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Let h = 1/ N be the mesh size, and let xi = ih (0 i N ) be the mesh points. For convenience, we introduce the ﬁnite difference operators δh2 and Ph as follows: δh2 u (xi ) = u (xi −1 ) − 2u (xi ) + u (xi +1 ), h Ph u (xi ) = 2 12 1 N − 1, i u (xi −1 ) + 10u (xi ) + u (xi +1 ) , 1 i N − 1. 7) Using the following Numerov’s formula (cf. 6) that −δh2 v (xi ) + Ph γ (xi ) v (xi ) = Ph g xi , u (xi ) + O h6 , v (x0 ) = T (α ), T u (xi ) = v (xi ), v (x N ) = T (β). 10) where u i and v i represent the approximations of u (xi ) and v (xi ), respectively, and γi = γ (xi ), g i (u i ) = g (xi , u i ).

1) and −δh2 v i = Ph f (xi , u i ), v 0 = T (α ), T (u i ) = v i , 1 i N − 1, v N = T (β). 2), we show the convergence of (u i , v i ) to (u (xi ), v (xi )) as h → 0. -M. 1. Assume that u i , u (xi ) ∈ Λ ⊂ I for each i and some interval Λ in R, and assume f u (x, ξ ) k(η) < π 2, 0 1, (ξ, η) ∈ Λ × Λ. 4) where c is a positive constant independent of h. Proof. Let e i = u (xi ) − u i and e i = v (xi ) − v i . 5) where ξi and ξi are two intermediate values between u (xi ) and u i . Deﬁne T E = (e 1 , e 2 , .

Comp. 11 (1957) 257–261. A. W. G. J. McRae, An eﬃcient method for parametric uncertainty analysis of numerical geophysical model, J. Geophys. Res. 102 (1997) 21925–21932. [18] A. Taﬂove, Computational Electrodynamics – The Finite-Difference Time-Domain Method, Aztech House, Boston, 1995. C. N. Trefethen, The kink phenomenon in Fejer and Clenshaw–Curtis quadrature, Report no. 6/16, Oxford University Computing Laboratory, 2006. [20] N. Wiener, The homogeneous chaos, Amer. J. Math. 60 (1938) 897–936.

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