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Žªâï¡àì{¤¥ª ¡àì, 2001, ’®¬ 3, ‚ë¯ã᪠4

“„Š 517.549.8

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@U

@t =a

2

U + @U

@t

+f(x;t); (1)

£¤¥

U =

@ 2

U

@x 2

1 +

@ 2

U

@x 2

2 +

@ 2

U

@x 2

3 ;

a 2

, >0 | ¯®áâ®ï­­ë¥, § ¢¨áï騥 ®â £¥®¬¥âਨ ¯« áâ  ¨á¢®©áâ¢

䨫ìâàãî-饩 ¦¨¤ª®áâ¨.

‚®¡« áâ¨Q

T

=(0;T),fxj0<x

<l

; =1;2;3g¤«ïãà ¢­¥­¨ï

(1) à áᬮâਬ § ¤ çã

Uj

@!

=0; U(x;0)='(x): (2)

1. €¯à¨®à­ ï ®æ¥­ª . ãáâì áãé¥áâ¢ã¥â ॣã«ïà­®¥ à¥è¥­¨¥ § ¤ ç¨

(1){(2). ‘­ ç «  ¯®«ã稬 ¤«ï à¥è¥­¨ï § ¤ ç¨  ¯à¨®à­ãî ®æ¥­ªã ¢

¤¨ää¥-७樠«ì­®© ä®à¬¥. „«ï í⮣® 㬭®¦¨¬ãà ¢­¥­¨¥ (1)᪠«ïà­® ­  U.

(U

t

;U),a 2

(U;U),

@

@t U;U

=(f;U): (3)

à¥®¡à §ã¥¬áãç¥â®¬ £à ­¨ç­ëåãá«®¢¨©(2)ª ¦¤®¥á« £ ¥¬®¥¢ëà ¦¥­¨ï(3)

(f;U)6 1

4" kfk

2

2;

+"kUk 2

Z

4" kfk

2 "l

2

4" kfk

2;Qt +kU

2" kfk

2

2;Qt

„«ï ¤®ª § â¥«ìá⢠ á室¨¬®á⨠¬¥â®¤  ®â¥, ¯®«ã稬  ¯à¨®à­ãî ®æ¥­ªã ¤«ï

à¥è¥­¨ï § ¤ ç¨ (6), (7).

“¬­®¦¨¬ ãà ¢­¥­¨¥ (6) ᪠«ïà­® ­  y

t

(y

t ;y

t

),0:5a 2

((y+y); y

t

),(y

t ;y

t

)=(f;y

t

): (8)

à¥®¡à §ã¥¬ ª ¦¤®¥ á« £ ¥¬®¥ (8)á ãç¥â®¬ £à ­¨ç­ëå ãá«®¢¨© (7)

((y+y); y

t )=

Z

j

(y+y) y

t

dx=,

Z

j y

2

x dx

; (9)

£¤¥ y 2

x =y

2

x

1 +y

2

x

2 +y

2

x

3 ,

j

=(t

j

) | á¥ç¥­¨¥ Q

t

¯«®áª®áâìî t =t

j

=j,

(y

t ;y

t )=,

Z

j y

2

x

t

dx: (10)

®¤áâ ¢«ïï (9), (10)¢ ᮮ⭮襭¨¥ (8), ­ å®¤¨¬

ky

t k

2

2;

j

+0:5a 2

ky

x k

2

2;

j

t

+ky

t k

2

2;

j 6

1

4" kfk

2

2;

j

+"ky

t k

2

2;

j

: (11)

‚롥६ ¢ (11) " = 1

2

, § â¥¬ ¯à®¨­â¥£à¨à㥬 ¯®«ã祭­®¥ ¢ëà ¦¥­¨¥ ¯® t ®â

¤® T 0

. ’®£¤  ¯®«ã稬 âॡ㥬ãî ®æ¥­ªã

T 0

X

t= ky

t k

2

2;

j +a

2

ky

x k

2

2;

j +

T 0

X

t= ky

x

t (t)k

2

2;

j

6 T

0

X

t=

kf(t)k 2

2;

j +a

2

ky(x;0)k 2

2;

0 :

(12)

Ž¡®§­ ç¨¬ç¥à¥§z =y,u ¨¯®¤áâ ¢¨¬ y=z+u¢ãà ¢­¥­¨¥ (6),⮣¤ ­ ©¤¥¬

z

t

=0:5a 2

(z+z) +z

t + ;

z

0 =z

N =0;

z(x;0)=0;

(13)

£¤¥ = 0:5a 2

(U +

U)+U

t

+f ,U

t

= O( 2

): à¨¬¥­ïï ®æ¥­ªã (12) ª

§ ¤ ç¥ (1){(2)¤«ï¯®£à¥è­®á⨠(13)¯®«ã稬, ç⮢ ª« áᥤ®áâ â®ç­®£« ¤ª¨å

à¥è¥­¨© ¨á室­®© § ¤ ç¨ (1){(2) ¬¥â®¤ ®â¥ á室¨âáï ᮠ᪮à®áâìî O( 2

).

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