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Tham khảo tài liệu 'burden - numerical analysis 5e (pws, 1993) episode 3 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 11.3 Finite-Difference Methods for Linear Pro blems 595 1 z ỉ2 11-17 xf T-n y xi i - y xf-i --py Xpil 2h 6 for some 17 in Xj_ 15 Xf ị . The use of these centered-difference formulas in Eq. 11.15 results in the equation v x 1 - 2y xf y xz_i _ Fyfe1 - X ------ ------72----------- p M ----------- ------------- ộOh y í- h _ 2h h2 r xf - p y4J A Finite-Difference method with truncation error of order O fr results by using this equation together with the boundary conditions y ứ aandyri f3 to define w0 a wN l ff and 2w - W.-J. 1 1 . W . 1 W _. 1 . ----------7 ----- p xi -r i h 2h for each i 1 2 . A. In the form we will consider Eq. 11.18 is rewritten as - 1 2 2 h2 xi wi - wi -h2r xff and the resulting system of equations is expressed in the tridiagonal A X A-matrix form 11.19 Aw b where 2 ĩ2ạ Xí 1 A -1 -2 2 0 -1 0 .0 2 i2ợ x2 -1 2P x2 . . . 1 2 P XN .0 -1 - 2 p fN 2 W1 w2 W-1 -W v and -h2ffx 11 I w0 ỉ2r x2 -A2r x _i f h -hffxff I 1 - -p xff i 594 CHAPTER 11 Boundary-Vahie Problemsfor Ordinary Differential Equations The following theorem gives conditions under which the tridiagonal linear system 11.19 has a unique solution. Its proof is a consequence of Theorem 6.27 and is considered in Exercise 9. Theorem 11.3 Suppose thatp q and r are continuous on a b . If q x 0 on a b then the tridiagonal linear system 11.19 has a unique solution provided that h 2 L where L maxaSjcSfc p x . H E It should be noted that the hypotheses of Theorem 11.3 guarantee a unique solution to the boundary-value problem 11.14 but they do not guarantee that y c4 a z We need to establish that y4 is continuous on a b to ensure that the truncation error has order O h2y Algorithm 11.3 implements the Linear Finite-Difference method. ALGORITHM Linear Finite-Difference To approximate the solution of the boundary-value problem y p x yr q x y r x a X b y a a y b 3 INPUT endpoints a b boundary conditions a 3 integer 1. OUTPUT approximations IV to y Xf for each i 0 1 . N 1. Step 1 Set h b a N -I- 1 X a h ữỵ 2 .