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,,,

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: 4 5

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:

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1998.

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- - - - - - - - - - - - - - - - - - - 3

ࠠ - - - - - - - - - - - - - - - - - - - - - - 5

䠠 - - - - - - - - - - - - - - - - - - - - - - 8

- - - - - - - - - - - - - - - - - - - - - - 10

頠 - - - - - - - - - - - - - - - - - - - - - - 13

젠 - - - - - - - - - - - - - - - - - - - - - - 16

Š - - - - - - - - - - - - - - - - - - -

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-

j(x,y) - , . (DEF) :

d2j + d2j = 0

dx2 dy2

( ABGH) - :

d2j + d2j = 0

dx2 dy2

q - e;

enn - ;

Nd(x,y) - ;

Na(x,y) - ;

e0 -


0 D E

y

B G

C F

A H

x

.1.


:

j| BC = Uu

j| DE = U

j| FG = Uc

j| AH = Un

AB GH:

dj = 0 dj = 0

dy AB dy GH

OY

:

dj = 0 dj = 0

dy DC dy EF

-

:

j| -0 = j| +0

eok Ex |-0 - enn Ex |+0 = - Qss

Qss - ;

eok - ;

enn - .

+0 -0 CF . - .

  {(x,y) : 0 < x < Lx , 0 < y < Ly }

W={(x,y) : 0 < i < M1 , 0 < j < M2}

x0 =0 , y0=0, xM1 = Lx , yM2 = Ly

xi+1 = xi + hi+1 , yj+1 = yj+ rj+1

i = 0,...,M1-1 j = 0,...,M2-1

.2.

yj yj+1/2 yj+1

xi-1

hi

xi- 1/2

xi

xi+1/2

hi+1

xi+1

yj-1 yj-1/2

rj


:

xi+ ½ = xi + hi+1 , i = 0,1,...,M1-1

2

yj+ ½ = yj + rj+1 , j = 0,1,...,M2-1

2

:

U(xi,yj) = Uij

I(xi+½,yj) = Ii+½,j

I(xi,yj+½) = Ii,j+½

:

Dj = - q (Nd + Na)

e0en

Q(x,y)

:

Vij = { (x,y) : xi- ½ < x < xi+ ½ , yj- ½ < y < yj+ ½ }

xi+ ½ yj+ ½ xi+ ½ yj+ ½

ò ò Dj dxdy = ò ò Q(x,y)dxdy

xi- ½ yj- ½ xi- ½ yj- ½

:

yj+½ xi+½

ò(Ex(xi+½,y) - Ex(xi-½,y) )dx + ò(Ey(x,yj+½) - Ey(x,yj-½))dy=

yj-½ xi-½

xi+ ½ yj+ ½

= ò ò Q(x,y)dxdy

xi-    ½ yj- ½

:

Ex(x,y) = - dj(x,y)

dx (*)

Ey(x,y) = - dj(x,y)

dy

x - .


yj-½ < y < yj- ½ Ex(xi + ½,yj) = Ei+ ½ ,j = const

yj-½ < y < yj- ½ Ex(xi - ½ ,yj) = Ei- ½ ,j = const (**)

xi-½ < x < xi+ ½ Ey(xi, yj + ½) = Ei,j+ ½ = const

xi-½ < x < xi+ ½ Ey(xi, yj ) = Ei,j - ½ = const

xi- ½ < x < xi+ ½

yj- ½ < y < yj+ ½ - Q(x,y) = Qij = const


(Ex)i+ ½ ,j - (Ex)i -½ ,j r*j + (Ey)ij+ ½ - (Ey)ij- ½ h*i = Qijh*i r*j

h*i = hi - hi+1 , r*j = rj - rj+1

2 2

i+ ½ ,j j(x,y) :

xi+1

òEx(x,yj)dx = - ji+1,j - jij

xi

(**) y=yj:

(Ex)i+ ½ ,j = - ji+1j - jij

hi+1

:

(Ey)i,j+ ½= - jij+1 - jij

rj+1

:


(Dj)ij = 1 j i+1,j - j ij - j i j - j i-1,j + 1 j i j+1 - j ij - j ij - j ij-1 =

h*i hi+1 hi r*j rj+1 rj

= Ndij + Naij

.3.

yj+1/2 yj+1

x-1

h-1

x-1/2

x1/2

h1

x1

yj-1 yj-1/2

rj

rj+1


SiO2

e1

Si y

en

x

V0j

yj+ ½ x ½

ene0 ò(Ex(x ½ ,y) - E+x(0,y))dy + ene0 ò (Ey(x,yj+ ½) - Ey(x,j- ½ ))dx =

yj- ½ 0

x ½ yj+½

= q ò ò (Nd + Na)dxdy

0 yj-½

V`0j

yj+ ½ x ½

ene0 ò(E-x(0,y) - Ex(x ,y))dy + ene0 ò (Ey(x,yj+½) - Ey(x,j-½))dx = 0

yj- ½ 0

E+x(0,y) E-x(0,y) -

.

:

ene0 dj + - e1e0 dj - = -Qss

dx dx

yj+½ x½

ò (ene0Ex(x½,y) - e1e0Ex(x,y) - Qss(y))dy + ene0ò (Ey(x,yj+½) + Ey(x,yj-½))dx +

yj-½ 0

0 x½ yj+½

+ e1e0 ò (Ey(x,yj+½) - Ey(x,yj-½))dx = q ò ò (Nd + Na)dxdy

x 0 yj-½

Ex Ey (**) Qss(y) = Qss = const yj-½ < y < yj+½ :

j+ = j- dj + = dj -

dy dy

+-

- -

:


ene0(Ex)½,j - e1e0(Ex)-½,j - Qss r*j + ene0h1 + e1e0h-1 . (Ey)0,j+½ - (Ey)0,j-½ =

2 2

= q (Nd0j - Na0j) h1r*j

2

:


1 ene0 jij -j0j - e1e0 j0j - jij + ene0h1 + e1e0h-1 j0,j+1 - j0j - j0j - j0,j-1 =

h* h1 h-1 2h*r*j rj+1 rj

= - q ( Nd0j - Na0j ) . h1 - Qss

2 h* h*

h* = h1 + h-1

2

, , , , .

:

LxxUmn + LyyUmn = j(xm,yn) (1)

Umn| = Y(smn) m,n = 1,2,...,M-1

:

 

d2U + d2U = j(x,y) 0<= x <=1

dx2 dy2 (2)

U| = Y(s) 0<= y <=1

(1) .

(1) , , , - .

U(x,y) (2) (x,y) , . j(x,y) Y(s) .

:

dV = d2V + d2V - j(x,y)

dt dx2 dy2

V| = Y(s) (3)

V(x,y,0) = Y0(x,y)

j Y (2), Y0(x,y) - .

j(x,y) Y(s) , , V(x,y,t) , V(x,y,t) t àOO U(x,y), (2). (2) (3) t, . .

(2) (3), (1) (2) (3).

, :

Up+1mn - Upmn = LxxUpmn + LyyUpmn - j(xm,yn)

t

Up+1mn| = Y(smn) (4)

U0mn = Y0xm,yn)

:

Up+1mn - Upmn = LxxUp+1mn + LyyUp+1mn - j(xm,yn)

t

Up+1mn| = Y(smn) (5)

U0mn = Y0(xm,yn)

Umn - Upmn = 1 [ LxxUmn + LyyUpmn - j(xm,yn)]

t 2

Up+1mn - Umn = 1 [ LxxUmn + LyyUp+1mn - j(xm,yn)]

t 2 (6)

Up+1mn| = Umn| = Y(smn)

U0mn = Y0(xm,yn)

, Y0(xm,yn) Up={Upmn} (4) .

Up+1 = {Up+1mn} (5) :

LxxUp+1mn + LyyUp+1mn - Up+1mn = j(xm,yn) - Upmn

t t (7)

Up+1mn| = Y(smn)

Up+1 = {Up+1mn} Up = {Upmn} (6) OX {Umn} n, OY {Up+1mn} m.

(4) (6) :

epmn = Upmn - Umn

Up = {Upmn} U = {Umn} (1).

{Umn} (1) :

Upmn - Umn = LxxUmn - j(xm,yn)

t

Umn| = Y(smn)

U0mn = Umn

(4) , epmn :

ep+1mn - epmn = Lxxepmn + Lyyepmn

t

ep+1mn| = 0 (9)

e0mn = Y0(xm,yn) - Umn

epmn p (p=0,1,...) .

:

dU = LU + f(x,t) , xÎG02 , tÎ[0,t0]

dt

U| = m(x,t) (1)

U(x,0) = U0(x)


LU = LU = (L1 +L2)U , LaU = d2U , a=1,2

dx2

G0a =G0 = {0<= xa <=la , a=1,2} - l1 l2, - G0 = G0 + .

G0 xa vh h1 = l1/N1 , h2 = l2/N2. nh - wh, , , vh = wh + nh.

La La:

Lay = yxaxa , L = L1 + L2

:

Aiyi-1 - Ciyi + Biyi+1 = -F , i=1,...,N-1

y0=m1 (2)

yn=mN

Ai > 0, Bi > 0, Ci > Ai + Bi

.

. vh , i2=0,1,2,...,N2, i1=1,2,...,N1. N1+1 N2+1 . N1+1, N2+1 - .

( ) (2) i2( i1), ( ), .. , (N1N2) . (2) .

y(x,t), .. y = yn y` = yn+1 y = yn+½ , t = tn+½ = tn+½ . n n+1 0.5t .

yn+½ - yn = L1yn+½ + L2yn + jn (3)

0.5t

yn+1 - yn+½ = L1yn+½ + L2yn+1 + jn (4)

0.5t

x = xi vh t=th > 0.

1 2, 1 2. (3),(4) :


y(x,0) = U0(x) , xÎvh (5)

, , :

yn+1 = mn+1 蠠 i1=0, i2=N2 (6)

yn+½ = m 蠠 i1=0, i2=N1 (7)

m = 1 (mn+1 + mn) - t L2(mn+1 - mn) (8)

2 4

.. , (3)-(8) (1). . (3) (4) :


2 y - L1 y = F , F = 2 y + L2 y + j

t      t (9)


2y` - L2 y` = F , F = 2 y + L1 y + j

t      t

:

xi = (i1h1 , i2h2)

F = Fi1,i2

y = yi1,i2

, , . (9) (2), ..:

 

 


1 yi1-1 - 2 1 + 1 yi1 + 1 yi1+1 = - Fi1

h21 h21 t h21

i1 = 1,...,N1-1 (10)

y =m i1 = 0,N1


1 y`i2-1 - 2 1 + 1 y`i2 + 1 y`i2+1 = - Fi2

h22 h22 t h22

i2 = 1,...,N2-1 (11)

y` = m` i2 = 0,N2

=n. òF, i2=1,...,N2-1 (10) y wh, F (11) i1=1,...,N1-1, y`=yn+1. n+1 n+2 , .. .

- , .

- :

L M N

y

K0

K1

x

II

I

III

.4


I : jk0,y = Un

t . jk+½i-1,y + 1 + t + t . jk+½ij - t . jk+½i+1y = Yij

2h*ihi 2h*ihi+1 2h*i2hi 2h*ihi+1

jk1,y = Un

 

Yij = jkij + t (Lyjkij + f kij )

2

Ly = 1 jkij+1 - jkij - jkij - jkij-1

r*j rj+1 rj


II: jij=U3

t . jk+½i-1,j + 1 + t + t . jk+½ ij - t jk+½i+1,j =

2h*ihi 2h*ihi+1 2h*ihi 2h*ihi+1

= jkij + t Lyjkij

2 , 0 < i < k0-1 L< j

 

eok . jk+½ i-1,j + - enn - eok . jk+½ ij + en . jk+½ i+1,j = Y*ij , i=k0

h*i-1 h*hi h*hi-1 h*ihi

t . jk+½i-1,j + 1 + t + t . jk+½ ij - t . jk+½i+1,j =

2h*ihi 2h*ihi 2h*ihi 2h*ihi+1

= jkij + t Lyjkij - f kij ,k0+1< i < k1

2

jk1,j = Un

...

III : jk0,j =Uc

t . jk+½i-1,j + 1 + t + t . jk+½ ij - t jk+½i+1,j =

2h*ihi 2h*ihi+1 2h*ihi 2h*ihi+1

= jkij + t Ly (jkij - f kij ), M+1 < j < N

2

jk1,j = Un

(I)-(III) OX.

y

V


.55

IV

VI

L M N

K0

K1

x

()

(IV)-(VI) OY.

1.  .., ..:

2.  .:

3.  ..:

4.  .., ..:

5.  .., ..:

6.  ..:

-

 

 

 

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