. , , ,

,,,

. —

( ). .

. . , , .

. , (), . , . , , , ( ). . .

.

:

2

U = f

G={ (x,y) : 0<=x<=a, 0<=y<=b }. G .

U = 0 Y

x=0 b

Uxxx = 0

x=0

G

Ux = 0

x=a

Uxxx = 0 0 a X

x=a

U = 0 U = 0

y=0 y=b

Uy = 0 Uxx + Uyy = 0

y=0 y=b y=b

.

.

G Wx Wy hx hy .

Wx={ x(i)=ihx, i=0,1...N, hxN=a }

Wy={ y(j)=jhy, j=0,1...M, hyM=b }

Uij=(x(i),y(j)) (i),y(j) G :


W={ Uij=(ihx,jhy), i=0,1...N, j=0,1...M, hxN=a, hyM=b }


W x=x(i) y=y(j).

W. W . . 0 A U f=AU . .

, . W - h R ..

W={Xi=a+ih, i=0, + 1, + 2...}

Yi=Y(Xi) , Xi W, :

L1Yi = Yi - Yi-1 , L2Yi=L1Yi+1

h

. :

L3Yi=Yi+1 - Yi-1 = (L1+L2)Yi

2h 2

A1, A2, A3 2 Lu=u . n- , n-1 , :

Yxxi=Yxi+1 - Yxi = Yi-1-2Yi+Yi+1

2

h h

Yxxi= Yxi+1-Yxi-1 = Yi-2 - 2Yi+Yi+ 2

2

2h 4h

. 3 .

.

. .

Ġ

.

:

AU = f

:

M

aijUj = fi , i=1,2...M

i=1

=(aij) (aii<>0) :

i (k+1) M (k)

aijYj + aijYj = fi , i=1,2...M

j=1 j=i+1

(k)

Yj - j k. .

(k+1)- i=1

(k+1) M (k)

a11Y1 = - a1jYj +f1

j=2

(k+1)

a11<>0 Y1. i=2 :

(k+1) (k+1) M (k)

a22Y2 = - a21Y1 - a2jYj + f2

j=3

(k+1) (k+1) (k+1) (k+1)

Y1 , Y2 ... Yi-1 . Yi :

(k+1) i-1 (k+1) M (k)

aiiYi = - aijYj - aijYj + fi (*)

j=1 j=i+1

(*) , . (*) Yi Yi.

, . aij , (*) M-1 .

2

2M - M .

m , , 2Mm-M .. M.

. A , :

A = D + L + U


0 0 . . . 0 0 a12 a13 . . . a1M

a21 0 0 0 a23 . . . a2M

a31 a32 0 0 .

L = . U= .

. .

. aM-1M

aM1 aM2 . . . aMM-1 0 0 0

D - .

(k) (k) (k)

Yk = ( Y1 ,Y2 ... YM ) k- . :

( D + L )Yk+1 + UYk = f , k=0,1...

:

( D + L )(Yk+1 - Yk) +AYk = f , k=0,1...

, aii - , , , aij i<>j - . Yi fi , aii.

Š ՠ

Yi=Y(i) i. Y(i) . , Y(i) - . .

.

:

dU = f(x) , x > 0

dx

:

Yi+1 - Yi = f(xi) , xi = ih, i=0,1...

h

Yi+1=Yi+hf(x), h - v={xi=ih, i=0,1,2...}. Yi=Y(i).

2

d U = f(x)

2

dx

:

2

Yi+1 - 2Yi + Yi+1 = yi , yi=h f i

fi = f(xi)

xi = ih

a U, U, U . .

Uij = U(i,j) .

Uxx + Uyy = f(x,y)

W :

Ui-1j - 2Uij+Ui+1j + Uij-1 - 2Uij+Uij+1 = fij

2 2

hx hy

hx - X

hy - Y

:

N

CijUj = fi i=0,1...N

j=0

U0, U1 ... UN . N .

i - , .. i = (i1 ... ip) :

ijUj =fi i Î W

jÎW

W. ij i, .

.. .

U=U(x,y)

y

M b

M-1

Uij j

j

1

0 1 2 i N-1 N=a x

i

G W . W Uij=U(xi,yj) ,

xi=x0+ihx

yi=y0+jhy

hx = a/N ,

hy = b/M ..

x0=y0

xi=ihx, yi=jhy, i=0...N

j=0...M

2

DU = f

(. ).

Uxij = Ui+1j - Uij , Uxi-1j = Uij - Ui-1j

hx hx

Uxxij = Ui-1j - 2Uij + Ui+1j

hx

Uxxxxij :

Uxxi-1j - Uxxij - Uxxij - Uxxi+1j

Uxxxxij = hx hx = Ui-2j - 4Ui-1j + 6Uij - 4Ui+1j + Ui+2j

4

hx hx

y :

Uyyyyij = Uij-2 - 4Uij-1 + 6Uij - 4Uij+1 +Uij+2

4

hy

Uxxyy :

Uxxij-1 - Uxxij - Uxxij - Uxxij+1

(Uxx)yyij = hy hy = Uxxij-1 - 2Uxxij +Uxxij+1 =

2

hy hy

= Ui-1j-1 - 2Uij-1 + Ui+1j-1 - 2 Ui-1j - 2Uij + Ui+1j + Ui-1j-1 - 2Uij+1 + Ui+1j+1

2 2 2 2 2 2

hxhy hxhy hxhy

DU = f

:

Ui-2j - 4Ui-1j + 6Uij - 4Ui+1j +Ui+2j +

4

hx

+ 2 Ui-1j-1 - 2Uij-1 + Ui+1j-1 - 4 Ui-1j - 2Uij +Ui+1j + 2 Ui-1j+1 -2Uij+1 + Ui+1j+1 +

2 2 2 2 2 2

hxhy hxhy hxhy

+ Uij-2 - 4Uij-1 + 6Uij - 4Uij+1 + Uij+2 = fij (*)

4

hy

i=1,2, ... N-1

j=1,2, ... M-1

. :

x=0 ~ i = 0

x=a ~ xN=a

y=0 ~ Yo=0

y=b ~ YM=b


1) =0 ( G)

U = 0

x=o

Uxxx = 0

x=o

Uoj=0

U-1j=U2j - 3U1j (1`)

2) =ࠠ ( G)

i=N

Ux = 0

x=a

Uxxx = 0

x=a  Ui+1j - Ui-1j = 0

2hx

UN+1j = UN-1j

UNj = 4 UN-1j - UN-2j (2`)

3

3) =0 ( G)

j=0

Ui ,-1 = Ui1

Ui0 = 0 (3`)

Uy = 0

y=o

U =0

y=o

4) =b

i=M

U = 0

y=b .. UiM=0 (**)

Uxx + Uyy =0 j=M (**)

UiM-1 = UiM+1

=b

UiM+1 = UiM-1

UiM = 0 (4`)

(*) W (1`)-(4`) G ( W)

(*),(1`) - (4`).

Uij = U(xi,yj)

xi = ihx

yj = jhy

hx = a/N ,

hy = b/M

x , N [0 , ] [0 , b]

2

DU = f

. W , .


1 Ui-2j - 4 + 4 Ui-1j + 6 - 8 + 6 Uij - 4 + 4 Ui+1j + 1 Ui+2j + 2Ui-1j-1 -

4 4 2 2 4 2 2 4 4 2 2 4 2 2

hx hx hxhy hx hxhy hy hx hxhy hx hxhy


- 4 + 4 Uij-1 + 2 Ui+1j-1 + 2 Ui-1j+1 - 4 + 4 Uij+1 + 2 Ui+1j+1 + 1 Uij-2 +

2 2 4 2 2 2 2 2 2 4 2 2 4

hxhy hy hxhy hxhy hxhy hy hxhy hy

+ 1 Uij+2 = f ij i=1 ... N-1, j=1 ... M-1

4

hy

U (1`) - (4`), 13 13- . :

(k+1) (k+1) (k+1) (k+1)

6 - 8 + 6 Uij = - 1 Uij-2 - 2 Ui-1j-1 + 4 + 4 Uij-1 -

4 2 2 4 4 2 2 2 2 4

hx hxhy hy hy hxhy hxhy hy

(k+1) (k+1) (k+1) (k)

- 2 Ui+1j-1 - 1 Ui-1j + 4 + 4 Ui-1j + 4 + 4 Ui+1j -

2 2 4 4 2 2 4 2 2

hxhy hx hx hxhy hx hxhy


(k) (k) (k) (k) (k)

- 1 Ui+2j - 2 Ui-1j+1 + 4 + 4 Uij+1 - 2 Ui+1j+1 - 1 Uij+2 + fij

4 2 2 2 2 4 2 2 4

hx hxhy hxhy hy hxhy hy

(k)

U (1`) - (4`). i=1, j=1 W. n = (N-1)(M-1).

.. 13 ( ) - .

j+2

j+1

j

j-1

: .

j-2

i-1

i

i+1

i+2

i-2

.

:

aq = 1 - G

b = 1 - G

N = 8 - [0,a]

M = 8 - [0,b]

h1 = aq/N - X

h2 = b/M - Y

:

u0 - U k-

u1 - U (k+1)-

a -

:

procedure Prt(u:masa) -

function ff(x1,x2: real):real - f (x1,x2)

procedure Koef -

:

u0 i=2 ... N , j=2 ... M. u1 u0. eps>0 . A, (k+1) , ( ) k.

: Borland Pascal 7.0

: 4 5

..

:

..

1997.

( ).

 

 

 

! , , , .
. , :