤɫɢɦɭɦ

ɡɧɚɯɨɞɢɬɶɫɹ

ɜ ɬɨɱɰɿ

xi xi 1

ɿ ɬɨɦɭ ɞɥɹ ɦɨɞɭɥɹ ɩɨɯɢɛɤɢ ɦɚɽɦɨ:

xi

2

f x )3 x d

¨

¸

, f x )3 x C[a,b]

d

.

M

4i § h2 ·2

M4i h4

M4h

4

©

¹

24

4

384

384

Ɂɜɿɞɫɢ ɨɬɪɢɦɚɽɦɨ ɬɟɨɪɟɦɭ:

a,b@ ɡɚɞɚɧɚ ɜ ɬɨɱɤɚɯ xi ɫɜɨʀɦɢ ɡɧɚɱɟɧɧɹɦɢ

Ɍɟɨɪɟɦɚ əɤɳɨ ɮɭɧɤɰɿɹ

f

x C4

yi

f xi , yic

f xi

,

, ɬɨ ɞɥɹ ɤɭɫɤɨɜɨ-ɤɭɛɿɱɧɨʀ ɟɪɦɿɬɨɜɨʀ

i

0,n

)3

¦

iMi

iMi

ɿɧɬɟɪɩɨɥɹɰɿʀ

n

0

yc

1

ɦɚɽ ɦɿɫɰɟ ɨɰɿɧɤɚ

x

y

x

x

i 0

x )3 x

d

f

C

M4h4

.

ɞɥɹ ɩɨɯɿɞɧɨʀ