I. .
.
II. .
, , ,
(1)
, (2)
;
, ;
;
.
,
. (3)
(2)
. (4)
, , (4) .
, , (1) , .
, (1) - .
; ;
;
; ;
; .
.
.
, (1)
. (5)
.
, (4).
,
;
, (4) :
(6)
(7)
,
. (8)
(8) , (6) (7), . , (7) , (8) , . , , .
( ) , , (5), , . (3)
(9)
, , .
. , .
|
|
. =0, ..
.
.
, , .
.
, .
, (7)
.
, (), , , , .
,
.
. (10)
, .
, , (7), , , , ..
, . (11)
, . , .
(11), . ,
.
, , ,
. (12)
, , ; , (7)
, .
, (11).
.
, (12).
.
, , .
, .
. , . , . , , .
,
,
- , .
. (5) 15 , . .
, (3) (9).
III. Maple 6.0.
> restart;
a:=1:
k:=3:
N:=4:
p:=N+k:
Digits:=6:
For i from -2 to 2 do
u[i,0]:=0;
od;
For j from -6 to 0 do
|
|
u[3,j]:=-0.02*p*a^2;
u[-3,j]:=-0.02*p*a^2;
od;
> for i from -2 to 2 do
u[i,-6]:=-1/2*p*(i*a/6)^2+0.1*p*a^2;
od;
> c:={}:
> for i from -2 to 2 do
For j from -5 to -1 do
c:=c union {20*u[i,j]-8*(u[i-1,j]+u[i+1,j]+u[i,j-1]+u[i,j+1])+2*(u[i-1,j-1]+u[i+1,j-1]+u[i-1,j+1]+u[i+1,j+1])+u[i-2,j]+u[i+2,j]+u[i,j-2]+u[i,j+2]=0};
od;
od;
> for j from -6 to -0 do
u[4,j]:=u[2,j]-2/15*p*a^2;
u[-4,j]:=u[-2,j]-2/15*p*a^2;
od;
> for i from -2 to 2 do
u[i,1]:=u[i,-1];
u[i,-7]:=u[i,-5];
od;
> b:={}:
> for i from -2 to 2 do
For j from -5 to -1 do
b:=b union {u[i,j]};
Od
od;
> d:=solve(c,b);
> for i from -3 to 3 do
sigma_y[i,0]:=36/a^2*(u[i+1,0]-2*u[i,0]+u[i-1,0]):
sigma_y[i,-6]:=36/a^2*(u[i+1,-6]-2*u[i,-6]+u[i-1,-6]):
od;
> for j from -6 to 0 do
sigma_y[3,j]:=36/a^2*(u[4,j]-2*u[3,j]+u[2,j]);
sigma_y[-3,j]:=36/a^2*(u[-2,j]-2*u[-3,j]+u[-4,j]);
od;
> for i from -3 to 3 do
sigma_x[i,0]:=36/a^2*(u[i,1]-2*u[i,0]+u[i,-1]);
sigma_x[i,-6]:=36/a^2*(u[i,-5]-2*u[i,-6]+u[i,-7]);
od;
> for j from -6 to 0 do
sigma_x[3,j]:=36/a^2*(u[3,j+1]-2*u[3,j]+u[3,j-1]);
sigma_x[-3,j]:=36/a^2*(u[-3,j+1]-2*u[-3,j]+u[-3,j-1]);
od;
> for i from -3 to 3 do
sigma_xy[i,0]:=36/(4*a^2)*(u[i+1,1]-u[i-1,1]-u[i+1,-1]+u[i-1,-1]);
sigma_xy[i,-6]:=36/(4*a^2)*(u[i+1,-5]-u[i-1,-5]-u[i+1,-7]+u[i-1,-7]);
od;
> for j from -6 to 0 do
sigma_xy[3,j]:=36/(4*a^2)*(u[4,j+1]-u[2,j+1]-u[4,j-1]+u[2,j-1]);
sigma_xy[-3,j]:=36/(4*a^2)*(u[-2,j+1]-u[-3,j+1]-u[-2,j-1]+u[-4,j-1]);
od;