BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
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98 Iulian Agape et al<br />
// l<br />
i−l l−<br />
li−<br />
1<br />
f () 1 = mi−<br />
1<br />
+ mi<br />
, i = 1,<br />
n ,<br />
L L<br />
n n<br />
where: L/n – the network intervals length; m i – the second derivate of function<br />
f(l) values, in the network nodes<br />
m<br />
( )<br />
//<br />
i f l i<br />
(4)<br />
= i = 0,<br />
n , (5)<br />
Integrating twice the Eq. (4) it will be obtained<br />
( ) ( )<br />
3 3<br />
li −l l−li−<br />
1 li −l l−li−<br />
1<br />
i−1<br />
i i i<br />
f () l = m + m + E + F , i=<br />
1, n<br />
L L L L<br />
6 6<br />
n n n n<br />
,<br />
(6)<br />
where: E i , F i are integration constants, determined from<br />
f(l i-1 ) = ξ i ; f(l i ) = ξ i+1 , i = 1,<br />
n . (7)<br />
Customizing the Eq. (6) for l = l i and l = l i-1 will obtain:<br />
2<br />
⎧mi<br />
⎛ L ⎞ + Fi<br />
=<br />
⎪ ⎜ ⎟<br />
⎪ 6 ⎝ n ⎠<br />
⎨<br />
6<br />
,<br />
i+<br />
1<br />
2<br />
⎪mi<br />
−1<br />
⎛ L ⎞ + Ei<br />
= ξi<br />
⎪<br />
⎩<br />
⎜ ⎟<br />
⎝ n ⎠<br />
ξ<br />
i = 1,<br />
n ,<br />
(8)<br />
and, replacing the constants F i , E i in (6)<br />
( ) ( )<br />
3 3 2<br />
li −l l−l ⎡<br />
i−1 mi−<br />
1 ⎛ L ⎞ ⎤li<br />
− l<br />
f()<br />
l = mi−<br />
1<br />
+ mi + ⎢ξi<br />
−<br />
L L ⎜ ⎟ ⎥ +<br />
6 6 ⎢⎣<br />
6 ⎝ n ⎠ ⎥ L<br />
⎦<br />
n n<br />
n<br />
2<br />
⎡ mi<br />
⎛ L ⎞ ⎤l−<br />
l<br />
+ ⎢ξi+<br />
1<br />
− ⎜ ⎟ ⎥<br />
⎢ 6 ⎝ n ⎠ L<br />
⎣<br />
⎥⎦<br />
n<br />
i−1<br />
The first derivative of the function is<br />
()<br />
f l<br />
, i = 1,<br />
n .<br />
m ( l −l) m ( l−l ) ξ −ξ ( m −m 1)<br />
L<br />
=− + + −<br />
2 L 2 L L 6 n<br />
n n n<br />
2 2<br />
i−1 i i i− 1 i+<br />
1 i i i− ,<br />
(9)<br />
(10)<br />
i = 1,<br />
n .<br />
It allows to determine the lateral limits for the derivative in the network<br />
nodes, for the arguments l 1 , l 2 , …, l n-1 ,