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¤«ï â¥áâ¨à®¢ ­¨ï ¯à¨¬¥­ï¥¬®£® ¬¥â®¤ .



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à¥¤¯®«®¦¨¬, çâ® ¢ ¯àאַ㣮«ì­®© á¨á⥬¥ ª®®à¤¨­ â xOyz ç áâì ¯à®áâà ­á⢠, ®£à ­¨ç¥­­ ï ãá«®¢¨ï¬¨ 0  x  L, ,B (x)=2  y  B (x)=2,
,H0 (x)  z  0 ¨ § ¯®«­¥­­ ï ¢®¤®©, ¯à¥¤áâ ¢«ï¥â £®à­®¥ ¢®¤®åà ­¨«¨é¥
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¯¥à¥¬¥­­ãî è¨à¨­ã ¢®¤®åà ­¨«¨é .
 áᬮâਬ ¢®«­®¢®¥ ¤¢¨¦¥­¨¥ ¢®¤ë, ¢ë§¢ ­­®¥ ⥬, çâ® á ¡¥à¥£  x = L ¢
¢®¤®åà ­¨«¨é¥ ¢â®à£áï ®¡¢ «ì­®-®¯®«§­¥¢ë© ¬ áᨢ ¨«¨ ¯®â®ª ᥫ¥¢®£® «¨¡®
« ¢¨­­®£® å à ªâ¥à . ‚ ¯à¨¡«¨¦¥­¨¨ «¨­¥©­®© ⥮ਨ ¬¥«ª®© ¢®¤ë ¢®«­®¢®¥
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@H
@V
= ,g
;

@t
@x
B

(1)

@H @BH0V
+
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@t
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£¤¥ V (x; t) | á।­ïï ¯® è¨à¨­¥ ª ­ì®­  ᪮à®áâì ¤¢¨¦¥­¨ï ¢®¤ë, B = B (x) |
è¨à¨­  ª ­ì®­ , H0 (x) | £«ã¡¨­  ¢®¤ë ¢ ¢®¤®åà ­¨«¨é¥ ¯à¨ ­¥¢®§¬ã饭­®¬
á®áâ®ï­¨¨, H (x; t) | ¢®§¬ã饭¨¥ £«ã¡¨­ë ¢ १ã«ìâ â¥ ¢â®à¦¥­¨ï.
‚¢¥¤¥¬ äã­ªæ¨î  = (x; t), ¯®¤®¡­ãî ¯®â¥­æ¨ «ã ᪮à®áâ¨, á«¥¤ãî騬
®¡à §®¬
1 @

@
:
V (x; t) = ; H (x; t) = ,

@x

g @t

‹¥£ª® § ¬¥â¨âì, çâ® ¤¨ää¥à¥­æ¨ «ì­®¥ ãà ¢­¥­¨¥ (1) ®â­®á¨â¥«ì­® ä㭪樨
(x; t) ¯à¥¢à é ¥âáï ¢ ⮦¤¥á⢮,   ãà ¢­¥­¨¥ (2) ¯à¨­¨¬ ¥â ¢¨¤





@ 2
@ 2
dH0 H0 dB @
,
gH0 2 , g

+
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2
@t
@x
dx
B dx @x

(3)

 ç «ì­ë¥ ¨ £à ­¨ç­ë¥ ãá«®¢¨ï ¤«ï à áᬠâਢ ¥¬®© § ¤ ç¨ § ¯¨èãâáï á«¥¤ãî騬 ®¡à §®¬:
@(x; t)
= 0 ¯à¨ t = 0;
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(x; t) =

@t

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j = 0;

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@
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@x x=L

(5)

£¤¥ V (t) | ᪮à®áâì ¢â®à¦¥­¨ï.
’ ª¨¬ ®¡à §®¬, ¬®¤¥«ì ¯à¥¤áâ ¢«ï¥â ­ ç «ì­®-ªà ¥¢ãî § ¤ çã (3){(5) ¤«ï
¤¨ää¥à¥­æ¨ «ì­ëå ãà ¢­¥­¨© ⥮ਨ ý¬¥«ª®©þ ¢®¤ë.

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2. —¨á«¥­­ë© ¬¥â®¤ à¥è¥­¨ï ­ ç «ì­®-ªà ¥¢®© § ¤ ç¨
¯®¢¥àå­®áâ­ëå £à ¢¨â æ¨®­­ëå ¢®«­

  ¯«®áª®á⨠(x; t) ¢¢¥¤¥¬ à ¢­®¬¥à­ãî á¥âªã f(ih; jk); i = 0; 1:::; n;

k = 0; 1; :::; mg, £¤¥ h; k | è £¨ á¥âª¨ ¯® ­ ¯à ¢«¥­¨ï¬ x ¨ t ᮮ⢥âá⢥­­®.
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k = nT, ; x = ,h=2; t = ,k=2. ‡¤¥áì T | à áᬠâਢ ¥¬ë© ¯¥à¨®¤ ¢à¥¬¥­¨
(á); m; n | ª®«¨ç¥á⢮ 㧫®¢ á¥âª¨ ¤«ï ¯¥à¥¬¥­­ëå x; t ᮮ⢥âá⢥­­®.
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ᮢ®ªã¯­®áâì 㧫®¢ á¥âª¨, â® ¥áâì ¬­®¦¥á⢮ f(xi; tj ) : 1 
i  m , 1; 1  j  n , 1g, ¯®«®¦¨¬ xi = (i , )h; tj = (j , )k.
„«ï «î¡®£® 㧫  (xi; yj ) 2
­ ¯¨è¥¬ ãà ¢­¥­¨¥,  ¯¯à®ªá¨¬¨àãî饥
¤¨ää¥à¥­æ¨ «ì­®¥ ãà ¢­¥­¨¥ (3)
i;j , 2i;j + i;j, = gH i ;j , 2i;j + i, ;j
;i
k
h
(6)

,

i
;j
i
,

;j
+ gci
= 0;
2h
£¤¥ i;j = (xi ; tj ), H ;i = H (xi ), ci = dHdx0 (xi) + H (xi ) B x dB
dx (xi ).
Š ª ¨§¢¥áâ­® [2], ª®­¥ç­®-à §­®áâ­®¥ ãà ¢­¥­¨¥ (6)  ¯¯à®ªá¨¬¨àã¥â ¨á室­®¥ ãà ¢­¥­¨¥ (3) á ¯®à浪®¬ O(h ; k ).
‡ ¯¨è¥¬ ãà ¢­¥­¨ï,  ¯¯à®ªá¨¬¨àãî騥 ­ ç «ì­ë¥ ãá«®¢¨ï (4){(5). à¨
í⮬ §­ ç¥­¨¥ ä㭪樨 ¢ à áç¥â­®© â®çª¥ ¡ã¤¥¬ § ¬¥­ïâì á।­¨¬  à¨ä¬¥â¨ç¥áª¨¬. ’ ª¨¬ ®¡à §®¬, ¤«ï â®çª¨ (xi ; 0) = (xi; t = ) ¨¬¥¥¬
i; + i; = 0; i; , i; = 0:
(7)
2
k
’¥¯¥àì ­¥®¡å®¤¨¬® § ¯¨á âì £à ­¨ç­ë¥ ãá«®¢¨ï ¤«ï ª®­¥ç­®-à §­®áâ­®© á奬ë
¢ â®çª å (0; tj ) = (x = ; tj ) ¨ (L; tj ) = (xm, = ; tj ), ᮮ⢥âá⢥­­®,
1

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1

( i)

2

1 2

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1

1 2

0

1 2

1;j , 0;j = 0; m,1;j , m;j = V ;
j
h
h

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£¤¥ Vj = V (tj ).
à¨¢¥¤¥­­ ï á奬  (6){(8) ï¥âáï ®©; ãá⮩稢®áâì ᮡ«î¤ ¥âáï ¯à¨
¢ë¯®«­¥­¨¨ ãá«®¢¨ï, ª®â®à®¥ ¯à¥¤áâ ¢«ï¥â ᮡ®© ®£à ­¨ç¥­¨¥ ­  è £¨ ¯®
¯à®áâà ­á⢥­­®© ¨ ¢à¥¬¥­­®© ª®®à¤¨­ â¥:
(1 , hk gH ;i  21 :





2




0 

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„«ï ç¨á«¥­­®£® ¨áá«¥¤®¢ ­¨ï ¯®áâ ¢«¥­­®© § ¤ ç¨ ¢ ª ç¥á⢥ ¢å®¤­ëå ¤ ­­ëå
¨á¯®«ì§®¢ «¨áì ¯ à ¬¥âàë ‡ à ¬ £áª®£® ¢®¤®åà ­¨«¨é  (‘Ž-€). —¨á«¥­­ë¥
१ã«ìâ âë à¥è¥­¨ï á¨á⥬ë (6){(8) ¯à¥¤áâ ¢«¥­ë ¢ ¢¨¤¥ £à ä¨ª  ­  à¨á. 1.
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t, c.

‘ª®à®áâì ¢â®à¦¥­¨ï §¤¥áì § ¤ ­  ä㭪樥©
 (t) =

V



V0

0

ï¤ë 1, 2, 3 ᮮ⢥âáâ¢ãîâ §­ ç¥­¨ï¬

¯à¨
¯à¨
V0

t



t0 ;

t > t0 :

= 2,

V0

= 4 ᮮ⢥âá⢥­­®.

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2. ƒ ¢ãਭ Œ. Š. ‹¥ªæ¨¨ ¯® ¬¥â®¤ ¬ ¢ëç¨á«¥­¨©.|Œ.:  ãª , 1977.|247 c.

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