II. FINITE DIFFERENCE METHOD 1 Difference formulae

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The numerical computation can be performed by a program of the following type (the language and environment R can be taken from CRAN, or www.Rproject.org) programma
y 5 10 15 20 25 y’’+y=x^2, y(−1)=3, y(1)=5 −1.0 −0.5 0.0 0.5 1.0 Figure 2: The difference method applied to a problem y ′′ (x) + y(x) = x 2 , with Dirichlet conditions y(a) = α, y(b) = β. EX. <strong>II</strong>.2B - Let us consider the Dirichlet problem y ′′ (x) + x 2 y(x) = sin(x) y(¡4π) = 0, y(4π) = 0 It is solved by a difference scheme, building an R function with output in Figure 4. As the above table suggests, the following theorem shows that, if the solution is sufficiently regular, the truncation error of the solution is O(h 2 ). Thm. <strong>II</strong>.2.2 (Convergence) Let y(x) be the solution of the problem with Dirichlet conditions ¡y ′′ (x) + q(x)y(x) = f(x), y(a) = α, y(b) = β with q(x) and f(x) continuous functions on [a, b] and q(x) ¸ 0. Assume, moreover, that the function y(x) has bounded fourth derivative, jy (4) (x)j · M for any x 2 [a, b]. If the approximation of y(xi) obtained by the difference method is denoted by ¯yi, then for some C independent of h, jy(xi) ¡ ¯yij · Ch 2 , i = 1, 2, ..., n ¡ 1 9 x
Page 1 and 2: Computational Methods and advanced
Page 3 and 4: Let us apply Taylor’s formula. 1
Page 5 and 6: errore(h) errore totale h errore di
Page 7: By hypothesis (2), A satisfies: z T
Page 11 and 12: i.e. the method satisfies a converg
Page 13 and 14: the world: ”All things were creat
Page 15 and 16: This scheme, which is known as upwi
Page 17 and 18: iterabile
Page 19 and 20: u t Figure 6: The explicit method (
Page 21 and 22: is transformed into the difference
Page 23 and 24: t(y) trasposta del vettore colonna
Page 25 and 26: sinusoide source("sinusoide.txt") >
Page 27 and 28: qchisq(0.99,df=10) quantile 0.99 di
Page 29 and 30: ___________________________________

II. FINITE DIFFERENCE METHOD 1 Difference formulae

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