BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

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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 ,
Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 99 / mi L ξi+ 1 − ξi mi − mi− 1 L limsup l→l f ( l) = + − , i 2 n L 6 n n (11) / mi L ξi+ 2 − ξi+ 1 mi+ 1 − mi L liminf l→l f ( l) =− + − . i 2 n L 6 n n From the first derivative f / (l) continuity condition in the network nodes, (n-1) equations will be obtained L L n 2 L n ξi+ 2 −ξi+ 1 ξi+ 1 −ξi mi− 1 + mi + mi+ 1 = − , i= 1, n . 6 3 n 6 L L (12) n n From the limit condition upon the second derivative (condition 4) it will be obtained m 0 = m n = 0 (13) equation that gives a natural cubic interpolative character to the function f(l). From the Eq. (12) and (13), we obtain a system of linear equation with the unknown m 1 , m 2 , …,m n Am = Hξ , (14) where: the square matrix A (n-1 rows and n-1 columns) and the vectors m and ξ take the forms ⎛2L L ⎞ ⎜ 0 ... 0 0 3n 6n ⎟ ⎜ ⎟ ⎜ L 2L L ⎟ ⎜ ... 0 0 ⎟ ⎛m1 ⎞ ⎜6n 3n 6n ⎟ ⎜m ⎟ 2 A = ⎜ L 2L ⎟; m = ⎜ ⎟; ⎜0 ... 0 0 ⎟ ... m3 ⎜ 6n 3n ⎟ ⎜m ⎟ ⎝ n − 1 ⎠ ⎜..................................................... ⎟ ⎜ ⎟ L 2L (15) ⎜0 0 0 ... ⎟ ⎝ 6n 3n ⎠ ξ ⎛ξ1 ⎜ ξ ⎜ 2 ⎜... ξ ⎜ξn ⎜ ⎝ = 3 ⎜ ⎟ ξ n + 1 ⎞ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠
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BULETINUL INSTITUTULUI POLITEHNIC D
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PETRU CÂRLESCU, VASILE DOBRE, IOAN
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