BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

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