ChM_teory2

Ƀɨɝɨ ɯɚɪɚɤɬɟɪɢɫɬɢɱɧɟ ɪɿɜɧɹɧɧɹ
qn 4qn 1 5qn 2 0,
q2 4q 5 0, q1 1, q2	5.

əɜɧɢɣ ɞɜɨɤɪɨɤɨɜɢɣ ɦɟɬɨɞ ɧɚɣɜɢɳɨɝɨ ɫɬɟɩɟɧɹ – ɧɟɫɬɿɣɤɢɣ ɦɟɬɨɞ ɨɫɤɿɥɶɤɢ

q2 5.

Ɍɟɨɪɟɦɚ 1 ɇɟɯɚɣ ɛɚɝɚɬɨɤɪɨɤɨɜɢɣ ɦɟɬɨɞ ɡɚɞɨɜɨɥɶɧɹɽ ɤɨɪɟɧɟɜɿɣ ɭɦɨɜɿ ɬɚ ɩɪɚɜɚ

ɱɚɫɬɢɧɚ ɡɚɞɨɜɨɥɶɧɹɽ ɭɦɨɜɭ Ʌɿɩɲɢɰɹ,

f (t,u)

d L

u

v

, ɬɨɞɿ ɦɚɽ ɦɿɫɰɟ

f (t,v)

ɨɰɿɧɤɚ ɬɨɱɧɨɫɬɿ

§

max

\k

·

zn

yn

u(tn )

zn

©

zk

¹

(4)

ɞɟ \k

d M¨ max

¸ ,

– ɩɨɯɢɛɤɚ ɚɩɪɨɤɫɢɦɚɰɿʀ.

k 0,m 1

k

0,n 1

ɛ)

f

u ,

0.

ȼ

ɰɶɨɦɭ

ɜɢɩɚɞɤɭ

ɪɨɡɜ’ɹɡɨɤ

ɡɚɞɚɱɿ

Ʉɨɲɿ

(1)

u(t)

u0e (t

t0 )

o 0

ɿ ɜɿɧ ɽ ɚɫɢɦɩɬɨɬɢɱɧɨ ɫɬɿɣɤɢɦ ɜɿɞɧɨɫɧɨ 0. ɋɥɿɞ ɨɱɿɤɭɜɚɬɢ,

tof

ɳɨ ɱɢɫɟɥɶɧɢɣ ɦɟɬɨɞ ɦɚɽ ɬɭ ɠ ɜɥɚɫɬɢɜɿɫɬɶ. Ɍɨɛɬɨ

yn

yn 1

y0

u0

(5)

Ⱦɥɹ ɹɜɧɨɝɨ ɦɟɬɨɞɭ ȿɣɥɟɪɚ

1 W .

yn 1

yn Wf (Fn , yn )

yn

yn

(1

)yn

qyn , q

Ɍɨɦɭ (5) ɦɚɽ ɦɿɫɰɟ ɬɿɥɶɤɢ

ɹɤɳɨ

q

1 ɚɛɨ

1

q 1.

ɉɪɚɜɚ

ɧɟɪɿɜɧɿɫɬɶ

ɜɢɤɨɧɭɽɬɶɫɹ ɞɥɹ ɞɨɜɿɥɶɧɨɝɨ W , ɚ ɥɿɜɚ

ɬɿɥɶɤɢ ɞɥɹ W

2

.

Ɇɟɬɨɞ ɪɨɡɜ’ɹɡɚɧɧɹ ɡɚɞɚɱɿ Ʉɨɲɿ ɧɚɡɢɜɚɽɬɶɫɹ ɭɦɨɜɧɨ ɫɬɿɣɤɢɦ, ɹɤɳɨ ɜɿɧ

ɫɬɿɣɤɢɣ

ɩɪɢ

W d

0 .

əɤɳɨ

ɠ

ɜɿɧ

ɫɬɿɣɤɢɣ

ɞɥɹ

0,

ɬɨ ɬɚɤɢɣ

ɦɟɬɨɞ

ɧɚɡɢɜɚɽɬɶɫɹ ɚɛɫɨɥɸɬɧɨ ɫɬɿɣɤɢɦ.

Ɍɚɤɢɦ ɱɢɧɨɦ ɹɜɧɢɣ ɦɟɬɨɞ ȿɣɥɟɪɚ – ɭɦɨɜɧɨ ɫɬɿɣɤɢɣ. Ⱦɥɹ ɧɟɹɜɧɨɝɨ

ɦɟɬɨɞɭ ȿɣɥɟɪɚ:

yn 1

f (tn 1, yn 1 )

yn 1 .

yn

yn

1

1

Ɂɜɿɞɫɢ

ɡɚɩɢɲɟɦɨ

yn 1

1

W

yn ,

0

q

1

W

1,

0 . Ɍɚɤɢɦ

ɱɢɧɨɦ

ɧɟɹɜɧɢɣ ɦɟɬɨɞ ȿɣɥɟɪɚ ɚɛɫɨɥɸɬɧɨ ɫɬɿɣɤɢɣ ɧɚ ɬɟɫɬɨɜɨɦɭ ɪɿɜɧɹɧɧɿ.

ɜ)

f

u , Re

0. Ɋɨɡɝɥɹɧɭɬɢ ɫɢɫɬɟɦɭ

du

Au&,

dt

/

ɞɟ

A

– ɦɚɬɪɢɰɹ

ɩɪɨɫɬɨʀ

ɫɬɪɭɤɬɭɪɢ,

ɬɨɛɬɨ

H :

HAH 1

diag(

i ),

i

i (A). ɉɨɩɟɪɟɞɧɹ ɫɢɫɬɟɦɚ ɡɜɨɞɢɬɶɫɹ ɞɨ ɬɚɤɨʀ:

dv&

/v&, ɚɛɨ

dvi

ɞɟ

vi

vi0e it

vi0eRe

ivi ,

i 1,n ,

Im

i .

it cosM t

jsin M t ,

–

ɤɭɬ

ɞɥɹ ɹɤɨɝɨ tgM

dt

dt

Re

i

Ⱦɨɫɥɿɞɢɦɨ ɹɜɧɢɣ ɬɚ ɧɟɹɜɧɢɣ ɦɟɬɨɞ ȿɣɥɟɪɚ ɧɚ ɬɚɤɨɦɭ ɬɟɫɬɨɜɨɦɭ ɩɪɢɤɥɚɞɿ.

121

əɜɧɢɣ ɦɟɬɨɞ ȿɣɥɟɪɚ: yn 1 qyn , q						Ɋɢɫ. 13					Re z iImz . ɍɦɨɜɚ
						1				1 z , z
	q		d 1 ɜɢɤɨɧɭɽɬɶɫɹ ɞɥɹ ɜɧɭɬɪɿɲɧɨɫɬɿ ɤɨɥɚ ɪɚɞɿɭɫɚ 1 ɡ ɰɟɧɬɪɨɦ ɜ ɬɨɱɰɿ (-1, 0)
(ɪɢɫ. 13):						1 W i				1, i.
				qi
Ɂɜɿɞɫɢ				W d						W0 ,
						2
						2Re	i

i

Ɍɚɤɢɦ ɱɢɧɨɦ ɹɜɧɢɣ ɦɟɬɨɞ ȿɣɥɟɪɚ ɭɦɨɜɧɨ ɫɬɿɣɤɢɣ ɿ ɞɥɹ ɬɟɫɬɨɜɨɝɨ ɪɿɜɧɹɧɧɹ ɡ

Re	0.
	Ⱦɥɹ ɧɟɹɜɧɨɝɨ ɦɟɬɨɞɭ ȿɣɥɟɪɚ q		1	.
	1		z

ɍɦɨɜɚ					d	Ɋɢɫ.14
			q			1 ɜɢɤɨɧɭɽɬɶɫɹ ɞɥɹ ɡɨɜɧɿɲɧɨɫɬɿ ɤɨɥɚ ɪɚɞɿɭɫɚ	R	1, ɡ ɰɟɧɬɪɨɦ ɜ
ɬɨɱɰɿ	z0					(1,0) (ɪɢɫ. 14), ɜ ɬɨɣ ɱɚɫ, ɹɤ ɬɨɱɤɢ zi	i	ɥɟɠɚɬɶ ɜ ɥɿɜɿɣ
ɩɿɜɩɥɨɳɢɧɿ						Ɂɧɚɱɢɬɶ ɭɦɨɜɚ ɫɬɿɣɤɨɫɬɿ ɡɚɞɨɜɨɥɶɧɹɽɬɶɫɹ ɞɥɹ ɞɨɜɿɥɶɧɢɯ .

Ɍɚɤɢɦ ɱɢɧɨɦ ɧɟɹɜɧɢɣ ɦɟɬɨɞ ȿɣɥɟɪɚ ɽ ɚɛɫɨɥɸɬɧɨ ɫɬɿɣɤɢɣ. ɇɟɞɨɥɿɤ ɧɟɹɜɧɨɝɨ ɦɟɬɨɞɭ ȿɣɥɟɪɚ - ɧɢɡɶɤɚ ɬɨɱɧɿɫɬɶ: p 1.

Ʉɚɠɭɬɶ, ɳɨ ɦɟɬɨɞ ɪɨɡɜ’ɹɡɚɧɧɹ ɡɚɞɚɱɿ Ʉɨɲɿ ɽ A -ɫɬɿɣɤɢɦ, ɹɤɳɨ ɨɛɥɚɫɬɶ

ɣɨɝɨ ɫɬɿɣɤɨɫɬɿ	ɞɥɹ ɬɟɫɬɨɜɨɝɨ ɪɿɜɧɹɧɧɹ f		u , Re	0 ɜɤɥɸɱɚɽ ɜɫɸ
ɩɿɜɩɥɨɳɢɧɭ Re z	0, z	W .

Ɂɚɞɚɱɚ 41 ɉɨɤɚɡɚɬɢ , ɳɨ ɦɟɬɨɞ ɬɪɚɩɟɰɿʀ ɪɨɡɜ’ɹɡɚɧɧɹ ɡɚɞɚɱ Ʉɨɲɿ ɽ A -ɫɬɿɣɤɢɦ. Ɍɟɨɪɟɦɚ 2 (Ⱦɚɥɤɜɿɫɬ). əɜɧɢɣ ɥɿɧɿɣɧɢɣ ɛɚɝɚɬɨɤɪɨɤɨɜɢɣ ɦɟɬɨɞ ɧɟ ɦɨɠɟ ɛɭɬɢ A -ɫɬɿɣɤɢɦ. ɉɨɪɹɞɨɤ ɧɟɹɜɧɨɝɨ A - ɫɬɿɣɤɨɝɨ ɦɟɬɨɞɭ ɧɟ ɦɨɠɟ ɛɭɬɢ ɜɢɳɢɦ

ɡɚ 2. ɇɚɣɛɿɥɶɲ ɬɨɱɧɢɣ ɫɟɪɟɞ ɰɢɯ ɦɟɬɨɞɿɜ ɽ ɦɟɬɨɞ ɬɪɚɩɟɰɿɣ.

122

2 . ɐɟ ɞɭɠɟ ɠɨɪɫɬɤɚ ɭɦɨɜɚ. Ɍɨɦɭ

Ɂɚɞɚɱɚ Ʉɨɲɿ (1) ɧɚɡɢɜɚɽɬɶɫɹ ɠɨɪɫɬɤɨɸ ɞɥɹ t

t0 ,T@, ɹɤɳɨ Re i 0 ɬɚ

max

Re

i (t)

S(t)

i

1,

min

Re

i (t)

ɞɟ i

i

i (t) ɜɥɚɫɧɿ ɡɧɚɱɟɧɧɹ ɦɚɬɪɢɰɿ əɤɨɛɿ ɩɪɚɜɨʀ ɱɚɫɬɢɧɢ ɪɿɜɧɹɧɧɹ

§ f

·m

An

©

wu j

¹i, j 1

¨ w i (tn )¸ .

ɏɚɪɚɤɬɟɪɧɚ ɨɫɨɛɥɢɜɿɫɬɶ ɪɨɡɜ’ɹɡɤɿɜ ɠɨɪɫɬɤɨʀ ɡɚɞɚɱɿ Ʉɨɲɿ ɽ ɧɚɹɜɧɿɫɬɶ

ɤɨɦɩɨɧɟɧɬ e it , ɹɤɿ ɩɨɜɿɥɶɧɨ ɡɦɿɧɸɸɬɶɫɹ (ɤɨɥɢ

Re

i

min

Re i

) ɬɚ ɲɜɢɞɤɨ

i

ɫɩɚɞɚɸɬɶ (ɤɨɥɢ

Re i

i

). ɋɬɿɣɤɿɫɬɶ ɹɜɧɢɯ ɦɟɬɨɞɿɜ (Ɋɭɧɝɟ-Ʉɭɬɬɚ,

max

Re

i

ɹɜɧɨɝɨ ɦɟɬɨɞɭ ȿɣɥɟɪɚ) ɜɢɡɧɚɱɚɽɬɶɫɹ ɤɨɦɩɨɧɟɧɬɚɦɢ, ɹɤɿ ɲɜɢɞɤɨ ɡɦɿɧɸɸɬɶɫɹ. ȼ ɬɨɣ ɱɚɫ ɹɤ ɜ ɬɨɱɧɨɦɭ ɪɨɡɜ’ɹɡɤɭ ɜɨɧɢ ɲɜɢɞɤɨ ɩɪɹɦɭɸɬɶ ɞɨ ɧɭɥɹ ɿ ɞɚɸɬɶ ɦɚɥɢɣ ɜɤɥɚɞ. ɋɚɦ ɪɨɡɜ’ɹɡɨɤ ɡɦɿɧɸɽɬɶɫɹ ɩɨɜɿɥɶɧɨ ɪɚɡɨɦ ɡ ɩɨɜɿɥɶɧɢɦɢ@ ɤɨɦɩɨɧɟɧɬɚɦɢ. ɥɟ ɭɦɨɜɚ ɫɬɿɣɤɨɫɬɿ ɹɜɧɢɯ ɦɟɬɨɞɿɜ ɧɚ ɜɫɶɨɦɭ ɩɪɨɦɿɠɤɭ t0 ,T

ɩɨɜɢɧɧɚ ɛɭɬɢ ɨɪɿɽɧɬɨɜɚɧɚ ɧɚ ɤɪɨɤ W

max	i
i

ɬɚɤɿ ɫɢɫɬɟɦɢ ɧɚɡɢɜɚɸɬɶɫɹ ɠɨɪɫɬɤɢɦɢ.

Ɂɚɫɬɨɫɭɜɚɧɧɹ A - ɫɬɿɣɤɢɯ ɦɟɬɨɞɿɜ ɡɧɿɦɚɽ ɩɪɨɛɥɟɦɭ ɫɬɿɣɤɨɫɬɿ: ɤɪɨɤ ɦɨɠɧɚ ɦɿɧɹɬɢ ɬɿɥɶɤɢ ɡ ɭɦɨɜɢ ɜɢɤɨɧɚɧɧɹ ɦɚɥɨʀ ɩɨɯɢɛɤɢ ɧɚ ɨɞɧɨɦɭ ɤɪɨɰɿ. ɥɟ ɬɨɱɧɿɫɬɶ A- ɫɬɿɣɤɢɯ ɦɟɬɨɞɿɜ ɧɟ ɛɿɥɶɲɟ 2. ɓɨ ɪɨɛɢɬɢ? ɉɨɫɥɚɛɥɸɸɬɶ ɨɡɧɚɱɟɧɧɹ A -ɫɬɿɣɤɨɫɬɿ.

	Ɋɢɫ. 15
	Ɇɟɬɨɞ ɪɨɡɜ’ɹɡɚɧɧɹ ɡɚɞɚɱɿ Ʉɨɲɿ ɧɚɡɢɜɚɸɬɶ A( ) - ɫɬɿɣɤɢɦ 0 d D d					,
					2
	ɹɤɳɨ ɨɛɥɚɫɬɶ ɫɬɿɣɤɨɫɬɿ ɦɟɬɨɞɭ ɞɥɹ ɬɟɫɬɨɜɨɝɨ ɪɿɜɧɹɧɧɹ ɡ
			f	u , Re	0,
	ɜɤɥɸɱɚɽ ɱɚɫɬɢɧɭ ɩɿɜɩɥɨɳɢɧɢ Re z ( z	), ɳɨ ɥɟɠɢɬɶ	ɜ	ɫɟɪɟɞɢɧɿ ɦɿɠ
	ɩɪɨɦɟɧɹɦɢ, ɳɨ ɭɬɜɨɪɸɸɬɶ ɡ ɜɿɫɫɸ Im z ɤɭɬɢ	(ɪɢɫ. 15). Ɂɚɭɜɚɠɢɦɨ, ɳɨ A -
	ɫɬɿɣɤɿɫɬɶ = A(0) - ɫɬɿɣɤɿɫɬɶ !

ɇɚɣɛɿɥɶɲ ɩɨɲɢɪɟɧɢɦɢ ɞɥɹ ɪɨɡɜ’ɹɡɭɜɚɧɧɹ ɠɨɪɫɬɤɢɯ ɫɢɫɬɟɦ ɽ ɦɟɬɨɞɢ Ʉɭɪɬɿɫɚ-ɏɟɪɲɮɿɥɶɬɚ:

123

m	W
k 0
¦	ak	yn k	f (tn , yn ) .

Ɉɛɥɚɫɬɿ ʀɯ ɫɬɿɣɤɨɫɬɿ ɞɥɹ ɪɿɡɧɢɯ m ɩɪɢɜɟɞɟɧɚ ɧɚ ɪɢɫ. 16:

Ɋɢɫ. 16

ɐɟ ɧɟɹɜɧɿ m - ɤɪɨɤɨɜɿ ɦɟɬɨɞɢ ɡ b0 1, bk 0 k 1,m (ɿɧɲɿ ɧɚɡɜɢ: ɦɟɬɨɞɢ

ɞɢɮɟɪɟɧɰɿɸɜɚɧɧɹ ɧɚɡɚɞ, ɱɢɫɬɨ ɧɟɹɜɧɿ ɦɟɬɨɞɢ).

Ɂɚɞɚɱɚ 42 Ɇɟɬɨɞɨɦ ɧɟɜɢɡɧɚɱɟɧɢɯ ɤɨɟɮɿɰɿɽɧɬɿɜ ɩɨɛɭɞɭɜɚɬɢ ɦɟɬɨɞ ɞɢɮɟɪɟɧɰɿɸɜɚɧɧɹ ɧɚɡɚɞ ɞɥɹ m 3.

Ⱦɥɹ ɠɨɪɫɬɤɢɯ ɡɚɞɚɱ ɬɪɟɛɚ ɜɢɤɨɪɢɫɬɨɜɭɜɚɬɢ ɪɟɚɥɿɡɚɰɿɸ ɧɟɹɜɧɢɯ ɦɟɬɨɞɿɜ ɡɚ ɞɨɩɨɦɨɝɨɸ ɿɬɟɪɚɰɿɣɧɨɝɨ ɦɟɬɨɞɭ ɇɶɸɬɨɧɚ ɡ ɜɢɛɨɪɨɦ ɩɨɱɚɬɤɨɜɨɝɨ

ɧɚɛɥɢɠɟɧɧɹ ɡɚ ɹɜɧɨɸ ɫɯɟɦɨɸ.
Ⱦɥɹ ɠɨɪɫɬɤɢɯ ɡɚɞɚɱ ɬɚɤɨɠ ɡɚɫɬɨɫɨɜɭɸɬɶ ɧɟɹɜɧɿ ɦɟɬɨɞɢ Ɋɭɧɝɟ-Ʉɭɬɬɚ.							ɥɟ
ɞɥɹ ʀɯ ɪɟɚɥɿɡɚɰɿʀ ɬɪɟɛɚ ɪɨɡɜ’ɹɡɭɜɚɬɢ ɫɢɫɬɟɦɭ ɡ m						N ɧɟɥɿɧɿɣɧɢɯ ɪɿɜɧɹɧɶ,	m –
ɤɿɥɶɤɿɫɬɶ ɫɬɚɞɿɣ, N – ɪɨɡɦɿɪɧɿɫɬɶ ɫɢɫɬɟɦɢ. Ɍɚɛɥɢɰɹ Ȼɚɬɱɟɪɚ ɰɢɯ ɦɟɬɨɞɿɜ ɦɚɽ
ɜɢɝɥɹɞ	§	D1		E11	E1m	·
	¨					¸
	¨					¸
	¨			Em1		¸.
	¨Dm				Emm ¸
	©					¹
				p1	pm

11. Ɇɟɬɨɞɢ ɪɨɡɜ’ɹɡɚɧɧɹ ɤɪɚɣɨɜɢɯ ɡɚɞɚɱ ɞɥɹ ɡɜɢɱɚɣɧɢɯ ɞɢɮɟɪɟɧɰɿɣɧɢɯ ɪɿɜɧɹɧɶ

ɉɨɱɧɟɦɨ ɡ ɩɨɫɬɚɧɨɜɤɢ ɤɪɚɣɨɜɢɯ ɡɚɞɚɱ.

	1). ɇɟɥɿɧɿɣɧɚ ɞɜɨɬɨɱɤɨɜɚ ɤɪɚɣɨɜɚ ɡɚɞɚɱɚ
				dU	&	&
					F	x,U	a	x
				dx
			M U a ,U b				d
			&
Ɍɭɬ	u1,...&,um T&		&
&
U		, uk	uk x , F f1,..., fm T ,				fk	fk
Mk	Mk U a ,U	b , d	d1,...,dm , dk - ɱɢɫɥɚ.

2). Ʌɿɧɿɣɧɚ ɞɜɨɬɨɱɤɨɜɚ ɤɪɚɣɨɜɚ ɡɚɞɚɱɚ

b			(1)
	,		(2)
&		&	M1,...,Mm T ,
x,U		M

124

dU

&

&

A x U

x

F

x

(3)

dx

aij x im, j 1,

&

B1U a B2U b

d

(4)

A x

x ,

ɱɢɫɥɨɜɿ

ɦɚɬɪɢɰɿ

F

f1,..., fm ,

fk

fk

B1, B2

-

&

m u m ; d - ɜɟɤɬɨɪ.

Ʉɪɚɣɨɜɿ ɭɦɨɜɢ (2) ɿ (4) ɧɚɡɢɜɚɸɬɶɫɹ ɧɟɪɨɡɞɿɥɶɧɢɦɢ. ɑɚɫɬɨ

ɡɭɫɬɪɿɱɚɸɬɶɫɹ ɪɨɡɞɿɥɟɧɿ ɤɪɚɣɨɜɿ ɭɦɨɜɢ (ɧɚɩɪɢɤɥɚɞ, ɞɥɹ ɥɿɧɿɣɧɨʀ ɡɚɞɚɱɿ):

C1 m k u m-

C1U a

d1, C2U b

d2 ,

(4/)

ɞɟ

ɦɚɬɪɢɰɹ,

C2

k

m-

ɦɚɬɪɢɰɹ;

rangC1

m k ,

rangC2 k ; d1

m

k - ɜɟɤɬɨɪ,

d2 k - ɜɟɤɬɨɪ.

Ⱦɨ (3),(4/) ɡɜɨɞɢɬɶɫɹ ɤɪɚɣɨɜɚ ɡɚɞɚɱɚ ɞɥɹ ɪɿɜɧɹɧɶ ɜɢɳɢɯ ɩɨɪɹɞɤɿɜ. ɇɟɯɚɣ

1

m

ɡɚɞɚɧɚ ɤɪɚɣɨɜɚ ɡɚɞɚɱɚ:

°

m

1

­u

x

p

x u m

x

...

p

x u x

f

x

°

m 1

®

Di1u

m

1

a

...

Dimu a

Pi

,i

1,m

k .

¯

Ei1u

b ...

Eim

b

Qi ,i

1, k

ȼɨɧɚ ɡɜɨɞɢɬɶɫɹ ɞɨ ɡɚɞɚɱɿ (3), (4) ɡ

·

§

¨

0

1

0

..

0

¸

¨

0

0

1

..

0

¸

A x

¨

¸

¨ ..

..

..

..

..

¸ ,

¨

¸

¨

0

0

0

..

1

¸

© pm pm 1 pm 2 ..

p1

¹

j

i,m

j 1,m

C1

Dij i

, C2

Eij

i

,

&

1,m k

1,k

P1,...,Pm

k T ,

Q1,...,Qk T .

d1

d2

ȼɜɚɠɚɽɦɨ, ɳɨ ɜɫɿ ɡɚɞɚɱɿ ɦɚɸɬɶ ɽɞɢɧɿ ɪɨɡɜ’ɹɡɤɢ. Ɋɨɡɝɥɹɧɟɦɨ

ɦɟɬɨɞɢ

ɪɨɡɜ’ɹɡɚɧɧɹ ɰɢɯ ɡɚɞɚɱ.

11.1. Ɇɟɬɨɞ ɫɬɪɿɥɶɛɢ [ɅɆɋ, 288 – 290],[ȻɀɄ, 434 – 435]

Ɋɨɡɝɥɹɧɟɦɨ ɥɿɧɿɣɧɭ ɤɪɚɣɨɜɭ ɡɚɞɚɱɭ ɡ ɧɟɪɨɡɞɿɥɟɧɢɦɢ ɤɪɚɣɨɜɢɦɢ

ɭɦɨɜɚɦɢ:

dU

&

&

A x U

x

F x

(1)

dx

B2U b

B1U a

d

(2)

Ɇɟɬɨɞ ɫɬɪɿɥɶɛɢ ɡɜɨɞɢɬɶ ɤɪɚɣɨɜɭ ɡɚɞɚɱɭ ɞɨ ɩɨɫɥɿɞɨɜɧɨɫɬɿ ɡɚɞɚɱ Ʉɨɲɿ.

Ⱦɥɹ ɰɶɨɝɨ ɪɨɡɜ’ɹɠɟɦɨ (m&+1) ɡɚɞɚɱɭ Ʉɨɲɿ:

&

dY0

&

&

A x Y0

F x , Y0

a

0

(3)

dx

125

&

&

&

&

&

Gij mj 1, i

dYi

A x Yi x ,

Yi

a

Gi ,

Gi

1,m

(4)

&

dx

Ɇɚɬɪɢɰɹ Y(x)

ɧɚɡɢɜɚɽɬɶɫɹ

ɮɭɧɞɚɦɟɧɬɚɥɶɧɨɸ

ɦɚɬɪɢɰɟɸ

Yi (x) i

1,m

ɨɞɧɨɪɿɞɧɨʀ ɫɢɫɬɟɦɢ (1). Ɋɨɡɜ’ɹɡɨɤ (1) ɲɭɤɚɽɦɨ ɭ ɜɢɝɥɹɞɿ:

&

&

m

&

Y x

Y0 x

¦ciYi x

(5)

ȼɿɧ ɡɚɞɨɜɨɥɶɧɹɽ (1) ci . ɋɚɦɿ ci

i

1

ɡɧɚɯɨɞɹɬɶɫɹ ɡ (2):

d ɚɛɨ

B1¨§Y0

a ¦ciYi

a ¸· B2

¨§Y0

b

¦ciYi b ¸·

©

&

&

¹

©

&

&

¹

&

i

i

>B1 B2Y b @c&

d

B2Y0 b

&

(6)

Ɋɨɡɜ’ɹɡɭɸɱɢ ɋɅ

Ɋ (6) ɡɧɚɯɨɞɢɦɨ ci . Ɂɚ ɽɞɢɧɿɫɬɸ Y x U x .

Ⱥɥɝɨɪɢɬɦ

Ⱥ1.

1) Ɋɨɡɜ’ɹɡɭɽɦɨ ɡɚɞɚɱɭ Ʉɨɲɿ (3), ɡɧɚɯɨɞɢɦɨ Y0 b .

2) Ɋɨɡɜ’ɹɡɭɽɦɨ m ɡɚɞɚɱ Ʉɨɲɿ (4), ɡɧɚɯɨɞɢɦɨ Y b .

3) Ɋɨɡɜ’ɹɡɭɽɦɨ ɋɅ

Ɋ (6), ɳɨ ɞɚɽ ɧɚɦ ci ,i

1,m.

4) Y x

U x ɡɧɚɯɨɞɢɦɨ ɡ (5).

1

ɋɤɥɚɞɧɿɫɬɶ ɰɶɨɝɨ

ɚɥɝɨɪɢɬɦɭ

ɫɩɿɜɩɚɞɚɽ

ɡɿ

ɫɤɥɚɞɧɿɫɬɸ

ɪɨɡɜ‘ɹɡɚɧɧɹ m

ɡɚɞɚɱ Ʉɨɲɿ.

əɤɳɨ ɤɪɚɣɨɜɿ ɭɦɨɜɢ ɪɨɡɞɿɥɟɧɿ

C1Y a d1, C2Y b d2 ,

ɬɨ ɦɨɠɧɚ ɡɦɟɧɲɢɬɢ&ɤɿɥɶɤɿɫɬɶ ɡɚɞɚɱ Ʉɨɲɿ, ɹɤɿ ɧɟɨɛɯɿɞɧɨ ɪɨɡɜ‘ɹɡɚɬɢ. Ⱦɥɹ ɰɶɨɝɨ ɩɨɛɭɞɭɽɦɨ ɜɟɤɬɨɪ V0 ɬɚɤɢɣ, ɳɨ

&	0
C V		d	1	(7)
1

ɐɟ ɡɚɜɠɞɢ ɦɨɠɧɚ ɡɪɨɛɢɬɢ, ɨɫɤɿɥɶɤɢ ɤɿɥɶɤɿɫɬɶ ɪɿɜɧɹɧɶ ɦɟɧɲɟ ɤɿɥɶɤɨɫɬɿ

ɧɟɜɿɞɨɦɢɯ. Ⱦɚɥɿ ɛɭɞɭɽɦɨ Vi ,i

1,k

ɬɚɤɿ, ɳɨ

&

C V i

0, i

1,k

1

Ɂɧɨɜɭ ɰɟ ɦɨɠɧɚ ɡɞɿɣɫɧɢɬɢ ɛɨ rangC1

m

k .

Ɋɨɡɜ’ɹɠɟɦɨ ɡɚɞɚɱɿ Ʉɨɲɿ:

&

0

&

&

&

&

dY

0

AY

0 F , Y 0 a

V

&dx

&

&

a

&

dY i

AY i , Y i

V i , i 1,k .

dx

ɋɬɚɥɿ ci ɡɧɚɯɨɞɢɦɨ ɡ ɭɦɨɜɢ ɜɢɤɨɧɚɧɧɹ ɞɪɭɝɨʀ ɤɪɚɣɨɜɨʀ ɭɦɨɜɢ.

Ⱥɥɝɨɪɢɬɦ Ⱥ2.

1). Ɋɨɡɜ’ɹɡɭɽɦɨ ɋɅ Ɋ (8) – (9). 2). Ɋɨɡɜ’ɹɡɭɽɦɨ ɡɚɞɚɱɭ Ʉɨɲɿ (9). 3). Ɋɨɡɜ’ɹɡɭɽɦɨ k ɡɚɞɚɱ Ʉɨɲɿ (10).

(8)

(9)

(10)

126

4). Ɋɨɡɜ‘ɹɡɭɽɦɨ ɋɅ Ɋ:		ª			0				º
&			&				k	&		&
B2Y b		¬				b	i 1		¼
	C2 «Y						¦ciY b »			d2 .	(11)
5). Ɋɨɡɜ’ɹɡɨɤ	x
&		&					k	&
Y		Y		0 x ¦ciY i x .							(12)
						i	1

Ɉɫɤɿɥɶɤɢ ɞɥɹ 1 ɬɚ 2 ɪɨɡɜ’ɹɡɨɤ ɡɚɞɚɱɿ Ʉɨɲɿ ɲɭɤɚɽɬɶɫɹ ɱɢɫɟɥɶɧɨ, ɬɨ ɦɚɽɦɨ

ɮɚɤɬɢɱɧɨ ɡɧɚɱɟɧɧɹ	&
	Y i xn , n 0, N , xn a,b@.

Ȳɯ ɬɪɟɛɚ ɡɚɩɚɦ‘ɹɬɨɜɭɜɚɬɢ ɞɥɹ (12). Ⱦɥɹ ɡɚɩɨɛɿɝɚɧɧɹ ɰɶɨɝɨ ɭ ɜɢɩɚɞɤɭ

ɪɨɡɞɿɥɟɧɨʀ ɤɪɚɣɨɜɨʀ ɡɚɞɚɱɿ ɡɚɩɢɲɟɦɨ ɚɥɝɨɪɢɬɦ

3.

Ⱥɥɝɨɪɢɬɦ Ⱥ3.

1). Ɋɨɡɜ’ɹɡɭɽɦɨ (8) – (9).

i xn , ɚ

2). Ɋɨɡɜ’ɹɡɭɽɦɨ ɡɚɞɚɱɭ Ʉɨɲɿ (9).

3). Ɋɨɡɜ’ɹɡɭɽɦɨ ɡɚɞɚɱɿ Ʉɨɲɿ (10) ɿ ɧɟ

ɡɚɩɚɦ’ɹɬɨɜɭɽɦɨ Y

i xN

b .

ɡɧɚɯɨɞɢɦɨ ɬɿɥɶɤɢ Y

Y

i

4). Ɋɨɡɜ’ɹɡɭɽɦɨ ɋɅ

Ɋ (11).

5). Ɋɨɡɜ’ɹɡɭɽɦɨ ɳɟ&ɨɞɧɭ ɡɚɞɚɱɭ Ʉɨɲɿ:

dY

&

&

&

k

&

AY

F , Y

a

V 0

¦ciV i .

dx

Ɍɨɞɿ ɡɚ ɮɨɪɦɭɥɨɸ (12) Y x U x .

i

1

Ɂɪɨɡɭɦɿɥɨ, ɳɨ “ɫɬɪɿɥɹɬɢ”, ɬɨɛɬɨ ɩɨɱɢɧɚɬɢ ɪɨɡɜ‘ɹɡɭɜɚɬɢ ɡɚɞɚɱɭ Ʉɨɲɿ,

ɬɪɟɛɚ ɡ ɬɨɝɨ ɛɨɤɭ, ɞɟ ɡɚɞɚɧɨ ɛɿɥɶɲɟ ɤɪɚɣɨɜɢɯ ɭɦɨɜ.
ɋɭɬɬɽɜɢɣ ɧɟɞɨɥɿɤ ɚɥɝɨɪɢɬɦɿɜ!. ɋɟɪɟɞ ɜɥɚɫɧɢɯ ɡɧɚɱɟɧɶ A x , ɹɤ ɩɪɚɜɢɥɨ,
ɽ ɬɚɤɿ, ɳɨ Re i x	0. Ɍɨɞɿ ɥɿɧɿɣɧɨ ɧɟɡɚɥɟɠɧɿ ɪɨɡɜ’ɹɡɤɢ ɡɚɞɚɱɿ Ʉɨɲɿ

ɧɚɪɨɫɬɚɸɬɶ ɟɤɫɩɨɧɟɧɰɿɚɥɶɧɨ. ɐɟ ɩɪɢɡɜɨɞɢɬɶ ɞɨ ɧɚɪɨɫɬɚɧɧɹ ɩɨɯɢɛɨɤ

ɡɚɨɤɪɭɝɥɟɧɶ ɬɚ ɩɨɝɚɧɨ ɨɛɭɦɨɜɥɟɧɨʀ ɦɚɬɪɢɰɿ ɫɢɫɬɟɦɢ (6) ɚɛɨ (11) (ɪɨɡɜ‘ɹɡɤɢ

i x ɫɬɚɸɬɶ ɦɚɣɠɟ ɥɿɧɿɣɧɨ ɡɚɥɟɠɧɿ).

Y

Ɍɨɦɭ >a,b@ ɪɨɡɛɢɜɚɸɬɶ ɧɚ ɩɪɨɦɿɠɤɢ

1, xp , p

, ɿ ɪɨɡɜ’ɹɡɭɸɬɶ

xp

1,M

ɡɚɞɚɱɭ Ʉɨɲɿ ɧɚ ɩɿɞɩɪɨɦɿɠɤɚɯ,

ɚ ɜ ɤɿɧɰɿ

x

xp ɨɪɬɨɝɨɧɚɥɿɡɭɸɬɶ ɨɬɪɢɦɚɧɿ

ɪɨɡɜ’ɹɡɤɢ.

Ɂɪɨɡɭɦɿɥɨ, ɳɨ ɞɥɹ x

&

&

b ,

ɹɤɿ

b ɨɬɪɢɦɭɸɬɶ ɧɟ Y i

b , ɚ ɞɟɹɤɿ W i

ɡɚɥɟɠɚɬɶ

ɜɿɞ

&

b

ɬɚ ɜɿɞɩɨɜɿɞɧɢɯ

ɩɟɪɟɬɜɨɪɟɧɶ

ɨɪɬɨɝɨɧɚɥɿɡɚɰɿʀ.

Ɂ

ʀɯ

Y i

&

&

b ɬɚ “ɩɪɨɝɨɧɹɸɬɶ” ɰɿ ɭɦɨɜɢ ɞɥɹ ɜɫɿɯ

ɞɨɩɨɦɨɝɨɸ ɩɨ W i

b ɨɛɱɢɫɥɸɸɬɶ Y i

&

a

&

a

&

. Ɍɚɤɚ ɿɞɟɹ ɦɟɬɨɞɚ ɨɪɬɨɝɨɧɚɥɶɧɨʀ ɩɪɨɝɨɧɤɢ

ɡɧɚɱɟɧɶ Y

Y 0

¦ciY i a

Ƚɨɞɭɧɨɜɚ, ɳɨ ɲɢɪɨɤɨ ɡɚɫɬɨɫɨɜɭɽɬɶɫɹ ɧɚ ɩɪɚɤɬɢɰɿ. 11.2. Ɇɟɬɨɞ ɩɪɢɫɬɪɿɥɤɢ

127

ɐɟ ɦɟɬɨɞ ɞɥɹ ɪɨɡɜ‘ɹɡɚɧɧɹ ɤɪɚɣɨɜɨʀ ɡɚɞɚɱɿ ɞɥɹ ɧɟɥɿɧɿɣɧɢɯ ɪɿɜɧɹɧɶ ɚɧɚɥɨɝɿɱɧɢɣ ɦɟɬɨɞɭ ɫɬɪɿɥɶɛɢ.

Ɋɨɡɝɥɹɧɟɦɨ ɤɪɚɣɨɜɭ ɡɚɞɚɱɭ ɡ ɪɨɡɞɿɥɟɧɢɦɢ ɤɪɚɣɨɜɢɦɢ ɭɦɨɜɚɦɢ:

dU

&

&

F

x,U

,a

x b,

(1)

dx

a

k 1, m

ui

ci ,i

,

(2)

Mi

U b

di ,i

1, k .

(3)

ɉɪɢ x

a ɧɟɜɿɞɨɦɿ k ɩɨɱɚɬɤɨɜɢɯ ɭɦɨɜ ui

a ,i

. Ȼɭɞɟɦɨ ʀɯ ɲɭɤɚɬɢ.

1,k

Ɋɨɡɜ’ɹɠɟɦɨ ɡɚɞɚɱɭ Ʉɨɲɿ:

­dY

&

&

°

F x,Y ,a x b;

a

°Y

C

ci

im 1.

® dx

,

¯

&

&

fi c1,...,ck

i

b;c1

,...,ck

di

0, i 1,k .

0

ɞɟ ci ,i

1,k

ɧɟɜɿɞɨɦɿ. Ȳɯ ɲɭɤɚɽɦɨ ɡ ɤɪɚɣɨɜɨʀ ɭɦɨɜɢ (3):

ɐɟ ɫɢɫɬɟɦɚ ɧɟɥɿɧɿɣɧɢɯ ɪɿɜɧɹɧɶ.

Ɂɚɞɚɽɦɨ ɩɨɱɚɬɤɨɜɿ ɡɧɚɱɟɧɧɹ

ci ,i

. Ɂɚ

1,k

ɹɤɢɦɨɫɶ ɿɬɟɪɚɰɿɣɧɢɦ ɦɟɬɨɞɨɦ ɡɧɚɯɨɞɢɦɨ ʀʀ ɪɨɡɜ‘ɹɡɨɤ. ɇɚɣɡɪɭɱɧɿɲɟ ɜɢɤɨɪɢɫɬɨɜɭɜɚɬɢ ɦɟɬɨɞ ɫɿɱɧɢɯ.

Ɇɟɬɨɞ ɩɪɢɫɬɪɿɥɤɢ ɧɚɣɛɿɥɶɲ ɩɪɨɡɨɪɨ ɜɢɝɥɹɞɚɽ ɞɥɹ k 1. ȼ ɰɶɨɦɭ

ɜɢɩɚɞɤɭ ɧɚɦ ɧɟɨɛɯɿɞɧɨ ɡɧɚɣɬɢ ɬɿɥɶɤɢ c1. ȼɢɤɨɪɢɫɬɚɽɦɨ ɦɟɬɨɞ ɞɿɥɟɧɧɹ ɧɚɜɩɿɥ.

Ɂɧɚɣɞɟɦɨ c 0

ɬɚɤɟ, ɳɨ

M1

0

1

y& b;c

d1 0 ,

1

ɬɚ c(1)

ɬɚɤɟ, ɳɨ

M1

1

1

y& b;c

d1 0.

Ɍɨɞɿ ɜɢɛɢɪɚɽɦɨ

1

c11 .

c12

2

c10

1

y b;c1

Ɂ ɬɪɶɨɯ c10 ,c11 ,c12 ɜɢɛɢɪɚɽɦɨ ɬɚɤɟ

, ɳɨ

1

d1 ɦɿɧɹɽ ɡɧɚɤ. ɉɪɨɰɟɫ

ɩɪɨɞɨɜɠɭɽɦɨ ɞɨ ɜɢɤɨɧɚɧɧɹ ɭɦɨɜɢ

ɞɟ H

M1

y& b,c k

d1

H ,

1

0 - ɡɚɞɚɧɚ ɬɨɱɧɿɫɬɶ.

11.3. Ɇɟɬɨɞ ɥɿɧɟɚɪɢɡɚɰɿʀ [ɅɆɋ, 293 – 294], [ȻɀɄ, 441 – 442]

Ɋɨɡɝɥɹɧɟɦɨ ɡɚɞɚɱɭ:

dU

&

&

F

x,U

a

x b,

(1)

dx

&

d .

(2)

M U a ,U b

128

Ɇɟɬɨɞ ɥɿɧɟɚɪɢɡɚɰɿʀ ɞɥɹ ɡɚɞɚɱɿ (1) ɰɟ ɚɧɚɥɨɝ ɦɟɬɨɞɭ ɇɶɸɬɨɧɚ ɞɥɹ ɫɢɫɬɟɦ

ɧɟɥɿɧɿɣɧɢɯ

ɪɿɜɧɹɧɶ.

ɇɟɯɚɣ

Y 0

x

–

ɞɟɹɤɟ ɧɚɛɥɢɠɟɧɧɹ.

ɉɨɛɭɞɭɽɦɨ

ɣɨɝɨ

ɭɬɨɱɧɟɧɧɹ

&

x ɞɨ ɬɨɱɧɨɝɨ ɪɨɡɜ’ɹɡɤɭ U x :

.

Z 0

&

U x

Y 0 x

Z 0 x

&

&

&

&

&

dZ

0

dY 0

Ɂ (1)

ɦɚɽɦɨ

) F x,V

Z

0 x F x,Y 0

.

Ɂɚɦɿɧɸɸɱɢ

ɫɟɪɟɞɧɽ

dx

dx

&

x ɧɚ

&

x ɨɬɪɢɦɚɽɦɨ ɥɿɧɿɣɧɟ ɪɿɜɧɹɧɧɹ:

ɡɧɚɱɟɧɧɹ V

Y 0

&

&

&

&

&

&

dZ

0

dY 0

) F x,Y 0

Z 0 F x,Y 0

(3)

dx

dx

ɧɚɥɨɝɿɱɧɨ:

&

b

&

&

&

&

b

&

&

&

)a Y 0

a ,Y 0

·

Z

0 a )b Y 0

a ,Y

0 b Z 0

d

M& Y

0 a ,Y 0 b

(4)

Ɍɭɬ ) F

§ wFi

–

ɦɚɬɪɢɰɹ

əɤɨɛɿ

ɩɪɚɜɨʀ

ɱɚɫɬɢɧɢ

F x,U ;

¨

¸

§

© wui

¹i, j

§

·

1,m

wMi

wMi

·

&

&

) a

¨

¸

,

)b

¨

¸

-

ɦɚɬɪɢɰɿ əɤɨɛɿ ɞɥɹ M

U a ,U b

¨

u

a

¸

¨

u

b

¸

© w j ¹i, j 1,m

© w

j

¹i, j 1,m

ɩɨ ɤɪɚɣɨɜɢɦ ɭɦɨɜɚɦ ɜ ɬɨɱɤɚɯ x

a ɬɚ x

b ɜɿɞɩɨɜɿɞɧɨ. Ɂɚɞɚɱɚ (3)-(4) ɥɿɧɿɣɧɚ

	ɿ ɪɨɡɜ’ɹɡɭɽɬɶɫɹ ɦɟɬɨɞɨɦ ɫɬɪɿɥɶɛɢ (ɡ ɨɪɬɨɝɨɧɚɥɿɡɚɰɿɽɸ). Ɋɨɡɜ‘ɹɡɚɜɲɢ ɰɸ
	ɡɚɞɚɱɭ, ɦɚɽɦɨ ɧɚɫɬɭɩɧɟ ɧɚɛɥɢɠɟɧɧɹ Y1 x			Y 0 x Z 0 (x). ɐɟɣ				ɩɪɨɰɟɫ
	ɩɪɨɞɨɜɠɭɽɦɨ ɞɨ ɜɢɤɨɧɚɧɧɹ ɭɦɨɜɢ ɬɨɱɧɨɫɬɿ				Z k (x)		H .
	ɇɟɞɨɥɿɤɢ ɦɟɬɨɞɭ:	&
	1) ɇɚɹɜɧɿɫɬɶ ɩɨɯɿɞɧɨʀ	dY 0	ɜ ɩɪɚɜɿɣ ɱɚɫɬɢɧɿ. Ɉɫɤɿɥɶɤɢ ɪɨɡɜ’ɹɡɨɤ ɡɚɞɚɱ
		dx

Ʉɨɲɿ ɱɢɫɟɥɶɧɢɣ, ɬɨ ɞɥɹ ʀʀ ɨɛɱɢɫɥɟɧɧɹ ɬɪɟɛɚ ɡɚɫɬɨɫɨɜɭɜɚɬɢ ɮɨɪɦɭɥɢ ɱɢɫɟɥɶɧɨɝɨ ɞɢɮɟɪɟɧɰɿɸɜɚɧɧɹ. ɐɟ ɦɨɠɟ ɩɪɢɜɟɫɬɢ ɞɨ ɜɟɥɢɤɢɯ ɩɨɯɢɛɨɤ ɡɚ ɪɚɯɭɧɨɤ ɧɟɫɬɿɣɤɨɫɬɿ ɡɚɞɚɱɿ ɱɢɫɟɥɶɧɨɝɨ ɞɢɮɟɪɟɧɰɿɸɜɚɧɧɹ.

2)Ɂɛɿɠɧɿɫɬɶ ɡɚɥɟɠɢɬɶ ɜɿɞ ɜɢɛɨɪɭ Y 0 .

11.4.Ɇɟɬɨɞ ɩɪɨɞɨɜɠɟɧɧɹ ɡɚ ɩɚɪɚɦɟɬɪɨɦ [ɅɆɋ, 293 – 294]

ɋɭɬɬɽɜɢɦ ɧɟɞɨɥɿɤɨɦ ɦɟɬɨɞɭ ɥɿɚɧɟɪɿɡɚɰɿʀ ɽ ɧɟɨɛɯɿɞɧɿɫɬɶ ɡɚɞɚɜɚɬɢ ɯɨɪɨɲɟ ɩɨɱɚɬɤɨɜɟ ɧɚɛɥɢɠɟɧɧɹ ɬɚ ɱɢɫɟɥɶɧɟ ɞɢɮɟɪɟɧɰɿɸɜɚɧɧɹ ɩɨɩɟɪɟɞɧɶɨɝɨ

ɧɚɛɥɢɠɟɧɧɹ. Ɋɨɡɝɥɹɧɟɦɨ ɦɟɬɨɞ, ɹɤɢɣ ɩɨɡɛɚɜɥɟɧɢɣ ɰɢɯ ɧɟɞɨɥɿɤɿɜ.
Ɋɨɡɝɥɹɧɟɦɨ ɡɚɞɚɱɭ ɡɧɚɯɨɞɠɟɧɧɹ ɜɟɤɬɨɪɚ U x					ui	i		, ɳɨ ɡɚɞɨɜɨɥɶɧɹɽ
							1,n
ɭɦɨɜɚɦ:
	dU	&	&
		F	x,U	a	x	b ,			(1)
	dx
&				d .					(2)
M U a ,U b

ɇɟɯɚɣ ɪɨɡɜ‘ɹɡɨɤ ɰɿɽʀ ɡɚɞɚɱɿ ɿɫɧɭɽ ɬɚ ɽɞɢɧɢɣ. Ɋɨɡɜ‘ɹɠɟɦɨ ɡɚɞɚɱɭ Ʉɨɲɿ

129

dY

&

&

&

&

F

x,Y

,

Y(a)

Y

0

(3)

&

dx

0 ɡɞɿɣɫɧɢɦɨ ɬɚɤ, ɳɨɛ ɛɭɥɨ ɡɚɞɨɜɨɥɶɧɹɥɨɫɹ ɹɤ ɦɨɠɧɚ ɛɿɥɶɲɚ ɤɿɥɶɤɿɫɬɶ

ȼɢɛɿɪ Y

ɡ ɤɪɚɣɨɜɢɯ ɭɦɨɜ (2). ɇɚɩɪɢɤɥɚɞ, ɹɤɳɨ

Mi U a ,U b

ui (a), ɬɨ ɜɢɛɢɪɚɽɦɨ

y0

d

i

.

& &

i

&

Ɉɛɱɢɫɥɢɦɨ d 0

a ,Y b . əɤɳɨ d 0

d ,

ɬɨ

Y

ɥɟ, ɹɤ ɩɪɚɜɢɥɨ,

M Y

U .

d 0

d

ɿ ɬɨɦɭ

ɧɟɨɛɯɿɞɧɨ

ɭɬɨɱɧɸɜɚɬɢ

ɩɨɱɚɬɤɨɜɟ ɧɚɛɥɢɠɟɧɧɹ.

Ɋɨɡɝɥɹɧɟɦɨ

ɩɚɪɚɦɟɬɪɢɱɧɭ ɤɪɚɣɨɜɭ ɡɚɞɚɱɭ

dV

&

&

F

x,V

,

a

x

b ,

(4)

dx

d (1

&

&

(5)

M V a ,V b

)d 0 ,

ɹɤɚ

ɡɚɥɟɠɢɬɶ

ɜɿɞ ɩɚɪɚɦɟɬɪɚ

,

&

Y(x), ɚ

:

V

V x,

əɫɧɨ, ɳɨ V x,0

&

&

V x,1

U .

ɋɩɪɨɛɭɽɦɨ ɩɪɨɞɨɜɠɢɬɢ ɪɨɡɜ‘ɹɡɨɤ ɡɚɞɚɱɿ (4), (5) ɜɿɞ ɜɿɞɨɦɨɝɨ Y(x)

ɞɨ

&

ɩɨ

ɲɭɤɚɧɨɝɨ U(x). Ⱦɥɹ ɰɶɨɝɨ ɩɪɨɞɢɮɟɪɟɧɰɿɸɽɦɨ (4), (5)

:

d

wVi

n

wFi

Vj

w

¦j 1

wu j

w

,

dx

n

wMi

wVj (a)

n

wMi

Vj (b)

¦j 1

¦j 1

0

wu j (a)

w

wu j (b)

w

di

di .

ɉɨɡɧɚɱɢɦɨ

&

wV

. Ɍɨɞɿ ɨɫɬɚɧɧɸ ɫɢɫɬɟɦɭ ɦɨɠɧɚ ɡɚɩɢɫɚɬɢ ɭ ɜɢɝɥɹɞɿ:

Z

w

&

&

&

dZ

)F x,V

Z ,

a

x

b,

(4)

dx

&

V (a),V(b) Z(b)

&

&

)a V (a),V(b) Z(a) )b

d

d 0 ,

(5)

wV&

&

&

&

0

.

(6)

§

·

w

Z,

V (x,0)

Y

ɞɟ

)F

F

-

əɤɨɛɿ

ɩɪɚɜɨʀ

ɱɚɫɬɢɧɢ ɪɿɜɧɹɧɧɹ

(1)

F(x.U);

¨ w

i

¸

ɦɚɬɪɢɰɹ

§

© wu j

¹i, j

1,n

wMi

·

&

&

)a

¨

¸

-

ɦɚɬɪɢɰɹ

əɤɨɛɿ

ɥɿɜɨʀ ɱɚɫɬɢɧɢ

M U a ,U b

ɤɪɚɣɨɜɨʀ

© wu j (a)

¹i, j

§

wMi

·

1,n

ɭɦɨɜɢ (2) ɩɨ ɩɟɪɲɨɦɭ ɚɪɝɭɦɟɧɬɭ U a ;

)b

¨

¸

-

ɦɚɬɪɢɰɹ əɤɨɛɿ

© wu j (b) ¹i, j

&

&

&

1,n

ɤɪɚɣɨɜɨʀ ɭɦɨɜɢ (2) ɩɨ ɞɪɭɝɨɦɭ ɚɪɝɭɦɟɧɬɭ U b .

ɥɿɜɨʀ ɱɚɫɬɢɧɢ M U a ,U b

130