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‚« ¤¨ª ¢ª §áª¨© ¬ â¥¬ â¨ç¥áª¨© ¦ãà­ «
Žªâï¡àì{¤¥ª ¡àì, 2001, ’®¬ 3, ‚ë¯ã᪠4

“„Š 517.946

Œ…’Ž„ Š‚€‡ˆ‹ˆ…€ˆ‡€–ˆˆ …˜…ˆŸ …‹ŽŠ€‹œŽ‰
…‹ˆ…‰Ž‰ ƒ€ˆ—Ž‰ ‡€„€—ˆ
‚. ‡. Š ­ç㪮¥¢, €. ”.  ¯á®

“áâ ­®¢«¥­  à ¢­®¬¥à­ ï á室¨¬®áâì à¥è¥­¨© ª¢ §¨«¨­¥©­ëå ¯à¨¡«¨¦¥­¨© ª à¥è¥­¨î
­¥«¨­¥©­®£® ®¡ëª­®¢¥­­®£® ¤¨ää¥à¥­æ¨ «ì­®£® ãà ¢­¥­¨ï á ­¥«®ª «ì­ë¬¨ £à ­¨ç­ë¬¨ ãá«®¢¨ï¬¨. à¥¤«®¦¥­ à §­®áâ­ë© ¬¥â®¤ à¥è¥­¨ï ª¢ §¨«¨­ à¨§®¢ ­­®© ­¥«®ª «ì­®© § ¤ ç¨.

1. ‘室¨¬®áâì ¬¥â®¤  ª¢ §¨«¨­¥ à¨§ æ¨¨ à¥è¥­¨ï £à ­¨ç­®© § ¤ ç¨.

 áᬮâਬ ­¥«¨­¥©­®¥ ®¡ëª­®¢¥­­®¥ ¤¨ää¥à¥­æ¨ «ì­®¥ ãà ¢­¥­¨¥

L[U ] 



dU

d
k(x)
dx
dx



, g(x)U (x) = f (U ); 0 < x < b;

(1)

á ­¥«®ª «ì­ë¬ £à ­¨ç­ë¬ ãá«®¢¨¥¬

U (0) = 0; U (b) =
£¤¥

m
X

k U (k );

(2)

k > C > 0, g (x) > 0, 0 < 1 < 2 <


< m < b, k

| ­¥ª®â®àë¥ ¨§¢¥áâ­ë¥

k=1

¯®áâ®ï­­ë¥.
ãáâì
­®è¥­¨¥¬

fUn g ¯®á«¥¤®¢ â¥«ì­®áâì, ª®â®à ï ®¯à¥¤¥«ï¥âáï ४ã७â­ë¬ á®®âL[Un ] = f (Un,1 ) + (Un , Un,1 )f 0 (Un,1 ); 0 < x < b;

£¤¥

Un (0) = 0; Un (b) =

m

X
k=1

k U (k );

(3)

(4)

U0 (x) | ­¥ª®â®à®¥ ­ ç «ì­®¥ ¯à¨¡«¨¦¥­¨¥ à¥è¥­¨ï ­¥«®ª «ì­®© ª¢ §¨«¨­¥©­®© § ¤ ç¨.
„«ï ¤®ª § â¥«ìá⢠ á室¨¬®á⨠¯®á«¥¤®¢ â¥«ì­®áâ¨

fUn g

¯à¨¡«¨¦¥­¨©

à¥è¥­¨ï § ¤ ç¨ (3){(4) ª à¥è¥­¨î ¨á室­®© § ¤ ç¨, ¢®á¯®«ì§ã¥¬áï ¬¥â®¤®¬
ª¢ §¨«¨­¥ à¨§ æ¨¨ [1]. „«ï í⮣® § ¬¥â¨¬, çâ® ¥á«¨ à¥è¥­¨¥ £à ­¨ç­®© § ¤ ç¨

c 2001 Š ­ç㪮¥¢ ‚. ‡.,  ¯á® €. ”.

4{24

‚. ‡. Š ­ç㪮¥¢, €. ”.  ¯á®

(1){(2) áãé¥áâ¢ã¥â, â® ®­® íª¢¨¢ «¥­â­® à¥è¥­¨î ­ £à㦥­­®£® ­¥«¨­¥©­®£®
¨­â¥£à «ì­®£® ãà ¢­¥­¨ï [2, 3]

U (x) =

Zb

G(x;  )f (U )d + '(x)

0

m
X
k=1

k U (k );

(5)

£¤¥ G(x;  ) | äã­ªæ¨ï ƒà¨­  ¯¥à¢®© ªà ¥¢®© § ¤ ç¨, '(x) | à¥è¥­¨¥ ªà ¥¢®©
§ ¤ ç¨: L['] = 0, '(0) = 1, '(b) = 0.
‚ëç¨â ï ¨§ ¨­â¥£à «ì­®£® ¯à¥¤áâ ¢«¥­¨ï (n + 1)-®© § ¤ ç¨  ­ «®£¨ç­®¥
¯à¥¤áâ ¢«¥­¨¥ ¤«ï n-®© § ¤ ç¨ ¨ ¯®«ì§ãïáì ¨§¢¥áâ­®© ⥮६®© ® á।­¥¬, ¯®«ã稬
b

Z
1
Un+1 (x) , Un (x) = 2 G(x;  )(Un , Un,1 )2 f 00 (U )d
Zb

0

(6)

+ G(x;  )(Un+1 , Un )f 0(Un )d

0

+ '(x)

m
X
k=1

k [Un+1 (k ) , Un (k )]; Un,1 6 U 6 Un :

à¨ ¢ë¯®«­¥­¨¨ ãá«®¢¨ï

A=1,

m
X
k=1

k '(k ) 6= 0;

(7)

¨§ (6) ¨¬¥¥¬

Un+1 (x) , Un (x)
=

Zb

0





G(x;  ) (Un , Un,1 )2f 00(U ) + (Un+1 , Un )f 0(Un ) d
b

m Z
X

1
+ '1 (x) A
G(k ;  ) 12 (Un , Un,1 )2 f 0(U ) + (Un+1 , Un )f 0(Un ) d:
k=1


(8)



0

„«ï ¯®«ã祭¨ï ãá«®¢¨ï á室¨¬®á⨠¯®á«¥¤®¢ â¥«ì­®á⨠ª¢ §¨«¨­¥©­ëå
¯à¨¡«¨¦¥­¨© fUn g ¯à¨¬¥¬ á«¥¤ãî騥 ®¡®§­ ç¥­¨ï:
0(U )j) = m;
max
j
G
(
x;


)
j
=
B;
max
max
(
j
f
(
U
)
j
;
j
f
06x;6b
jU j61

(9)

Œ¥â®¤ ª¢ §¨«¨­¥ à¨§ æ¨¨
max jf 00 (U )j = K; s = 1 + b
jU j61

m
X
k=1

4{25

jk =Aj:

(10)

’®£¤ , ¥á«¨ ¢ë¯®«­¥­® âॡ®¢ ­¨¥
M = 1 , mBsb > 0;
(11)
â® ¨§ ¯à¨­æ¨¯  ¬ ªá¨¬ã¬  ¤«ï ä㭪樨 '(x) ¨ c ãç¥â®¬ ¯à¨­ïâëå ®¡®§­ ç¥­¨©

(9){(10), ¯®«ã稬 ¨§ (8) ®æ¥­ªã
jUn+1 (x) , Un (x)j 6 q1 jUn (x) , Un,1 (x)j2;
(12)
£¤¥ q1 = q2 =M , q2 = kBsb=2.
ˆ§ (12) ®ª®­ç â¥«ì­® á«¥¤ã¥â ®æ¥­ª 

2
max jU (x) , Un+1 (x)j 6 q1 amax
jU (x) , Un,1 (x)j ;
(13)
a6x6b n+1
6x6b n
â. ¥. ¥á«¨ á室¨¬®áâì ¯®á«¥¤®¢ â¥«ì­®á⨠fUn g ª¢ §¨«¨­¥©­ëå ¯à¨¡«¨¦¥­¨©
¨¬¥¥â ¬¥áâ®, â® ®­  ª¢ ¤à â¨ç­ ï.
‘ ¤à㣮© áâ®à®­ë, ®¡ëç­ ï ¨­¤ãªæ¨ï ¯®ª §ë¢ ¥â, çâ® á室¨¬®áâì ¡ã¤¥â
§ ¢¨á¥âì ®â ¢¥«¨ç¨­ë q1 0max
jU (x) , U0 (x)j, ª®â®à ï ¡ã¤¥â ¬¥­ìè¥ ¥¤¨­¨æë,
6x6b 1
­ ¯à¨¬¥à, ¯à¨ ¤®áâ â®ç­® ¬ «®¬ b. à¨ í⮬ ¤«ï á室¨¬®á⨠fUn g ¤®áâ â®ç­® ¬ «®á⨠0max
jU (x) , Un (x)j å®âï ¡ë ¯à¨ ®¤­®¬ n. ‘«¥¤®¢ â¥«ì­®,
6x6b n+1
¤ ¦¥ ¥á«¨ § ¤ ­­ë© ®â१®ª [0; b] ®ª ¦¥âáï ¡®«ì訬, ¬®¦­® ­ ¤¥ïâìáï, çâ® § 
áç¥â 㤠筮£® ¢ë¡®à  ­ ç «ì­®£® ¯à¨¡«¨¦¥­¨ï U0(x) ¬®¦­® ¤®¡¨âìáï ¬ «®áâ¨
jU1(x) , U0(x)j ¨ ¤®¡¨âìáï á室¨¬®á⨠ª¢ §¨«¨­¥©­ëå ¯à¨¡«¨¦¥­¨©.
‚ëç¨â ï ¨§ (5) ¨­â¥£à «ì­®¥ ¯à¥¤áâ ¢«¥­¨¥ n-®© § ¤ ç¨ (3), (4), ­¥ âà㤭®
 ­ «®£¨ç­ë¬ ®¡à §®¬ ¯®«ãç¨âì ®æ¥­ªã

2
max jU (x) , Un (x)j 6 q2 0max
jU (x) , Un (x)j ;
(14)
06x6b
6x6b
â. ¥. ¥á«¨ ¨¬¥¥â ¬¥áâ® á室¨¬®áâì ª¢ §¨«¨­¥©­ëå ¯à¨¡«¨¦¥­¨© fUn (x)g ª à¥è¥­¨î U (x) ¨­â¥£à «ì­®£® ãà ¢­¥­¨ï (5), â® ®­  ⮦¥ ª¢ ¤à â¨ç­ ï.
’ ª¨¬ ®¡à §®¬, ¥á«¨ qi < 1 (i = 1; 2), â® ¯®á«¥¤®¢ â¥«ì­®áâì ª¢ §¨«¨­¥©­ëå ¯à¨¡«¨¦¥­¨© fUn g à ¢­®¬¥à­® á室¨âìáï ª ä㭪樨 U (x), ïî饩áï à¥è¥­¨¥¬ ­¥«¨­¥©­®£® ­ £à㦥­­®£® ¨­â¥£à «ì­®£® ãà ¢­¥­¨ï (5), ª®â®à®¥
íª¢¨¢ «¥­â­® ¨áá«¥¤ã¥¬®© § ¤ ç¥ (1), (2). Šà®¬¥ ⮣®, ¨§ ®æ¥­ª¨ (14), ª ª
®¡ëç­®, á«¥¤ã¥â ¥¤¨­á⢥­­®áâì.
2.  §­®áâ­ë© ¬¥â®¤ à¥è¥­¨ï ª¢ §¨«¨­¥©­®© § ¤ ç¨.  áᬮâਬ
à §­®áâ­ë© ¬¥â®¤ à¥è¥­¨ï ª¢ §¨«¨­¥ à¨§®¢ ­­®© ­¥«®ª «ì­®© n-®© § ¤ ç¨


dU
d
n
(15)
H [Un ]  dx k(x) dx , g(x)Un (x) = ,f (Un,1 ); 0 < x < b;

4{26

‚. ‡. Š ­ç㪮¥¢, €. ”.  ¯á®

Un (0) = 0; Un (b) =

m

X

k=1

k Uk (k );

(16)

£¤¥ g(x) = g(x) + f 0 (Un,1 (x)) > 0; f (Un,1 ) = f (Un,1) , f 0(Un,1 )Un,1 .
®«ì§ãïáì ®¡é¥© ⥮ਥ© à §­®áâ­ëå á奬 [4] ¨ á«¥¤ãï [5], ¯®áâ ¢¨¬ ¢ ᮮ⢥âá⢨¥ ¤¨ää¥à¥­æ¨ «ì­®© ­¥«®ª «ì­®© § ¤ ç¥ (15), (16) á«¥¤ãîéãî à §­®áâ­ãî § ¤ çã ¯®à浪  O(h2):
(ayx)x;i , di yi + 'i = 0; i = 1; 2 : : : ; N , 1;
m

x
,
y (0) = 0; y (b) = k yik ik+1 k
h
k=1
X



(17)

, yik+1 k ,h xik ;


(18)

£¤¥ y(n) = Un (xi ), xi = ih, h > 0 | è £ á¥âª¨, xik < k < xik+1 .
¥è¥­¨¥ à §­®áâ­®© § ¤ ç¨ (17), (18) ¨é¥âáï ¢ ¢¨¤¥
yi = pi + yN qi ;

(19)

£¤¥ pi ¨ qi ­ å®¤ïâáï ¨§ à¥è¥­¨ï ᮮ⢥âáâ¢ãîé¨å ª« áá¨ç¥áª¨å à §­®áâ­ëå
§ ¤ ç
(20)
(aPx )x , dP = ,'; P (0) = 0; PN = 0;
(aqx )x , dq = 0; q(0) = 0; qN = 1
(21)
å®à®è® ¨§¢¥áâ­ë¬ ¬¥â®¤®¬ ¯à®£®­ª¨,   ¢ëà ¦¥­¨¥ ¤«ï yN ¨¬¥¥â ¢¨¤




m  P xik ,k + P
 ,x
ik+1 k h ik
k=1 k ik h

i :
yN = h P m
1 , k=1 k Pik xikh,k + Pik+1 k ,hxik
P

(22)

à¨ í⮬, ¥á«¨ ¢ë¯®«­¥­ë ãá«®¢¨ï k(x), g(x) 2 C 2 [0; b], k(x) > C > 0,
g(x) > 0 ¢áî¤ã ­  [0; b], â® à¥è¥­¨¥ y (x) à §­®áâ­®© ª¢ §¨«¨­¥©­®© n-®© § ¤ ç¨
áãé¥áâ¢ã¥â ¨ ¥¤¨­á⢥­­® ¤«ï «î¡®£® h > 0 ¨ ¯à¨ h ! 0 áâ६¨âáï ª à¥è¥­¨î
ᮮ⢥âáâ¢ãî饩 ª¢ §¨«¨­¥©­®© ¤¨ää¥à¥­æ¨ «ì­®© § ¤ ç¨ Un (x) á® ¢â®àë¬
¯®à浪®¬ â®ç­®á⨠¯® è £ã h ¢ à ¢­®¬¥à­®© ¬¥âਪ¥.
ˆâ¥à æ¨®­­ë© ¯à®æ¥áá, ®á­®¢ ­­ë© ­  à¥è¥­¨¨ n-®© ª¢ §¨«¨­¥©­®© ­¥«®ª «ì­®© § ¤ ç¨, ¯à®¤®«¦ ¥âáï ¤® â¥å ¯®à, ¯®ª  ­¥ ¡ã¤¥â ¤®á⨣­ãâ  âॡ㥬 ï
â®ç­®áâì, â. ¥ ­¥ ¢ë¯®«­¨âáï ãá«®¢¨¥

 ( +1)


n
y
max
i
06i6N



, yi n  < ":
( )

Œ¥â®¤ ª¢ §¨«¨­¥ à¨§ æ¨¨

4{27

‹¨â¥à âãà 

1. ¥««¬ ­ ., Š « ¡  . Š¢ §¨«¨­¥ à¨§ æ¨ï ¨ ­¥«¨­¥©­ë¥ ªà ¥¢ë¥ § ¤ ç¨.|Œ.: Œ¨à,
1963.
2.  åã襢 €. Œ.  £à㦥­­ë¥ ãà ¢­¥­¨ï ¨ ¨å ¯à¨«®¦¥­¨ï // „¨ää¥à¥­æ. ãà ¢­¥­¨ï.|
1983.|’. 19, ü 1.|‘. 86{93.
3. Š ¬ª¥ . ‘¯à ¢®ç­¨ª ¯® ®¡ëª­®¢¥­­ë¬ ¤¨ää¥à¥­æ¨ «ì­ë¬ ãà ¢­¥­¨ï¬.|Œ.:  ãª ,
1971.
4. ‘ ¬ à᪨© €. €. ’¥®à¨ï à §­®áâ­ëå á奬.|Œ.:  ãª , 1983.
5. ˜å ­ãª®¢ Œ. •. Ž á室¨¬®á⨠ࠧ­®áâ­ëå á奬,  ¯¯à®ªá¨¬¨àãîé¨å ­¥«®ª «ì­ë¥ ªà ¥¢ë¥ § ¤ ç¨ ⨯  ¨æ ¤§¥ | ‘ ¬ à᪮£®.-‚ á¡.: ’¥§¨áë ¤®ª« ¤®¢ ¢á¥á®î§­®£® ­ ãç­®£®
ᮢ¥é ­¨ï ýŒ¥â®¤ë ¬ «®£® ¯ à ¬¥âà þ,  «ì稪, 1983, ‘. 163.

£.  «ì稪

‘â âìï ¯®áâ㯨«  27 ¤¥ª ¡àï 2001 £.