численные методыА.Б. САМОХИН, В.В. ЧЕРДЫНЦЕВ, А.А. ВОРОНЦОВ

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6. ɑɂɋɅȿɇɇɕȿ ɆȿɌɈȾɕ Ɋȿɒȿɇɂə ɈȻɕɄɇɈȼȿɇɇɕɏ

ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼɇȿɇɂɃ

Ɋɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɭɪɚɜɧɟɧɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɪɚɡɪɟɲɢɦɵɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɟɪɜɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɫ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ

(x0 , y0 ) :

dy	f (x, y) .	(6.1)
dx

ɋɭɳɟɫɬɜɭɟɬ ɬɟɨɪɟɦɚ Ʉɨɲɢ ɨ ɟɞɢɧɫɬɜɟɧɧɨɫɬɢ ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɪɢ ɡɚɞɚɧɧɵɯ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢ ɹɯ (x0 , y0 ) Ƚɟɨɦɟɬɪɢɱɟɫɤɢ f (x, y) ɨɩɪɟɞɟɥɹɟɬ ɩɨɥɟ ɧɚɩɪɚɜɥɟɧɢɣ

ɧɚ ɩɥɨɫɤɨɫɬɢ (x, y) ɚ ɪɟɲɟɧɢɹ ɨɛɵɤɧɨɜɟɧɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨ ɝɨ ɭɪɚɜɧɟɧɢɹ ɈȾɍ - ɢɧɬɟɝɪɚɥɶɧɵɟ ɤɪɢɜɵɟ

ɑɢɫɥɟɧɧɵɟ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ Ʉɨɲɢ ɞɥɹ ɈȾɍ ɨɫɧɨɜɚ ɧɵ ɧɚ ɬɨɦ ɱɬɨ ɪɟɲɟɧɢɟ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɪɚɡɥɨɠɟɧɢɹ ɜ ɪɹɞ Ɍɟɣɥɨɪɚ ɫ ɥɸɛɨɣ ɫɬɟɩɟɧɶɸ ɬɨɱɧɨɫɬɢ

Ɇɟɬɨɞ ɪɚɡɥɨɠɟɧɢɹ ɜ ɪɹɞ Ɍɟɣɥɨɪɚ

Ɋɟɲɟɧɢɟ ɢɳɟɬɫɹ ɜ ɜɢɞɟ				ycc(x0 )
y(x) y(x0 ) yc(x0 )(x x0 )				ycc(x0 )		(x x0 )2	.	(6.1.1)
y(x) y(x0 ) yc(x0 )(x x0 )						(x x0 )2	.	(6.1.1)
					2!
Ɏɭɧɤɰɢɨɧɚɥɶɧɵɟ ɡɚɜɢɫɢɦɨɫɬɢ y(k ) (x)ɢɡɜɟɫɬɧɵ
		c			f (x, y) ,
		y (x)			f (x, y) ,
cc	wf	wf wy			fx f y f ,			(6.1.2)
y (x)	wx	wy wx			fx f y f ,			(6.1.2)
	wx	wy wx
ccc			2		f yy f y ( fx ff y )) ɢ ɬ ɞ
y (x) ( fxx 2 ffxy f					f yy f y ( fx ff y )) ɢ ɬ ɞ

ɗɬɨɬ ɦɟɬɨɞ ɩɪɢɜɨɞɢɬ ɤ ɝɪɨɦɨɡɞɤɢɦ ɜɵɪɚɠɟɧɢɹɦ ɞɥɹ ɩɪɨɢɡ ɜɨɞɧɵɯ ɢ ɜ ɨɫɧɨɜɧɨɦ ɢɫɩɨɥɶɡɭɸɬɫɹ ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɞɪɭɝɢɯ ɱɢɫ ɥɟɧɧɵɯ ɦɟɬɨɞɨɜ

Ɉɛɳɚɹ ɫɯɟɦɚ ɦɟɬɨɞɚ Ɋɭɧɝɟ—Ʉɭɬɬɵ

Ɉɞɧɨɲɚɝɨɜɵɟ ɦɟɬɨɞɵ ɩɨɡɜɨɥɹɸɬ ɩɨɥɭɱɢɬɶ ɡɚɞɚɧɧɭɸ ɬɨɱ ɧɨɫɬɶ ɢɫɩɨɥɶɡɭɹ ɬɨɥɶɤɨ ɩɪɟɞɵɞɭɳɟɟ ɡɧɚɱɟɧɢɟ y(x) ɂɡɦɟɧɟɧɢɟ y(x) ɧɚ ɲɚɝɟ h ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜ ɜɢɞɟ ɤɜɚɞɪɚɬɭɪɧɨɣ ɮɨɪɦɭɥɵɬɢɩɚ Ƚɚɭɫɫɚ

x h	n
'y y(x h) y(x) ³ f (x, y)dx \| ¦ AiIi ,
x	i 0

ɝɞɟ I	hf (x D	h, y E	I	0	E I ...E I		) .
i	i	i,0		0	i,1 1	i,i 1` i 1

Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ Ai , Di ɢ Ei ɤɜɚɞɪɚɬɭɪɧɚɹ ɫɭɦɦɚ

ɪɚɡɥɚɝɚɟɬɫɹ ɜ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦ h. ɉɨɥɭɱɟɧɧɨɟ ɪɚɡɥɨɠɟɧɢɟ ɫɪɚɜɧɢɜɚɟɬɫɹ ɫ ɪɹɞɨɦ Ɍɟɣɥɨɪɚ

c	1	cc	2	....	(6.2.1)
'y(x) y (x)h	2	y (x)h		....	(6.2.1)

ȼ ɨɛɳɟɦ ɜɢɞɟ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɨɥɭɱɢɬɶ ɬɪɭɞɧɨ ɩɨɷɬɨɦɭ ɪɚɫɫɦɨɬɪɢɦ ɧɚɢɛɨɥɟɟ ɭɩɨɬɪɟɛɢɬɟɥɶɧɵɟ ɮɨɪ ɦɭɥɵ ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɹ

I0 (x)	hf (x, y) ,
I (x)	hf (x D h, y E			I		) ,		(6.2.2)
1	1	1,0			0
I2 (x)	hf (x D2h, y E2,0I0 E2,1I1) ,
………………………………………….
I (x)	hf (x D	h, y E	2,0		I		... E I	) .
i	i		2,0		0		i,i 1 i 1

Ʉɜɚɞɪɚɬɭɪɧɭɸ ɮɨɪɦɭɥɭ ɪɚɡɥɚɝɚɟɦ ɜ ɪɹɞ ɩɨ h:

'y h( A A ) f h2 A (D f					x	E f	y		f )
0	1		1	1		1,0					(6.2.3)
h3 / 2 A (D2 f			2D E f		f E2 f					f ) O(h4 ),
		xx		xy				yy
1 1			1 1,0			1,0
g x , g y , g xx , g yy	– ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ x ɢ y g(x, y).

ɉɨɥɭɱɟɧɧɨɟ ɪɚɡɥɨɠɟɧɢɟ ɫɪɚɜɧɢɜɚɟɬɫɹ ɫ ɪɹɞɨɦ Ɍɟɣɥɨɪɚ Ɋɚɫɫɦɨɬɪɢɦ ɧɟɫɤɨɥɶɤɨ ɱɚɫɬɧɵɯ ɫɥɭɱɚɟɜ

Ɇɟɬɨɞɵ Ɋɭɧɝɟ–Ʉɭɬɬɵ ɧɢɡɲɢɯ ɩɨɪɹɞɤɨɜ

6.3.1. Ɇɟɬɨɞ ɗɣɥɟɪɚ

ɂɧɚɱɟ ɟɝɨ ɧɚɡɵɜɚɸɬ ɦɟɬɨɞɨɦ ɥɨɦɚɧɵɯ ɢ ɜ ɫɢɥɭ ɩɪɨɫɬɨɬɵ ɱɚɫ ɬɨ ɢɫɩɨɥɶɡɭɸɬ

ȼ ɤɜɚɞɪɚɬɭɪɧɨɣ ɮɨɪɦɭɥɟ ɨɝɪɚɧɢɱɢɜɚɟɦɫɹ ɨɞɧɢɦ ɫɥɚɝɚɟɦɵɦ

'y y(x h) y(x) A0I0 hf (x, y) .

(6.3.1.1)

ɂɧɬɟɝɪɚɥɶɧɚɹ ɤɪɢɜɚɹ ɡɚɦɟɧɹɟɬɫɹ ɥɨɦɚɧɨɣ ɥɢɧɢɟɣ ɫɨɫɬɨɹɳɟɣ ɢɡ ɩɪɹɦɨɥɢɧɟɣɧɵɯ ɨɬɪɟɡɤɨɜ ȼɵɛɢɪɚɟɬɫɹ ɲɚɝ h ɢ ɡɧɚɱɟɧɢɟ ɮɭɧɤ ɰɢɢ ɜ ɬɨɱɤɟ x x h ɢɳɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ y(x h) y f (x, y)h , ɬ ɟ ɜ ɢɧɬɟɝɪɚɥɶɧɨɦ ɭɪɚɜɧɟɧɢɢ f(x,y ɡɚɦɟɧɹɟɬɫɹ ɧɚ ɤɨɧɫɬɚɧɬɭ

Ɉɲɢɛɤɢ ɦɟɬɨɞɚ v h2 , ɬɚɤ ɤɚɤ ɜ ɪɹɞɟ Ɍɟɣɥɨɪɚ ɨɬɛɪɚɫɵɜɚɸɬɫɹ ɜɬɨɪɵɟ ɩɪɨɢɡɜɨɞɧɵɟ

Ɇɟɬɨɞ ɬɪɚɩɟɰɢɣ ɢ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ

ɗɬɨ ɩɨɩɭɥɹɪɧɵɟ ɦɟɬɨɞɵ ɢɧɚɱɟ ɢɯ ɧɚɡɵɜɚɸɬ ɦɟɬɨɞ Ʉɨɲɢɗɣɥɟɪɚ ɢ ɦɨɞɢɮɢɰɢɪɨɜɚɧɧɵɣ ɦɟɬɨɞ ɗɣɥɟɪɚ ɢɯ ɨɲɢɛɤɚ | h3

ɉɪɟɞɫɬɚɜɥɟɧɢɟ 'y		A0I0 A1I1 ɩɨɡɜɨɥɹɟɬ					ɫɪɚɜɧɢɬɶ ɞɜɚ ɩɟɪ
ɜɵɯ ɫɥɚɝɚɟɦɵɯ ɜ ɪɚɡɥɨɠɟɧɢɢ ɫ ɪɹɞɨɦ Ɍɟɣɥɨɪɚ
A A 1, D A				1	, E A		1	.
A A 1, D A					, E A			.
0	1	1	1	2	10	1	2
				2			2

ɉɨɥɭɱɟɧɵ ɬɪɢ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɱɟɬɵɪɟɯ ɧɟɢɡɜɟɫɬɧɵɯ ɱɬɨ ɹɜ ɥɹɟɬɫɹ ɨɛɳɢɦ ɫɜɨɣɫɬɜɨɦ ɦɟɬɨɞɚ Ɋɭɧɝɟ–Ʉɭɬɬɵ Ɍɨ ɟɫɬɶ ɞɥɹ ɤɚɠ ɞɨɝɨ ɩɨɪɹɞɤɚ ɬɨɱɧɨɫɬɢ ɫɭɳɟɫɬɜɭɟɬ ɦɧɨɠɟɫɬɜɨ ɜɵɱɢɫɥɢɬɟɥɶɧɵɯ ɫɯɟɦ:

						D	E	1	, A	1 A .
						D	E		, A	1 A .
						1	1,0	0		1
								2 A1
ɉɨɥɨɠɢɦ A1					1 2 ɦɟɬɨɞ ɬɪɚɩɟɰɢɣ ɬɨɝɞɚ
'y	1	(I	0	I )		h	( f (x, y) f (x h, y hf (x, y))) ,				(6.3.2.1)
	2		0		1	2

ɬɨ ɟɫɬɶ ɡɧɚɱɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣ ©ɩɨɞɩɪɚɜɥɹɟɬɫɹª ɡɧɚɱɟɧɢɟɦ ɜ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɨɩɪɟɞɟɥɟɧɧɨɣ ɬɨɱɤɟ

ȼ ɦɟɬɨɞɟ ɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜ A	1	ɬɨɝɞɚ A 0, D	E	1 .
1		0		2
ȼ ɷɬɨɦ ɫɥɭɱɚɟ
'y hf (x h / 2, y hf (x, y) / 2).				(6.3.2.2)

Ɇɟɬɨɞɵ Ɋɭɧɝɟ–Ʉɭɬɬɵ ɜɵɫɲɢɯ ɩɨɪɹɞɤɨɜ

ȼ ɦɟɬɨɞɟ Ɋɭɧɝɟ–Ʉɭɬɬɵ ɬɪɟɬɶɟɝɨ ɩɨɪɹɞɤɚ ɬɨɱɧɨɫɬɢ

'y A0I0 A1I1 A2I2 .

Ɋɚɡɥɚɝɚɹ ɜ ɪɹɞ ɩɨ h ɞɨ h 3 ɢ ɫɪɚɜɧɢɜɚɹ ɫ ɪɹɞɨɦ Ɍɟɣɥɨɪɚ (6.1.1 ), ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɭɸ ɫɢɫɬɟɦɭ ɢɡ ɲɟɫɬɢ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɜɨɫɶɦɢ ɧɟɢɡɜɟɫɬɧɵɯ

A0 A1		A2		1			½
A * D A			* D		2	1 / 2	°
1	1	2			2		°
	2	A2			2		°
A1 * D1		A2		* D2		1 / 3°
A2 * D1 *E21				1 / 6			¾
A2 * D1 *E21				1 / 6			°
E20 E21			D2				°
E20 E21			D2				°
E	D						°
10	1						°
10	1						¿

ɇɚɢɛɨɥɟɟ ɭɩɨɬɪɟɛɢɬɟɥɶɧɚ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɫɢɦɦɟɬɪɢɱɧɚɹ ɪɚɡɧɨɫɬɧɚɹ ɫɯɟɦɚ ɚɧɚɥɨɝ ɦɟɬɨɞɚ ɩɚɪɚɛɨɥ ɩɪɢ ɱɢɫɥɟɧɧɨɦ ɢɧ

ɬɟɝɪɢɪɨɜɚɧɢɢ A0 A2		1 / 6 ɬɨɝɞɚ
A 4 / 6	, D 1 / 2	, D		2		1, E 1 / 2						, E	20	1, E	21	2 .
1	1			2				10					20		21
	'y			1 (I			0	4I I			2	),
				6			0	1			2
			I0				hf (x, y) ,					/ 2) ,
	I hf (x h / 2, y I									0		/ 2) ,
	1									0
	I2	hf (x h, y I0 2I1) .
ȼ ɦɟɬɨɞɟ Ɋɭɧɝɟ-Ʉɭɬɬɵ						ɬɨɱɧɨɫɬɢ ɩɨɪɹɞɤɚ h4									ɩɨɥɭɱɚɟɬɫɹ
ɫɢɫɬɟɦɚ ɢɡ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɧɟɢɡɜɟɫɬɧɵɯ
ɇɚɢɛɨɥɟɟ ɭɩɨɬɪɟɛɢɬɟɥɶɧɵ ɞɜɟ ɜɵɱɢɫɥɢɬɟɥɶɧɵɟ ɫɯɟɦɵ
1. Ⱥɧɚɥɨɝ ɦɟɬɨɞɚ ɜ ɱɢɫɥɟɧɧɨɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɢ:
	'y	1	(I		0	3I 3I			2	I ) ,
		8			0			1	2			3

ɝɞɟ

			I0 hf (x, y) ,
	I hf (x h / 3, y I							0	/ 3) ,
	1							0
I2	hf (x 2h / 3, y I0 / 3 I1) ,
I hf (x h, y I					0		I I			2	) .
3					0			1		2
2. Ⱥɧɚɥɨɝ ɦɟɬɨɞɚ ɩɚɪɚɛɨɥ:
'y	1	(I	0	2I 2I		2	I ) ,				(6.4.1)
	6		0	1		2		3
ɝɞɟ I0 hf (x, y) ,
	I hf			(x h / 2, y I				0	/ 2) ,
	1							0
	I2	hf (x h / 2, y I1 / 2) ,
	I			hf (x h, y I				2	) .
		3						2

ɉɪɨɛɥɟɦɚ ɜɵɛɨɪɚ ɬɨɣ ɢɥɢ ɢɧɨɣ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɫɯɟɦɵ ɩɪɢ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɢ ɡɚɜɢɫɢɬ ɨɬ ɜɢɞɚ g(ɯ, ɭ) ɬɚɤ ɤɚɤ ɨɬ ɷɬɨɝɨ ɡɚɜɢ ɫɢɬ ɜɟɥɢɱɢɧɚ ɨɫɬɚɬɨɱɧɨɝɨ ɱɥɟɧɚ

Ɂɚɞɚɧɢɟ ɤ ɬɟɦɟ ɢ ɩɪɢɦɟɪ ɪɟɲɟɧɢɹ ɈȾɍ

ɇɚɣɬɢ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ Ʉɨɲɢ ɞɥɹ ɈȾɍ

yc x2 (K2 1) y , y(0) L ɧɚ ɢɧɬɟɪɜɚɥɟ >0, 2@. K ɢ L ɩɚɪɚɦɟɬɪɵ ɢɡ

ɬɚɛɥ Ɋɟɲɢɬɶ ɩɹɬɶɸ ɦɟɬɨɞɚɦɢ

Ɇɟɬɨɞ ɜɚɪɢɚɰɢɢ ɩɨɫɬɨɹɧɧɵɯ ɬɨɱɧɨɟ ɪɟɲɟɧɢɟɊɚɡɥɨɠɟɧɢɟ ɜ ɪɹɞ Ɍɟɣɥɨɪɚ ɞɨ ɱɟɬɜɟɪɬɨɝɨ ɩɨɪɹɞɤɚ

3.Ɇɟɬɨɞ ɗɣɥɟɪɚ

4.Ɇɟɬɨɞ ɬɪɚɩɟɰɢɣ Ʉɨɲɢ–ɗɣɥɟɪɚ

5.Ɇɟɬɨɞ Ɋɭɧɝɟ–Ʉɭɬɬɵ (6.4.1).

ɉɨɫɬɪɨɢɬɶ ɝɪɚɮɢɤɢ ɢ ɫɪɚɜɧɢɬɶ ɬɨɱɧɨɫɬɶ ɪɚɡɥɢɱɧɵɯ ɦɟɬɨɞɨɜ ɲɚɝ h 0,5.

ɉɪɢɦɟɪ Ʉ=3, L=2. yc x2 y , y(0) 2 .

ȼ ɦɟɬɨɞɟ ɜɚɪɢɚɰɢɢ ɩɨɫɬɨɹɧɧɵɯ ɪɟɲɟɧɢɟ ɢɳɟɬɫɹ ɜ ɜɢɞɟ y C(x) yɨɞɧ . Ɉɞɧɨɪɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ yc y 0 ɢɦɟɟɬ ɨɱɟɜɢɞɧɨɟ

ɪɟɲɟɧɢɟ y	ɋHx ɉɨɞɫɬɚɧɨɜɤɚ ɜ ɧɟɨɞɧɨɪɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɚɟɬ
ɨɞɧ	c		2		x
		x		e		ɉɨɫɥɟ ɢɧɬɟɝɪɢɪɨɜɚ
ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɚ C (x)
ɧɢɹ ɢ ɩɨɞɫɬɚɧɨɜɤɢ ɧɚɱɚɥɶɧɨɝɨ		ɭɫɥɨɜɢɹ ɩɨɥɭɱɢɦ y(x)

4ex x2 2x 2 .

Ɋɚɡɥɨɠɟɧɢɟ ɜ ɪɹɞ Ɍɟɣɥɨɪɚ ɩɪɨɜɨɞɢɬɫɹ ɜ ɬɨɱɤɟ ɯ = ȼɫɟ

ɩɪɨɢɡɜɨɞɧɵɟ ɜ ɷɬɨɣ ɬɨɱɤɟ ɢɡɜɟɫɬɧɵ:

y(0)

y (0)

2 ;

y (x)

2x y (x) ,

y (0)

ccc

(x)

2 y (x) ,

y (0)

(4)

(x)

ccc

(4)

(0)

4 .

y (x) , y

ccc

(4)

(0)

yT (x)

y(0) y (0)x

2x x

Ɇɟɬɨɞ ɗɣɥɟɪɚ Ɋɚɫɱɟɬ ɜɟɞɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟɆɟɬɨɞ Ʉɨɲɢ-ɗɣɥɟɪɚ ɦɟɬɨɞ ɬɪɚɩɟɰɢɣ ȼɧɚɱɚɥɟ ɪɚɫɫɱɢɬɵ

ɜɚɟɬɫɹ ɡɧɚɱɟɧɢɟ yk 1 yk hf (xk , yk ) ɤɨɬɨɪɨɟ ɡɚɬɟɦ ɢɫɩɨɥɶɡɭɟɬ

ɫɹ ɜ ɨɤɨɧɱɚɬɟɥɶɧɨɦ ɜɵɪɚɠɟɧɢɢ 5. Ɇɟɬɨɞ Ɋɭɧɝɟ-Ʉɭɬɬɵ ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɜɵɱɢɫɥɹɸɬɫɹ ɡɧɚɱɟ

ɧɢɹ ɩɪɨɢɡɜɨɞɧɨɣ ɜ ɩɪɨɦɟɠɭɬɨɱɧɵɯ ɬɨɱɤɚɯ ɢ ɢɫɩɨɥɶɡɭɸɬɫɹ ɜ ɨɤɨɧɱɚɬɟɥɶɧɨɦ ɜɵɪɚɠɟɧɢɢ ɫ ɡɚɞɚɧɧɵɦɢ ɜɟɫɚɦɢ

Ɋɟɡɭɥɶɬɚɬɵ ɪɚɫɱɟɬɨɜ ɩɪɟɞɫɬɚɜɥɟɧɵ ɜ ɬɚɛɥɢɰɟ

x	0	0,5	1	1,5	2
Ɍɨɱɧɨɟ ɪɟɲɟɧɢɟ	2	3,34488	5,87313	10,6768	19,5562

Ɋɹɞ Ɍɟɣɥɨɪɚ	2	3,3438	5,83333	10,3438	18

Ɇɟɬɨɞ ɗɣɥɟɪɚ	2	3	4,625	7,4375	12,2812

Ɇɟɬɨɞ Ʉɨɲɢ–ɗɣɥɟɪɚ	2	3,3125	5,72656	10,2432	18,4889

Ɇɟɬɨɞ Ɋɭɧɝɟ–Ʉɭɬɬɵ	2	3,34440	5,87111	10,6710	19,5423

7. ɑɂɋɅȿɇɇɈȿ Ɋȿɒȿɇɂȿ ɇȺɑȺɅɖɇɈ-ɄɊȺȿȼɕɏ ɁȺȾȺɑ ȾɅə ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼɇȿɇɂɃ ȼ ɑȺɋɌɇɕɏ ɉɊɈɂɁȼɈȾɇɕɏ

Ɋɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɩɪɨɫɬɟɣɲɢɟ ɭɪɚɜɧɟɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢ ɡɢɤɢ ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɝɨ ɩɚɪɚɛɨɥɢɱɟɫɤɨɝɨ ɢ ɷɥɥɢɩɬɢɱɟɫɤɨɝɨ ɬɢɩɨɜ ɫ ɧɚɱɚɥɶɧɨ-ɤɪɚɟɜɵɦɢ ɭɫɥɨɜɢɹɦɢ ɞɥɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟ ɧɢɣ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ Ⱦɍɑɉ Ɂɚɞɚɱɢ ɬɚɤɨɝɨ ɬɢɩɚ ɜɨɡɧɢ ɤɚɸɬ ɜ ɮɢɡɢɤɟ ɬɟɯɧɢɤɟ ɞɪɭɝɢɯ ɩɪɢɤɥɚɞɧɵɯ ɧɚɭɤɚɯ

Ⱦɥɹ ɱɢɫɥɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɧɚɱɚɥɶɧɨ-ɤɪɚɟɜɵɯ ɡɚɞɚɱ ɞɥɹ Ⱦɍɑɉ ɢɫɩɨɥɶɡɭɟɬɫɹ ɦɟɬɨɞ ɤɨɧɟɱɧɵɯ ɪɚɡɧɨɫɬɟɣ ɨɫɧɨɜɚɧɧɵɣ ɧɚ ɩɪɢɛɥɢ ɠɟɧɧɵɯ ɮɨɪɦɭɥɚɯ ɞɥɹ ɩɟɪɜɨɣ ɢ ɜɬɨɪɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɣ ɉɪɢ ɷɬɨɦ ɧɚɱɚɥɶɧɨ-ɤɪɚɟɜɚɹ ɡɚɞɚɱɚ ɡɚɦɟɧɹɟɬɫɹ ɧɚ ɫɟɬɨɱɧɵɟ ɭɪɚɜ ɧɟɧɢɹ ɫɜɹɡɵɜɚɸɳɢɟ ɡɧɚɱɟɧɢɹ ɢɫɤɨɦɨɣ ɮɭɧɤɰɢɢ ɜ ɭɡɥɚɯ ɫɟɬɤɢ

7.1. Ʉɨɧɟɱɧɵɟ ɪɚɡɧɨɫɬɢ

Ɉɛɥɚɫɬɶ ɪɟɲɟɧɢɹ ɧɚ ɩɥɨɫɤɨɫɬɢ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ ɧɚɩɪɢɦɟɪ

t,x	ɪɚɡɛɢɜɚɟɬɫɹ		ɧɚ ɞɢɫɤɪɟɬɧɭɸ ɫɟɬɤɭ	ɢɡ ɭɡɥɨɜ (ti , x j ),
i Z1,	j Z2 , Z1, Z2 ɩɨɞɦɧɨɠɟɫɬɜɚ ɰɟɥɵɯ ɱɢɫɟɥ ɇɚɩɪɢɦɟɪ ɜ
ɩɪɹɦɨɭɝɨɥɶɧɢɤɟ 0 d t d T ,0 d x d X ɭɡɥɵ ɫɟɬɤɢ
	ti i 't,	x j	j 'x, 0 d i d M ,0 d j d N ,		(7.1.1)
ɝɞɟ 't	T M , 'x	X N ɲɚɝɢ ɫɟɬɤɢ ɩɨ ɤɨɨɪɞɢɧɚɬɚɦ t			ɢ x ɫɨɨɬ
ɜɟɬɫɬɜɟɧɧɨ; M , N ɰɟɥɵɟ ɱɢɫɥɚ
ɇɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ u(t, x) ɭɱɚɫɬɜɭɸɳɚɹ ɜ ɤɪɚɟɜɨɣ ɡɚɞɚɱɟ
ɡɚɦɟɧɹɟɬɫɹ ɢɫɤɨɦɨɣ ɫɟɬɨɱɧɨɣ ɮɭɧɤɰɢɟɣ ui, j				u(ti , x j )	ɧɚ ɭɡɥɚɯ

ɫɟɬɤɢ ɑɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ ɤɨɨɪɞɢɧɚɬɚɦ ɡɚɦɟɧɹɸɬɫɹ ɫɨɨɬ ɜɟɬɫɬɜɭɸɳɢɦɢ ɤɨɧɟɱɧɵɦɢ ɪɚɡɧɨɫɬɹɦɢ ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɛɵɬɶ ɪɚɡ ɥɢɱɧɨɝɨ ɩɨɪɹɞɤɚ ɬɨɱɧɨɫɬɢ ɩɨ ɲɚɝɭ ɫɟɬɤɢ ɜɞɨɥɶ ɤɨɨɪɞɢɧɚɬɵ ɉɭɫɬɶ h ɲɚɝ ɫɟɬɤɢ ɜɞɨɥɶ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɤɨɨɪɞɢɧɚɬɵ f0

ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɬɨɱɤɟ (i, j), f1, f2 , f 1, f 2

ɩɨɫɥɟɞɭɸɳɢɟ ɢ ɩɪɟɞɵɞɭɳɢɟ ɡɧɚɱɟɧɢɹ ɫɟɬɨɱɧɨɣ ɮɭɧɤɰɢɢ ɩɨ ɞɚɧ ɧɨɣ ɤɨɨɪɞɢɧɚɬɟ Ɍɨɝɞɚ ɩɟɪɜɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɩɨ ɷɬɨɣ ɤɨɨɪɞɢɧɚɬɟ

ɦɨɠɟɬ ɛɵɬɶ ɡɚɦɟɧɟɧɚ ɩɪɚɜɨɣ ɢɥɢ ɥɟɜɨɣ ɤɨɧɟɱɧɨɣ ɪɚɡɧɨɫɬɶɸ ɩɨ ɪɹɞɤɚ O(h):

f /	( f	f	0	) / h,	f /	( f	0	f	1	) / h	(7.1.2)
0,	1		0		0,		0		1

ɢɥɢ ɰɟɧɬɪɚɥɶɧɨɣ ɤɨɧɟɱɧɨɣ ɪɚɡɧɨɫɬɶɸ ɩɨɪɹɞɤɚ O(h2 ):

f /	( f f	1	) / (2h).	(7.1.3)
0	1	1

Ɇɨɠɧɨ ɡɚɩɢɫɚɬɶ ɬɚɤɠɟ ɞɥɹ ɩɟɪɜɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɤɨɧɟɱɧɭɸ ɪɚɡɧɨɫɬɶ ɩɨɪɹɞɤɚ O(h4 ):

f /	( f	2	8 f	8 f	1	f	2	) / (12h) .
0		2	1		1		2

ɐɟɧɬɪɚɥɶɧɵɟ ɤɨɧɟɱɧɵɟ ɪɚɡɧɨɫɬɢ ɞɥɹ ɜɬɨɪɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɩɨɪɹɞɤɚ O(h2 ) ɢ O(h4 ) ɜɵɝɥɹɞɹɬ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ

			f //	( f 2 f			0	f	1	) / h2 ,
			0		1		0		1				(7.1.4)
													(7.1.4)
f //	( f	2	16 f		30 f	0		16 f			f	2	) / (12h2 ) .
0		2		1		0			1			2

ɇɚɩɪɢɦɟɪ ɞɥɹ ɩɟɪɜɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɩɨ ɜɪɟɦɟɧɢ ɦɨɠɧɨ ɩɪɢ ɧɹɬɶ ɩɪɚɜɭɸ ɤɨɧɟɱɧɭɸ ɪɚɡɧɨɫɬɶ ɩɨɪɹɞɤɚ O(h):

ɚ ɞɥɹ ɜɬɨɪɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɩɨ ɤɨɨɪɞɢɧɚɬɟ x ɰɟɧɬɪɚɥɶɧɭɸ ɤɨɧɟɱ ɧɭɸ ɪɚɡɧɨɫɬɶ ɩɨɪɹɞɤɚ O(h2 ):

w2u(t ,x)					ui, j 1 2ui, j ui, j 1
w2u(t ,x)					ui, j 1 2ui, j ui, j 1
i	x	x		\|		.
wx2	x	x	j	\|	('x)2	.
wx2			j		('x)2

7.2. Ƚɢɩɟɪɛɨɥɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ

ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɜɨɥɧɨɜɨɟ ɭɪɚɜɧɟɧɢɟ ɤɨɥɟɛɚɧɢɣ ɷɥɚɫɬɢɱɧɨɣ ɫɬɪɭɧɵ

u (x,t)	c2u	xx	(x,t) , x (0, X ) , t (0,T ) ,	(7.2.1)
tt		xx

ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɡɚɤɪɟɩɥɟɧɢɟ ɫɬɪɭɧɵ ɧɚ ɤɨɧɰɚɯ

u(0,t) ul (t) { L , u( X ,t) ur (t) { R ,

t [0,T ], (7.2.2)

ɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɩɟɪɜɨɝɨ ɪɨɞɚ ɞɥɹ ɮɭɧɤɰɢɢ u(x,t) ɨɬ ɤɥɨɧɟɧɢɟ ɨɬ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɢ ɜɬɨɪɨɝɨ ɪɨɞɚ ɞɥɹ ɟɺ ɩɪɨɢɡ ɜɨɞɧɨɣ ɫɤɨɪɨɫɬɢ ɨɬɤɥɨɧɟɧɢɹ

u(x,0) f (x) , ut (x,0) g(x) ɞɥɹ x [0, X ].

(7.2.3)

ɇɚ ɫɟɬɤɟ ɫɟɬɨɱɧɚɹ ɮɭɧɤɰɢɹ ui, j ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɫɨɨɬ ɧɨɲɟɧɢɹɦ ɫɥɟɞɭɸɳɢɦ ɢɡ ɢ :

2ui, j u

2ui, j

1, j

i, j

('t)2

('x)2

Ɉɛɨɡɧɚɱɢɦ q

ɉɨɫɥɟ ɧɟɛɨɥɶɲɢɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɩɨɥɭ

ɱɚɟɦ ɹɜɧɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɡɧɚɱɟɧɢɹ ɫɟɬɨɱɧɨɣ ɮɭɧɤɰɢɢ ɧɚ (i 1)-ɦ ɫɥɨɟ ɩɨ ɜɪɟɦɟɧɢ ɱɟɪɟɡ ɟɺ ɡɧɚɱɟɧɢɹ ɧɚ i -ɦ ɢ (i 1)-ɦ ɫɥɨɹɯ

u	2(1 q2 )u	q2 (u		u		) u		(7.2.4)
	i, j	i, j				i
i 1, j			1 i, j		1		1, j

ɞɥɹ j 1, 2,..., N 1.

ɋɟɬɨɱɧɵɣ ɲɚɛɥɨɧ ɜɵɱɢɫɥɟɧɢɣ ɩɨ ɮɨɪɦɭɥɟ ɹɜɥɹɟɬɫɹ ɩɹɬɢɬɨɱɟɱɧɵɦ ɬɚɤ ɤɚɤ ɫɜɹɡɵɜɚɟɬ ɦɟɠɞɭ ɫɨɛɨɣ ɩɹɬɶ ɫɨɫɟɞɧɢɯ ɭɡ ɥɨɜ ɫɟɬɤɢ ɜɨɤɪɭɝ ɬɨɱɤɢ (i, j) ɜɤɥɸɱɚɹ ɟɺ Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɜɵɱɢɫ ɥɟɧɢɹ ɩɨ ɮɨɪɦɭɥɟ ɛɵɥɢ ɭɫɬɨɣɱɢɜɵ ɧɟɨɛɯɨɞɢɦɨ ɜɵɩɨɥɧɟ ɧɢɟ ɫɨɨɬɧɨɲɟɧɢɹ q d1 Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɭɫɥɨɜɢɹ ɭɫɬɨɣɱɢɜɨɫɬɢ ɜɵɱɢɫɥɟɧɢɣ ɩɨ ɹɜɧɨɣ ɫɯɟɦɟ ɧɚɤɥɚɞɵɜɚɸɬ ɨɝɪɚɧɢɱɟɧɢɹ ɧɚ

ɲɚɝ ɩɨ ɜɪɟɦɟɧɢ ɩɪɢ ɡɚɞɚɧɧɨɦ ɲɚɝɟ ɩɨ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɤɨɨɪɞɢ ɧɚɬɟ

Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɡɧɚɱɟɧɢɣ ɫɟɬɨɱɧɨɣ ɮɭɧɤɰɢɢ ɧɚ ɞɜɭɯ ɧɚ ɱɚɥɶɧɵɯ ɫɥɨɹɯ i 0 ɢ i 1 ɢɫɩɨɥɶɡɭɟɦ ɡɚɞɚɧɧɵɟ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨ ɜɢɹ ɢ ɩɪɚɜɭɸ ɤɨɧɟɱɧɭɸ ɪɚɡɧɨɫɬɶ ɩɨɪɹɞɤɚ O('t)ɞɥɹ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɩɟɪɜɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɜ ɧɚɱɚɥɶɧɨɣ ɬɨɱɤɟ ɂɫɩɨɥɶ ɡɨɜɚɧɢɟ ɤɨɧɟɱɧɨɣ ɪɚɡɧɨɫɬɢ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɬɨɱɧɨɫɬɢ ɜɧɨɫɢɬ ɧɚ ɧɚɱɚɥɶɧɨɦ ɷɬɚɩɟ ɨɲɢɛɤɭ ɛɨɥɶɲɭɸ ɱɟɦ ɩɪɢ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɭɪɚɜɧɟɧɢɹ ɜɨ ɜɧɭɬɪɟɧɧɢɯ ɬɨɱɤɚɯ ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɚɟɦ ɞɥɹ ɡɧɚ ɱɟɧɢɣ ɧɚ ɫɥɨɹɯ i 0,1

u0, j f j , u1, j f j 't g j .	(7.2.5)

Ɋɢɫ 7.1

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