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: 1/4

 

( ). .

. . , , .

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

.

 

 

:

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