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Днепропетровский национальный университет им. Олеся Гончара

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MTMO / Приставка О.П., Приставка П.О., Ємел'яненко Т.Г., Мацуга О.М. Випадков_ процеси (correct)

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, , – . - ( ( (.

& εi , i = 1, N , h ,

( .

( ( (. %

2.4.1. 2' !0$ !

, ! ,

. /

’ . %

' % ! &$' " ! ', & 3 ( ,( " ( !# 4 5 &$! (#$.

# (

k = 2m + 1. - xi = x(ti )

( k : xi−m, , xi−1, xi, xi+1, , xi+m . ( ! (

: i − m, ,i −1,i,i + 1, ,i + m (

−m, ,−1,0,1, m ( . 2.5).

". 2.5. ."( # ( ! 4 5 &$! (#$ (k = 5 )

% [−m;m]		(
( p ( p < m )
p
f (t) = ajt j	, t [−m;m] ,	(2.4)

j=0

Θˆ = {aˆ0,aˆ1, ,aˆp} (

		m				p		2
S2 = xi+t − a jt j ,									(2.5)
					j=0
		t=−m			j=0
’(
		∂S2
		∂S2		= 0 ,	j = 0, p .				(2.6)
		∂aj		= 0 ,	j = 0, p .				(2.6)
		∂aj
, xi t = 0: xi = aˆ0 .
% (			f (t)		(				p ≤ 5.
'
f (t) = a	+ a t + a t2				+ a t3		, t [−m;m] .		(2.7)
0	1		2		3
# ( k = 5 .
(2.5) :
2	(xi+t − a0 − a1t − a2t2 − a3t3 )2 ,
S2 =	(xi+t − a0 − a1t − a2t2 − a3t3 )2 ,

t=−2

(2.6), , !

2	2	2
t = 0,	t2 = 10 ,	t3 = 0,
t=−2	t=−2	t=−2
2	2	2
t4 = 34 ,	t5 = 0,	t6 = 130,
t=−2	t=−2	t=−2

5	0	10

0	10	0
10	0	34
0	34	0

				2
				xi+ t
				t = −2
0	a			2
0	a
	ˆ	0		txi+ t
		0		txi+ t
34	aˆ1		= t = −2				.	(2.8)
0	aˆ			2
130		2		t2 xt
130	aˆ3			t = −2
				t = −2
				2
				2	3
				t		xi+ t
			t = −2

'’ (2.8) :

xN −1,

ˆ		1	(−3xi−2 + 12xi−1						+17xi +12xi+1 − 3xi+2 ),
ˆ
a0	= xi =	35
	ˆ		1					2	2		3
	a1 =				65 txi+t				−17 t xi+t			,
	a1 =				65 txi+t				−17 t xi+t			,
			72				t=−2		t=−2
			72				t=−2		t=−2
		ˆ		1				2	2	2		(2.9)
		ˆ		1				2	2	2
		a2 =				−2		xi+t + t xi+t ,
		a2 =				−2		xi+t + t xi+t ,
			14						t=−2
	ˆ		14				t=−2		t=−2
	ˆ		1					2	2		3
	a3 =				−17 txi+t + 5 t xi+t							.
	a3 =				−17 txi+t + 5 t xi+t							.
			72					t=−2	t=−2
			72					t=−2	t=−2

+ (2.9) , ! x1, x2 , xN

( (, , . # (2.7)t = −2;−1;1;2 (2.9).

(

f (−2) =

(69x

+ 4x

− 6x + 4x

− x

i−2

i−1

i+1

i+2

f (−1) =

(

+ 27x

+ 12x − 8x

+ 2x

i−2

i−1

(2.10)

f (1) =

(2x

− 8x

+ 12x +

27x

+ 2x

+ 2

i−2

i−1

i+1

f (2) =

(

−x

+ 4x

− 6x +

+ 69x

i−2

i−1

i+1

) i = 3 N − 2 , (2.10), (

x1 = f

(−2) =

(69x1 + 4x2 − 6x3 + 4x4 − x5 ),

x2 = f (−1) =

(2x1 + 27x2 + 12x3 − 8x4 + 2x5 ),

(2.11)

xN −1 = f

(1) =

(2xN −

4 − 8xN −3 + 12xN −2 + 27xN −1 + 2xN ),

xN = f

(2) =

(−xN −4 + 4xN −3 − 6xN −2 + 4xN −1 + 69xN ).

, ! aˆ0 = xi

[5] aˆ0 = 351 [−3,12,17,12,−3].

$ , (2.7) (

2.13, ( ( . 2.6).

". 2.6. # # $% !&$' % " ( -- ' !0$ ( ! 4 5 &$! (#$

! 2.13. (2.7) 0

xi		= aˆ0	(2.9) i = 3, N − 2	x1, x2 ,
xN −1,	xN (2.11).

	& 1. / (1981) , ! (

			( p = 2,5

k = 5; 7; 9; 11; 13; 15; 17; 19; 21.

- , (2.7) :

1)7

[7]aˆ0 = 1 [−2,3,6,7,6,3,−2];

					21
ˆ	1					3	3		3
a1 =			397 txi+t − 49 t							xi+t ;
a1 =			397 txi+t − 49 t							xi+t ;
1512					t=−3		t=−3
1512					t=−3		t=−3
ˆ		1				3	3	2
a2	=			−4 xi+t + t xi+t ;
a2	=			−4 xi+t + t xi+t ;
					t=−3		t=−3
		84			t=−3		t=−3
ˆ		1				3	3	3
a3	=				−7	txt + t xi+t .
a3	=				−7	txt + t xi+t .
		216				t=−3	t=−3
		216				t=−3	t=−3

2)9

[9]aˆ0 = 2311 [−21,14,39,54,59,54,39,14,−21];

815 txi+t − 59 t

xi+t ;

7128

t=−4

−20 xi+t + 3 t

xi+t ;

a2 =

924

t=−4

−59 txi+t + 5 t

xi+t .

t=−4

7128

* !# ' ' !0$ . % ( (

«-( 53 »,

6. " ( , (

( ( . 7

, ,

. $

, ,

. 4, $. 8 (1997).

)

{x(ti );i =

}

1, N

{Mi (ti , xi ); i = 1, N},

2, N −1},

–

– {Mi (ti , xi ); i =

, ! (

Mi−1MiMi+1 ,

i = 2, N −1:

Mi = 1(Mi−1 + Mi + Mi+1),

3(ti−1

+ ti + ti+1);

xi =

(xi−1

+ xi + xi+1),

ti ;

xi = xi +

xi ,

i = 2, N −1,

(2.12)

ti = ti +

2t = t

− 2t + t

i+1

2x = x

− 2x + x

i−1

i i+1

# ( M1, MN

, ! 0 :

	M0M1 = M1M2 ,	MN M N −1 = MN +1MN	(2.13)

M3M0 = M3M2 + M3M1,		MN −2M N +1 = MN −2MN −1 + MN −2M N . (2.14)
+	! (		M0(x0,y0),
MN+1(tN+1,xN+1):
1)	(2.13):
	t0 = 2t1 − t2,	tN +1 = 2tN − tN −1,	(2.15)
	x0 = 2x1 − x2,	xN +1 = 2xN − tN −1;	(2.15)
	x0 = 2x1 − x2,	xN +1 = 2xN − tN −1;
2)	(2.14):

(4t + t

− 2t

(4t

+ t

− 2t

N +1

N −1

N −2

(2.16)

x =

(4x + x −

(4x

+ x

− 2x

N +1

N −1

N −2

, ! (2.12)

1/3, « ’ » (0 ≤ α ≤ 1 3):

+ αΔ ti ,

= xi

+ αΔ xi ,

i = 1, N ,

(2.17)

ti = ti

/ α = 1 3, ,

α = 1 8

’

, ! α = 1 6 –

. * (2.17)

. &

L − L		≤ Aα (L N )2 ,	(2.18)

L , L – ,

Mi , Mi , i = 1, N ; A ( . 9 ( .

! 2.14 {Mi (ti , xi ); i = 1, N}.

" ( M0 (t0, x0 ),

MN +1 (tN +1, xN +1)

(2.15)

(2.16).

α [0;1 3] ((

ti ,

(2.17).

$(( :

L = li ,

i=2

li =

+ xi − xi−1

+ xi

ti − ti−1

− ti−1

− xi−1

$(( , ’ :

qti = ti − ti−1 ,

pti = qti+1 − 2qti + qti−1,

qxi = xi − xi−1,

pxi = qxi+1 − 2qxi + qxi−1,

i = 2, N .

" (

A = liTi ,

qti pti + qxi pxi

i=2

6. * ( (2.18). &

( , ( .

* ( . 2.7).

". 2.7. # # $% !&$' % " ( -- !# ' !0$ (α = 13)

$ ( ( ( ,

- .

2.4.2. *" '$ +# ( # ! % ' . "% 4 #$

& ,

, (( - .


%
	x(t)		h, , h
			x = (...xi−2, xi−1, xi, xi+1, xi+2...), i Z
x . # x(t)
	h	-		,

Sr,0 (x,t) = xiBr,h (t − (i + 0,5)h), r = 2,4;

i Z

S3,0 (x,t) = xiB3,h (t − ih),

i Z

Br,h (t) – - r - , ! ( ( (

		t+h 2	1	(	t		< h 2),

Br,h (t) =	1	Br−1,h (t)dt ,	B0,h (t) =
				(			≥ h 2).

	h t−h 2		0			t

%, - B2,h(t) ( :

	0,
	(3+ 2t h)2
	(3+ 2t h)2	8,
B2,h (t) =
3 4 − (2t h)2 4,
(3− 2t h)2		8,

t [− 3h2;3h2] ,

t [− 3h2;− h2],

(2.19)

t [− h2;h2], t [h2;3h2].

2 , !

- (

!. # (

. %, ! y = 2(t − (i + 0,5)h)h , y ≤ 1,

, (2.19), S2,0 (x,t)

S2,0 (x,t) = (xi−1 − 2xi + xi+1) y2 8 + (−xi−1 + xi+1) y4 + (xi−1 + 6xi + xi+1)8 . (2.20)

&, ! ( , ! ,

* . * (2.20)

, ! , (, . $ ,

- ,

																			r
											Sr,0 (x,t) = xi γi(,rc,0) yc ,																						(2.21)
																i	c=0
y =	2	(t − (i + 0,5)h),								y		≤ 1,			r = 2,4;						y =			2	(t − ih),				y	≤ 1,		r = 3;
	2																							2

	h																							h		−3					−1
	h																							h
							1		−2			1												1				3
							1		−2			1														−15			−3
		γ(	2,0	)		1											γ(	3,0		) =		1		23		−15			−3		3
					=			6 0				−2			;									23 15 −3 −3									;
					=			6 0				−2			;																		;
						8														48
								1	2		1													1			3	3			1
																								1			3	3			1
																1	−4			6		−4				1
																76 −88				24			8			−4
									γ(4,0)					1		76 −88				24			8			−4
											=			1		230	0		−60				0			6	.
											=					230	0		−60				0			6	.
													384			76	88			24		−8				−4
																76	88			24		−8				−4
																1	4			6			4			1
																1	4			6			4			1

& ( ,

-				h . & r = 2,4	y = 0
r = 3 y = 1 (2.21)
		1		i+1
				i+1
S2,0 (x,ih) =	1	6	(xi−1	xi xi+1) = γ 1,(Sj2,0 )xj ,	(2.22)
S2,0 (x,ih) =		6	(xi−1	xi xi+1) = γ 1,(Sj2,0 )xj ,	(2.22)
8		1		j=i−1
		1

						i+1							i+2
		S3,0 (x,ih) =				γ 1,(Sj3,0 )xj ,					S4,0 (x,ih) =		γ 1,(Sj4,0 )xj				,	(2.23)
						j=i−1							j=i−2
																	1
				1						1							76
	(S2,0 )		1	1			(S3,0 )		1	1				(S4,0 )		1	76
γ	(S2,0 )	=	1	6	;	γ	(S3,0 )	=	1	4		;	γ	(S4,0 )	=	1	230	.
γ		=		6	;	γ		=		4		;	γ		=		230	.
1			8			1			6				1			384
			8	1					6	1						384	76
				1						1							76
																	1
																	1
#						(						(

(2.22–2.23) ( (

. %, S2,0 S3,0

S2,0	(x,ih) = (xi−1	+ 6xi	+ xi+1)	8 ,	(2.24)
S3,0	(x,ih) = (xi−1	+ 4xi	+ xi+1)	6 ,

S4,0 – :

S4,0 (x,ih) = (xi−2 + 76xi−1 + 230xi + 76xi+1 + xi+ 2 )384.

xi = x i + x i , i Z ,

x i , x i – - . , ! x i

x i = Sr,0 (x,ih) = x i(Sr,0 ) , r = 2,3,4,

( :

x i(Sr,0 )

i+1

x i(S4,0 ) =

i+2

γ 1,(Sjr,0 )xj , r = 2,3,

γ 1,(Sj4,0 )xj ,

(2.25)

j=i−1

j=i−2

−1

−76

(S2,0 )

(S3,0 )

(S4,0 )

;

154

384

−1

−76

−1

& « »

. " (2.24). *, ! x(t),

- , . -

S2,0 (S2,0										(x,ih),ih)=											1			1		xi−2 +							3		xi−1 +					1	xi		+		3				1		xi−1 +						3	xi +			1		xi+1			+
S2,0 (S2,0										(x,ih),ih)=																xi−2 +									xi−1 +						xi		+								xi−1 +							xi +					xi+1			+
																						8 8									4									8					4 8											4			8
+		1			1		xi			+		3		xi+1			+		1	xi+ 2							=		1		xi−2 +							3		xi−1 +				19			xi +						3		xi+1				+	1		xi+2 =
+							xi			+				xi+1			+			xi+ 2							=				xi−2 +									xi−1 +							xi +								xi+1				+			xi+2 =
		8 8								4								8										64									16						32									16							64
				1							1																				1																					1
=	1			6						1			6			(x				x x								)		1				6		(x x x										)				1		6				(x x x									)		=


																	i	−2 i					−1 i																i−1 i			i+1															i		i+1 i+2
	8									8			1																	8				1																8
				1									1																					1																		1
							1
									6	+ 6
									6	+ 6																																								i+2
	=		1						1	+ 36+ 1							(xi−2									xi−1						xi					xi+1 xi+2 ) =													γ 1,(Sj2,0 (S2,0 ))xj ,																		(2.26)
	=								1	+ 36+ 1							(xi−2									xi−1						xi					xi+1 xi+2 ) =													γ 1,(Sj2,0 (S2,0 ))xj ,																		(2.26)
			64																																														j=i−2
												6 + 6																																					j=i−2
												6 + 6
																1
																1
																																					1
																																					12
																				(S2,0 (S2,0 ))														1			12
																	γ													=							38 .
																	γ													=							38 .
																		1															64
																																	64				12
																																					12
																																					1
																																					1
&																									Sr,0 (Sr,0 (x,ih),ih)= x i(Sr,0 (Sr,0 )),																																		(
(2.26) r = 3,4 ( « »
x i(S3,0 (S3,0 )) =															i+ 2										(S3,0 ))xj ,														x i(S4,0 (S4,0 )) =														i+ 4
x i(S3,0 (S3,0 )) =															γ 1,(Sj3,0										(S3,0 ))xj ,														x i(S4,0 (S4,0 )) =														γ 1,(Sj4,0 (S4,0 ))xj ,
															j=i−2																																					j=i−4
																																																											1
																																																											152
																						1																																					152
																						1																																					6236
																							8
								(S3,0 (S3,0 ))											1				8																	(S4,0 (S4,0 ))														1					35112
				γ													=					18 ;																γ										=											64454					.
				γ													=					18 ;																																										.
						1													36
																																							1												147456
																							8																												147456
																							8																																			35112
																							1																																				6236
																							1																																				6236

																																																											152
																																																											1
																																																											1

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