148 F. Beltrán-Carbajal et al. In order to eliminate the influence of the unknown constants ri,j, Eq. 14.7 are differentiated four times with respect to the complex variable s: 2n X kD0 .2n/Š .2n/Š kŠ.2n k/Š.2n k/Š s2n k d2n k ds2n k xi .s/ Ca2n 1 2n 1 X kD0 .2n/Š.2n 1/Š kŠ.2n k/Š.2n 1 k/Š s2n 1 k d 2n k ds2n k xi .s/ Ca2n 2 2n 2 X kD0 .2n/Š.2n 2/Š kŠ.2n k/Š.2n 2 k/Š s2n 2 k d2n k ds2n k xi .s/ C Ca1 1 X kD0 .2n/Š.1/Š kŠ.2n k/Š.1 k/Š s1 k d2n k ds2n k xi .s/ Ca0 d2n ds2n xi .s/ D0 (14.9) where d 0 ds 0 xi .s/ Dxi .s/. Next, to avoid differentiation with respect to time, Eq. 14.9 are multiplied bys 4 and transformed back to the time domain: 2n X kD0 . 1/2n k .2n/Š .2n/Š kŠ.2n k/Š.2n k/Š Z .k/ t0 . t/2n kxi .t/Ca2n 1 2n 1 X kD0 . 1/2n k .2n/Š.2n 1/Š kŠ.2n k/Š.2n 1 k/Š Z .1Ck/ t0 . t/2n kxi .t/ Ca2n 2 2n 2 X kD0 . 1/2n k .2n/Š.2n 2/Š kŠ.2n k/Š.2n 2 k/Š Z .2Ck/ t0 . t/2n kxi .t/ C Ca1 1 X kD0 . 1/2n k .2n/Š.1/Š kŠ.2n k/Š.1 k/Š Z .2n 1Ck/ t0 . t/2n kxi .t/ Ca0. 1/ 2nZ .2n/ t0 . t/2nxi .t/ D0 (14.10) where t Dt t0, and Z .N/ t0 .t/ are iterated integrals of the form Z t t0 Z 1 t0 : : : Z N 1 t0 . N/d N : : :d 1 with Z .1/ t0 .t/ D Z t t0 . / d , Z .0/ t0 .t/ D .t/ andNa positive integer. The integral-type equations (14.10), after some more integrations, leads to the following linear system of equations: A.t/™ DB.t/ (14.11) where ™ D a0 a1 an 1 an T denotes the parameter vector to be identified, A(t) and B(t) are n n and n 1 matrices, respectively, described by AD 2 6 6 6 6 6 4 a11 a1;2 : : : a1n 1 a1n a21 a22 : : : a2n 1 a2n : : : an 11 an 12 : : : an 1n 1 an 1n an1 an2 : : : ann 1 ann 3 7 7 7 7 7 5 ; BD 2 6 6 6 6 6 4 b1 b2 : : : bn 1 bn 3 7 7 7 7 7 5 with
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