Unformatted text preview: sum of integrals over the subintervals (z0 z1),
(z1 z2), : : : , (z 1 z ). For simplicity, we'll also assume that A = B = 0 and that u(x)
and v(x) satisfy these conditions. Using (9.1.1), we integrate (9.1.9) by parts to obtain
M; (v LU ) = M Z b ;v u ; (pv) u + qvu]dx +
0 a 0 0 X
M j 3 =1 fv(x)u (x) + p(x)v(x)u(x)g jj;1 :
0 z
z Integrating the rst term in the integrand by parts once more
(v LU ) = Z b v ; (pv) + qv]udx +
00 a 0 X
M j =1 fv(x)u (x) ; v (x)u(x) + p(x)v(x)u(x)g jj;1 :
0 z 0 z We can write this result in a simpler form by using the inner product notation (9.1.8)
and by de ning the jump in a function q(x) at a point z as q(x)] = = lim q(z + ) ; q(z ; ):
0
x z (9.1.10) ! With this notation, we have
(v Lu) = (L v u) ;
T X1 M; j =1 vu ; v u + pvu] j
0 0 z (9.1.11a) where L is called the adjoint operator and satis es
T L v = v ; (pv) + qv:
T 00 (9.1.11b) 0 Let us simplify matters somewhat by assuming that u 2 C 1(a b) and v 2 C 0 (a b).
Then, (9.1.11a) becomes X1 M; j =1 v u] j = (v Lu) ; (L v u):
T 0 z (9.1.11c) De nition...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, The Land, Collocation, Ly, collocation points

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