chỉnh hóa một số phương trình tích chặp, chương 3

Chu(Jng

3: MQl s(/ bai loan quy v~ phuang trinh tfch chi;ip

CHu'dNG

3:

"

,..",

~

M9TSOBAITOANqUYVE
1-IIUdNG TIIINII rrtcll CII~I.

Qua

tdnh giai quye't nhi@u b~ti loan c1iav~t ly, ky thu~t nhu' cac b~ti to?

nhi~t, cac b~ti loan chuySn tru'ong ... phliong tdnh Hch ch?p (1ftl1tdnh nay, b~ng cong ql ly thuye't tniong va bien 06i Fourier, chung ta quy hai b~1i
loan nhi~t v@ phuong tdnh tich ch?p. D6 la bai toan nhi~t ngu'c;Jcthai gian
(Backward heat equation) va bai loan nhi~t trong 16 khoan tham do (Borchole
measurements heat equation).
3.1 Bdi tacm nhhT!tngllde thai gian
Tren mQt thanh d§n nhi~t dai va h,;lI1,gi{l sli cho biet Jlhi~t c1Qu d vi tri x tai
thai oiSm t nao c16,ch~ng h<;111
t = 1, 1a u(x, 1) = u(x). Ta can Om nhi~t dQ u 0 vi Iri x

Iuc t = 0, tlic 1a Om v(x) = u(x,O).
Trudc lien ta xet bai loan I1hi~t gia tf! c1au: Cho bilt

v(x)

= u(x,O). 11m

phfln

b6 nhi~t u(x,t) (1vi tri x vao thai c1iSm1.
Bai loan nhi~t gia tri c1auco phliong tdnh:

a2u au
~ax = -;at

(3.1)

XE R, t > o

{ u(x,O)= v(x). Tim u(x, t)?

."

' ?
Nh an xet: T a co tIleJl g la
th let u,
~

/

/

.

(-

00

.
v
.
au au I
a cae Ilam so b I c h an Iren micn
& ffl
.
.
'

--,-

/

'

~/

~

;:;

< x < + 00, t > 0) vI trang thl!c te Jlhi~t dQ eung nhli t6c (1Qbien thie n clla nbi~ t

(1Qtbeo khong gian va thai gian khong th@ tang den va h<;1l1.

8iJ c1~3.1. 1
Gia su u(x) 1a ham sO'bt ch?n tren R va lien t\ICt<;1ix = O. Kbi c16:
'

I1m

r~
-OCJ

n

I1X2

--.e--.u

11--+00 OCJ 7t

(

x )d x = U 0
(

)


Chuang 3: Mrlt so' bili tocin quy vi phu'cJn[?trinh rich ch('ip

-----

C/ui'ng minl1:

= ~.~ .e-""

£J~t f,,(x)
I

r:

TnI'dc he't ta chli'ng minh
oo

Tac6:

1=

-.e-lIx dx = 2 £
-00 n
)

--,e-I1X dx

n
~~-1
2

D6i bie'n sO' v = x, J;; ta dLI'QC:I

4
12 = -

~

n

[

00

e

-v

(x) = 0

,!i~AJ fl1

-00

2

f~

r:

(x)dx = 1 va

fl1

2

d v,

[

00

e

-y

)

= .Jn
2

2

roo

2

1) e ~v

dv

2

2
4
- ( v +y )
dy = dvdy
e
£ £
n))
-00

-00

Chuy€n sang tQa dQ c1,1'c:
7t
(0 ~ r < +00; 0 ~ (P ~ -)
2

V = r.COS(P
,
,

{ y = r.SlIHp

ta c6: I = n
"

M~t

khac

'-00

4 ~

2

1)2

d(p,

roo

2

1

e-r ,rdr

IIY1 f (x )
- lim

11-~I~1I

-

~

[

'im [11(x)dx

00
0011~00

1 = 1 00fl1(x)dx

=1

1

'

= 1I-~002\/
11m
r--2
I1X2
7tH.X ,e -

11-~00Fn,el1X2

=0
~

= 1. V~y

h. k. n. tren R

=0

(3.2)

Ham u(x) lien t~lct~i x = 0 ~ \if > 0, 38 >0 saGcho voi x ~ 8 thl
I

I

u(x) --

Vi u bi ch~n tren R nen ta d~t M = 2

I

E

u(O)1 <--

2

lIulloo

Tll' (3.2) SHYfa: Voi 8 >0 cO'dinh, 3 no E N sao cho \i n 2 no ta c6:

r

f (x)dx < ~
2M

~xl>/j II


ChU'cIng

3: Mell s{/ hdi loan quy v€ phU'(fng lrll1h ([eh eh~lp

Suyra: II~XJfl1(x).u(x)dx - U(o)!=1£:'[11(x).u(x)dx -£:fl1 (X).U(O)d~S;

s;r:f" (x)lu(x)- u(O)ldx = f f"(x)lu(x)-

u(O)ldx+

IxJ:58

+

f (, (x)lu(x)- u(O)ldx< ~ f f"(x)dx + M. f fn (x)dx <

Ixl>8
S

<-

,V~y

IXI:58

I

!~

Hx, ,

S

Ixl>8

S

f (x)dx+M--=-.I+-=s
2 -if) n
2M 2

2

r: f" (x)u(x)dx = u(O)

J!~XJ

86' d~ 3. 1.2
Hams6r(x,t,~,1:)=

-(X-~)2

1

thoa phu'dng tr)nh truy€n nhiet

.exp
2~n(t - 't)
[ 4(t - 1:) ]

(3.1) va c1~ng th((c:

o2r
or
o(i --7);
ClllJ'ng minh:

o2r

1. Ta clul'ngminh:
or

-1

-=
ax

-1
3 exp

ox

4.Jn(t -1:)2

-1

:

,

1-

4vn(t-1:)2

~ [.

or = ~
at

=

2.Jn

{

[

{. 2(t-1)2

-1

~

]

-.

-

)

2

'"

[ 4(t -1:) ]

exp--

.

.[

_(X-~)2

(X-~)2

J

.exp
2(t - 1:)

[

4(t -1:)

J}

_(X-~)2
4(t - 1:)

]

exp ~ix - ~)2 + -~

(X-~)2

,[
4vn(t -1:)2
o2r D[
=>--=ax 2 Ut

_ (X_!:

4(t -1:)

(x- ~)22(t - 1)

-1

3

or

= - at

-(X-_~)2

~=

=

oX2

(x-~).exp

,~
4vn(t-1:)2

o2r

--

(x - ~)2 exp --=-(x -:~

[ 4(t-1:) ] ..)t-1 4(t-1)
_(X-~)2

exp
2(t'-1:). ]
[ 4(t-1:) ]

V~y r (hc'SaplU(dng (dnh (rUY€llllhi~( (3.1)

-

4(t-1:)

]j


Chuang 3: M(jl sf)' bel; loan quy v€ phuang lrlnh tfch dl(lP

a2r

ar

2. Ta ch((ng lTIinh: a~ 2 ==- at

ar

x ~
-

-

==

a~

exp

[ 4(t - t) ]

4-Fn(t - t)3/2 .

a2r

1

-==
a~2

-exp

4l;;(t-t)3/2'

{

1
==

.exp

I
~
4\111:(t-t)2

ar ==~
at

t 2(t-t)2

I

a2r

[

.

~ .exp [ 4(t-1)

- II
J

.

(X-~)2

]{ 1- 2(t-~) }

ar
a1

0

==----

a~ 2

I

(X-~)2

]{ 2(t-t)

4(t-t)

_(X-~)2

4\111:(t-1)2

===> --

_(X-~)2

]}

r

1
==

_(X-~)2
_(X-~)2
(X-~)2
+
ex
p[ 4(t-t)
[ 4(t-t) ] 2(t-t)

-(X-~)2
-(X-~)2
(X-~)2
1
1
.cxp
.exp
~.
[ 4(t - t) ]
[ 4(t - t)2 ]
[ 4(t -1) ] ~

1

2J11:

- (x - ~)2

Menh d~ 3.1.3
Bai loan nhi9t gia td dfiu co nghi9lTIdliQcxac djnh bdi cong th('(c:

~2J;t fOOexp-

u(x, t) ==

[

-00

(x 4t

~)2

X E R, t > 0
1 v(~)d~;

CluIng min/1l.
D?t r(x, tf ~f 1:) ==

-- (x-=-~~
.exp

[ 4(t 1:) ]
2~n(t - 1:)

1

Voi (x, t) c6 (1jnh,ham s6

r

xac djnh vOl:

-

00

< ~<+oo;0 < 1 < t

Lay u (~, 1:) la lTIQtnghi9lTIcua plu(ong trlnh (3.1)

(3.3)

'C6 djnh (x,t) , (x ER, t> 0). Xet trliong vectO F:
,

a

a

F(~ t) == u(~,1:)--r(x,t,~,1:)-r(x,t,~,1:)-u(~,1:);U(~,l)r(X,t,~,1:)
,

a~
a~
(
(- 00< ~ < + 00; 0 < 1: < t)

J


Chuang3: Mf)l stYbai loan quy vi phuang lrlnh lfch ch912
J:)~t D= {(~,T)ER2 \-n<~

Kl hi~u aD la bien cua mi~n md D. Ap d~lI1gdjnh ly divergence cho tnrong
vectOF tren D, ta co:
->

(3.4)

fdiv F(~, T)d~dT = f<F(~, T),n(~, T) > ds

D

00

Nho b6 d~ 3.1.2 va (3.3) ta co:

a

.

of.
dlVF(~ T)=u--f,
a~ [ a~
= u or -

aT

au

a
a2f
+-(uf)=u--r-+a~] aT
a~2

a2u

a(uf)

a~ 2

aT

r au + a(ufL = 0
at

(3.5)

aT

T

->

t ,- - - - - - - - - - - - - - - - -

->

->

n1

~~ (BInh 1)

~~

Ll

L

~

~

n

-11

Ky hi~u nhl( trel1Bluh 1 ta co:
f< F(~,l),~(~,T)> ds = f< F,l~ > ds + f< F,l~ > ds + f< F,t;: > ds + f< F,t~:> cis(3.6)
aD

LI

L2

L]
4

->

Tiy (3.4), (3.5) va (3.6) ta SHYfa: I J < F(~,T),11;(~,T)> cis= 0
;=1L;

L'I


Chu'ong 3: M()t sc/ b!!i loan quy v~ phuong trlnh rich ch(lP

~

-t

TrenL,:

~=n,

-~

O
I"

=(1;0)

a

a

a~

a~

f<F,n, >cls= f [ lI(n,t)--f(x,t,n,t)-f(x,t,n,t)~lI(n,l)

LI

0

a

t"

t".
a
ft(x,t,n,l)--u(n,t)dt

= fu(n,t)--r(x,t,n,t)dta~

0

I"

1

X- n

f

= -4\1 In

0

- (x - n )
exp

3

-

(t-t)2

[

I"

1,

,:ro

f (t-t)2

.u(n, t)el1: -

I"

f-

1

f

2\1 n 0

(t-t)2

- (x - n )
,-exp
-,

[

2

,
4(t - t)

;::)
uU

]

.~(n,
a~

1)el1:

ve san cling ta ou'Qc:

(j

)

'"

2

'

I

.u(n, t)clt S
]1

exp [ - 4(t-t)

~3

0

oK

]

-. ( x-n

x-n

= 4\1nf-

JI

1

4(t - t)

Danh gia cac Hch phan
,

2

ell:

=

a~

0

l

4\1nI

X-11

-

/

f-}exp
0-

I-

(t-t)2

(X-H)

,2

,.u(n,
]

4(t-t)

1)<11

X-H

K

I

I

= 2~"t-t

DOlblensO:UI

D~t8=t-to

Do Ubj ch~n lien :3M]> 0 saG cho:

1
=> Jis

IX-III
2J~

f.e

r

too

-II~

(

( ))d

,U 11,1 u]

u1-

f

--I1~

1 e

""nix-II!
2l;

(u I) d Ur

'
,xn.::.::-.!.!!IX-I1
l2Ji'2J~ , l

--c/J

Ap d~lng ojnh 1;' hQi t\1bj ch~n ( 1.5, CI1l(ang 1) tren R, trong 06 :
2

f ( u ) =e-IlI
11

X

.

I

T a c6: J I
Do au
a~

0 khi n --+

--+

.

1

=

I

2-vn

f

-co

g(ul)=e-lIj

(3.7)

1: >

8 nen :3M 2 > 0 saG cho :

.

1

tf)

f

0

= M2" OOexp

n=1,2,3...f(u1)=Ova

00

bi chan va t --

,

J2

rl~~-nl
l 2 -Jl .2..;c- ]

,

( u I')

-x-n
(
I

"2
(t -- 1:)
-

exp

-

1:) ]

2

;:)

u

-u(t,'l)eh

[ 4(t - 'l) ] a~

(x - n)2_ ,x
[

[ 4(t

)

j

I"
'

S M2 fexp
[
0

(1)d'l.

010

Ap cI~lngojnh 1;' hOi t\1 bj ch~n (1.5, Chuang 1) tren R, trong (16:

-X-ll

(

,

)

2

4(t - T) ]

eh


.Chuang 3: MQ{ slYbiti {Danquy vi phuang {dnh tfch ch4p

f

'T

(x - n)2 .

'T ell
XrO,to!()
]

P[ 4(t-'T)

, 11( )
,

-

= ex

Ta co: J 2

~

0 khi n ~

n = 1,2,3...;f('T) = 0, g('T)= X[Oto j ('T)
(3.8)

Cf)

->

f < Ft 11I > cis --)- 0

Tli (3.7) va (3.8) ta SHYfa:

khi

11 --)- co

LI

'Ly lu~n tl(dng t~(nhu' tren Ll, ta cling co ke't que! tre11L, nhl( san:
-->

f
> ds ~ 0 khi n ~

Cf)

L,

--+

Tren

L2 :

'T

-t

= 0,

- 11 <

~< n,

n2 = (0;-1)

II

(

OCJ 1

11<F,112 >= -JU(~,O).r(x,t,~,O)d~

= -J2-fnt

~)2

exp [ - X4~

] V(~)'X[-II,II](~)d~

Do ham s6 v bj ch~n, ap d\l11gd1nh ly hQi t~1b1 ch~n (1.5, Chl(dng 1) tren R,
trong do:

.
_(X-~)2
_(X-~)2
(~) = exp - 4t
'X[-I1I1](~)'
n
=
1,2,3,
f(~)
=
g(~)
=
exp
.
4t
[
]
[

[II

ta dl(c;Jc:

1

--+

f< F,n2 > ds ~

I.

-

_(X-~)2

+OCJ

fexp

2-vnt -OCJ

, L2

[

]

4t

.v(~)d~ khi n ~

00

--+

TrenL4:

'T=to'
->

-n<~
n4 =(0;1)

11

f< F,n4 > ds = fu(~,to).r(x,t,~,to)d~ =
L~

-11
+OCJ

=

f 1
_OCJ2~n(t -

to)

exp

[

(x - ~)2

4(t - to)

.

] 'X[-Il,Il](~)lI(~,

to )cI~

A P d\lng dinh ly hQi t~lnoi LIen, trong d6:

f (~)=
11

f(~)

-(X _~)2

1

2~rc(t- to) ex { 4(t - to ] .X[-n,n](~)

= g(~) =

1
2~n(t-to)

exp

-

(x - ~)2

[ 4(t-to)

]

, n = 1, 2, 3 ...

-

]


Chu'ang 3: M(Jl srI hili loan quy v€ phuong trlnh tfch ch~lp

.

,

->

tadtiQc,

1

+00

J
>ds-+

J

(t-t)

-00 2~n

L,

.
khl n-+oo

- (x - ~)2

exp-

j

[ 4(t-to)

()

u(~,to)d~

Nhti v~y, tli (3,6) ta suy fa:
+00 1

f "ex -002-v71:t {

(~)2

x4t

1

v

B ~t

t-to =-,
n

+oo

{-4(t-to)
(x - ~)2]

;?'

-2

[

x-~
=-,ta

-ex
71:

{

4

,

n

luTI
-e
n-,oo-00 71:

fJ¥

]

{

n(x - ~)2

---

4

1
.u(~,t--)d~
j
11

1
11 -nt2
-e

71: iu(x-2tl't--~)dt111

f~

.u(~,t--)d~=
n

-00

taco:

1

-nt2

iu(x-2t[,t---)dt]

1

Suyra:

-

-ex
f~~.171:
2-00

+OO

1

Apd~1l1gkStquacuab6d~3,l.l,
+oo

.11(~,to)d~

'

): 2

~

f

,u(~,to)d~=-

- n(x -.., )

n

l

d tidc:

2

+OO

-00

1 +oo

I

ex I

",:,,:
' B 01 tJlenso:t

~)2

suyra:

-002~71:(t-to)

1

]

(

f
ex - x{ 4(t-to)
-002~71:(t-to)

,v(~)d~=

1

f

1

+00

u(x,t)=

=u(x,t)

n

- (x Sex
--2 71:t-00
4t
+00

.;

{

~)2

]

.v(~)d~;

x E R,t > 0

(3.9)

Bay 1a ham phan b6 nhi~t dQ (j V1tri x t~i thai di€m t > 0 cua b?ti tmll1 nhi~t
gia trj dfiu,
Bay giG ta xet bai tmln nhi~t ngl(c,1cthai gian:

a2u au
ax-2 -

[ u(x,l)
Tt( cong th((c (3,9), thay t

at

'

= u(x),

X E R, t > 0

TIm vex) = lI(X,O)?

= 1, ta dl(QCphu'dng trtnh tich pilau

Fredholm Im~i 1

(fin v):
+00

):2

- 1

- ( x-C;)
2';; -!exp [ -- 4
] ,v(~)d~ = HeX),x E R

(3.10)


Chu'(jng 3;' MQ[ so' bili loan quy v~ phuong [dnh tfch ch(ip

E>~t K, (x) = 2}; exp(_x: ). Khi do phuong trlnh (3.10) duQc vie't dtfoi cI~lI1g

tich ch~p:
(Kl*v) (x) = U (x)

3.2 Bai tocm nhh~t trong 16 khoan thorn do
Ta xem 16 khoan tren m~t da't nhti 1a nlia thanh cl5n nhi~t dai vo Iwn va t~li
di0m khoan ling voi tQa dO x

= 0,

Ngudi ta mu6n xac dinh nhi~t dO t?i t?i m?t c1a't(ling voi x = 0) vao thai di~111
t nao do. Do nhi€u lac dOng eua moi tnidng hen ngoai qua (HItl11ake't qua thu du'Qc
d~fa tren tinh loan tr~e tie'p se co sai sO',VI the' ngtfdi ta xae dinh nhi~t dO d vi tri
x> 0 t?i thdi di0m t, san do SHYra gia tri c~n tIm.

Truoe lien, ta xet b~tiloan nhi~t gia tri bien:
Cho bie't u(O,t) = v(t), TIm phan b6 nhi~t u(x,t) t?i vi tri x >0 vao thai c1i~mt.
B~d loan khong ma't Hnh t6ng quat ne'u gia sli ding t?i mQi vi tri x >0 dell c6
nhi~t dO bfing 0 t?i thdi di0m t =0,

. B~titmln nhi~t gia tfi bien co phuong trlnh:
au
a2u
-,
ax2
at
u(x,O) = 0 ,
u(O, t) = v(t);

x> 0

t >0

(3.11)

x> 0
t > 0,

?

TIm u ( x, t) ?

au au 1"
1 , sox
a caCUlill

"
,
I
TtfOn g tti n 1lU (I b al

loan n hlet gm tn (1au, ta g m tllet
u, - ,'A"

.

'

;1;

'?

'X

.'

ax

bi ch~n, Ngoai fa, ta md rOng ham sO'u(x,t) bfing each b6 sung:

u(x,t) = 0, x < 0, t >0
Khi c16:ham u xac dinh tren R x R+
B6 c1~3.2.1
E>~t

r(x, t,~, T)=

1
2~1t(t - T)

exp

-

(x -

~)2

[ 4(t - T) ]

(-CX)<~<+CX),

0<1
at

I
!


Chuang 3: MQl sf)'hili loan quy vJ phuang lrinh tich ch(ip
Khi do ham s6: G(x, t,~, 't) = r(x, t,~, 't)

-

r(x, t,-~, T)

thoa phuong trlnh truy~n nhi<$t(3.11) va co dc tich chat:
a2G

aG

-

G(x,t,O,t)= 0; ~

=-

a~

at

C/1l/'ng m;nh:
1. D€ thay G(x, t,O,t) = r(x, i,D,t) - r(x, t,O,t) = °
a2G

2. Ta chung minh

aG
-

ax

=

a2G
;-

ax+

-

exp

[

T) { 2(t - T)

- (x

~

X-

(x - ~)2

]

4(t - T)

(x + ~)2

+ ~)2

(x + ~)2

-

+ exp

(x -

~)2

7

exp
exp
2(t- T) [ 4(t - T) ] 4(t - T)2 [ 4(t - T) ]}
1

- (x - ~)2

exp

ao
at

T)~

][

[

{

3
-

J

4\1n(t - T)2

-

exp
{

- (x - ~)2

[

4(t - T) ]

-

[

1]

exp

=
(x + ~)2

-

+exp

4(t - T) . 2(t - T)

- 1

=

(x - ~)2

X+ ~

- (x - ~)2
exp
+
[ 4(t - T) ]
4(t - T)-

+.

(x + ~)2

-

.

[ 4(t - t) ] 2(t - T)}

4(t - t) ] 2(t - t)

-I

2~n(t -

4); (t -

+

{

t) {

1

1

at

ex - (x - ~)2 .

-

=

=

-

ax2 -

1

2~n(t -

aG

4(t -

- (x + ~)2

[ 4(t - T) ]}

T)'

(x + ~)2

---

][1

2(

t - T)

]}

+

1

- (x - ~)2 (x - ~)2
- (x + ~)2 (x + ~)2
exp
- exp[ 4(t - T) ] . 2(t - T)2}

2~n(t - T) { [ 4(t - T) ] . 4(t - T)2

-

_(X-~)2

]

-

3

J

-

4\1n (t - T) 2 {

exp --

[

4(t - T)

a2o
~-=-

ax2

.

(X-~)2

][ 2(t -

T)

-(x+~)2
-

1]

+ exp --

(X+~)2
-.-

[ 4(t - T) ]l].

,

2(t - T) ] }

aG

at

V~y G thoa phuong trlnh truy~n nhi<$t(3.11).
3. Ta chang minh : a2G = - aG
a~2

at

22


Chuang 3: M()t stYbdi roan quy vi phuO'ng trinh rich ch~lp

aG -

1

_(X-~)2

x-~

_(X+~)2

a~ - 2-JTC(t-T) { 2(t-T) exp [ 4(t-T)

=

a 2G 2 a~

1
4FnCt - T)~

1

+

exp

1

aG

aJ

=~

f

2-vTC

-

.1

t 2( t -

T)

-

4(t-T)

][

J exp

1

[

J

4Fn(t-T)2

2(t-J)

exp

(X-~)2
]

]

4(t-T)

a2G

-(x+~

. +

]}
2

j}

4(t-T)

_(X+~)2

_(X+~)2
exp

[

][

- 4(t-T)

[

4(t--T)

j

]j
(X+~ )

+ exp

[

'4(t-T)

J[

Menh de' 3.2.2
B~ti loan nhi~t gia tri bi(~nco nghi~m dt(Qc xac djnh beji rang thleC:

-x2

vCr)

i

u(x,t)=-, -~exp
2 -Vre)

~

(t

- 1:)2

[

4( t - 1:)J

CluIng minh:

Voi (x,t) cOdjnh, x> 0, t >0, ta d~t:
1
[(X,t,~,T)=

_(X-~)2

2.J re(t- T) exp [ 4(t - T) ]

( - 00 <

~< +00,

0 < T < t)

(11:, x>O

,t>O

2
]

':>

, 2(t-T)

0

t

J

=

aT

x

+1

+

_(X+~)2

]

2(t-T)

2(t-T)

.

_(X+~)2

-

1

]

--(X-_~)2

expo
[ 4(t-T)
.s.

(X-~)2

-

J

':>

aG

=:>-=-a~2

)

4( t - T)l

4(t-T)2

][1

].

4(t-T)

4(t - T)

l

1

.

-

+--sexp,

_(X-~)2
[

-

1 +
]

(X+~)2

4(t-T)

{

exp

2(t-T)

_(X-~)2

_(X+~)2

1

-

[ 4(t-T)

~

(X+~)2

[

[

(x - ~)2

-

2(t - T)

(X-~)2

4(t-T)

exp

3

2(t-T)2
-

[

{

4Fn(t-T)%

]

-(X-~)2

exp

]

4(t - T)

_(x+~)2

[

{

4FnCt-T)%

=

[

{

(x - 02

+

}

-(x+~)2

+(x+~)exp

]

4(t-T)

- (x - ~) 2

- exp

1

[

] '2(t-T)

] +exp [ 4(t-T)

-(X-~)2

1
~. (x-~),exp
{
4FnCt-T)2

x+~

-

]}


Chuang 3: M6t seJbd(todn quy vi phu'ang trinh tfch ch(zp

Xet ham Green cho bai loan Direchlet:
G(x, t,~, 1) = r(x, t,~, T)- r(x, t,-~, 1)
Lffy U(~,1 ) la mOt nghi~m cua phuong trlnh (3,11)

(3.12)

CO'dinh (x,t), x> 0 , t> 0, Xet tntong vect(i F:
F(~,1) = u(~, 1)~G(X,

(

,

a~

t,~, 1) - G(x, t,~, 1)~

a~

u(~, 1); u(~, 1).G(X, t,~, T)

]

(0< ~<+CO,O< 1< t)
D~t Q = ~~,1) E R21 0 < ~ < 11;0 < 1 < to < t; 11EN}

Ki hi~u 30 la bien clla mien md Q. Ap d\l1lgdjnh ly divergence cho tHrong
vectd F tren Q, ta co:
--,

f divF(~, 1)d~d1 =,Xlf <

n

(3.13)

F(~, 1), n(~, 1) > ds

Nho b6 d€ 3,2.1 va (3.12) ta co:
,

a

aG

au

dlVF(~ 1)=u---G'.
a~ [ a~

a~ ]

a(uG)

+--

a1

= u a2G -G a2u + a(uGl
a~ 2
a~ 2
ch

=- u aG - G au + a(uG)
a1

a1

a1

=0

(3.14)

l'

(Hlnh 2)
->
->

~

x

Ky hi~u nhl! tren Hinh 2, ta co:

11

~


ChLCang3: M()[ sf)' bili [Dan quy v~ phLCang trinh tfch ch4p
->

-->

f
-->

f<F,n, >ds+ f<F,112 >ds

un

s,

s)

-->

f< F, n

+

3

-->

> ds +

S,

f< F,

114

(3.15)

> ds

S4

4

Tli (3.13), (3.14) va (3.15) ta suy fa:

-->

L f<F(~,T),l1i(~,T) > cIs =0

(3.16)

i=1 s;

-->

Tren 51: T = 0, 0 <

~< n,

111

= (0;

-1)

Vi u(~,O) =0 nen ta co:
f< F,l~ > ds =
s,

-

I u(~,O).G(x, t,~,O)d~ = 0

(3.17)

-->

0 < T < to,

Tren 52: ~ =11,

112

= (1; 0)

CG
a
f<F,n2 >ds= 1 u(I1,T)--(x,t,n,T)dT-1)G(x,t,n,T)-u(n,T)ch=
a;
a;
-->

r"

r"

S2

1
.= A I

" x-n

'+V7T

i

1 ex

)

.

-(x-n)2
:.

(t-T)2
1

--

r"

{

4(t-T)

-(x-n)2

1

2[;; 1)) -exl-~t-T

--

[

1
)cl
(
un, T T+ A I

.

]

'+V7T

" x+n

i
)

(t-T)2

aIr"

-(x+n)2
~

ex!

{ 4(t-T) ]
-(x+n)2

1

-_exl--

.
4(t-T) ] -u(n,T)ch-+---=))
a;

2J7T) ft-T

LIen,T)dT

f

4(t-T) ]

a

-u(n,T)clT

a;

5ti dl,ll1gket qua oil tlnh tren Ll trong chling ll1jnh m~nh c1~3.1.3, ta SHYfa:
-->

J<F,112>ds
5,

-~Okhjn-~oo

(3.18)

->

Tren 53: T = to, 0 <

~ < n , 113 = (0;1)

-->

f< F,n] > ds = - [u(~, to)G(x, t,~, to)d~ =

S,

= [u(~, to)r(x, t,~, to)d~ - [u(C; to)r(x, t,~ to)d~

Thea ket qUC!oil tlnh tren L4 trong chling minh ll1~nh c1~3.1.3, ta co:


_Chu'o'ng3: M()l so' bcii loan guy v~ pluiong tdnh tlch ch(ip
11

11

~

U( c"to)

f~

0fl1(~,to)l(x,t,~,to)d~=

-n2

-1-0')

-0')

4(t-to)

~ 2

J d~

.

2

(~ )
uc,t
-x-)
[- n,n ](~)d~
2 ~ n ( t' 0 toO) exp [ 4((t - t~ ) ] X

f

=

-(x-c,)

,exp [

net-to)

.---

0
JU(~,lo)r(X,

M~c khac:

I,-~, 10)d~ =

~

HeX,t)

-112
net-to)
J Jl(~,
10)

khi n

.exr [-

-

--)- 00

(x ~ ~)2 d~

4(t

to)

]

1

J:

-I-

_(X+~)2
d
f
]
[
= _Cf)2~n(t-to) exp 4(t-to)
X[-n.n](~)~~
O')

u(~,to)

-IO') u(~,to)

f

~

_(X+~)2

exp

[ 4(t - to) ]d

_CI)2~n(t- to)

IV

vat t
.

'

kl1J n -~

,:?,

n

2'

oo

oo

00

' K
c1
to =- 1, va c1 01 bJen
soK t l = -,X + ~ talidc:

-

+

u(~,to)

f
_002~n(t-to)

;_(X+~)2
J: - 1 +
exp -c,--exp
[ 4(t--to) ]d
2-00 n

f~

+oo

~ fe
11

=

~

c,

-n

-l1li

-00

-11(X+~)2

[

---

4

J

~
.U(c"t--

.

)d c,

n

1

u(2t,-x,t---)dt,
n

Apch;ll1g k€t qua ci'1ab6c1€3,l.I,tac1l(Qc:
+OO
,

IHn

I~-en

11-'DO
-DO

I

-1112

(VJ (-x) < 0)

'u(2t,-x,t--)dt,=u(-x,t)=0

7t

n

->

V~y :

(3.19)

I< F,n, > ds -+ u(x,t) khi n-+ co
s)

Tren 54:

~ = 0,

->

0 < 1" < to ,
I"

I<F,114 >ds=s,

-,
114=(-1;0)

au.

l"au(O,1") ,

fu(O,1")--(X,t,O,'t:)d1" +
0

a~

f

0

U(x,t,O,1")ch
a~

t

,

v I U (x, t, 0,1")d 1"= 0, ta soy
-

ra

I

" au(O,1") --,
0

I"

au

IU(O, 1")-(x,
aJ:

°

c,

I"

(j(x, t, 0 ,1")d 1"= 0

a~

--x2

xv(1")

t,O, 1")d1"= J-~--~exp
I

°2\i7t(t-1")2-

-

. (h
[ 4 ( t - 1") ]


Chuang 3: M()[ so' bai loan quy vi phuang [dnh tich ch4p
x

-

2Fn [

-

oo

-- X2

v ('t )

00

(t-'t)2

('t)d't

[ ]

- exp [ 4(t-'t) ] X o,t_.!..

11

Ap d~lllg dinh 19 hQi tv bi ch~n, ta c6:
x ro vCt)
11)
3 ex

2-..;n

)

-

(t-1)2
"

V <;lY

-x

X

2

{ --4(t-1)

->

]

d1~-

-

K€t hQp tli (3.17) o€n (3.20),

J

i

-

2 n)

v( 't)
3

(t - 't)2

{ 4(t

2

{ 4(t-'t)

- X2

3exl.

(t - 1f

x

:1 ex

~

v(1)

!
2-..;n )

$4

HeX,t) =

!

(t-1)2

X

I

J < F ,n4>cs~--1

1

2-..;n)

-x

vCt)

'

J

ch kIll n~Cf)

'

- 1) ]d

1:

khIn~Cf)

(3.20)

ta SHY ra:

-x 2
exp

d't, x > 0, t > 0
[ 4( t - 't) ]

(3.2l)

Day 1a ham phan b6 tinh nhi~t dQ d vi tri x >0 VaG thai c1i@mt clJa bai tmln
nhi~t gia tri bien.
Bay gia ta xet bai loan nhi~t trong 16 khoan tham do:

a2u
ax 2 --

au

at'

u(x,O)= 0 ;

x> 0, t> 0
x>O
t>0

u(1,t) = net);

Tli cong thltc (3.21), thay x

,TIm vet) = tI(O,t)?

= 1, ta Ol(\jCphl(Ong

trlnh rich phan Fredholm

lo<;li 1

\In v):
1
I

2-vn

f

I

°

-1

Vel:)
3

:.

exp

[ 4(t-l:)

(t-l:)2

r

,D;,ltK2(t)=

ch = net),t > 0

(3.22)

.

1
-1
--.exp(-);t>
~
4t

,

]
0

2t2Fn

t

0

va md rQng ham s6 v voi v (t)

;t sO

= 0 khi t s

O. Khi d6 phl(Ong trlnh (3.22) dl(QC viet

du'oi d<;lngrich ch~p:
(K2

*

v) (t)

= net)