-

yj+ ½ x ½

ene0 ò(E-x(0,y) - Ex(x -½,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-½,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-½ 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 i1=0, i2=N2 (6)

yn+½ = m 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






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 <M

 

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

 


K0




K1

x




()

(IV)-(VI) OY.

 

 

 

 

 

 

 

 

 

1.  .., ..:

2.  .:

3.  ..:

4.  .., ..:

5.  .., ..:

6.  ..:


: 1, 2






           

2009 .