HW13suggestions - < x < 1 and 0...

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Jacobs University, Bremen School of Engineering and Science Prof. Dr. Lars Linsen, Orif Ibrogimov Spring Term 2010 Homework 13 120202: ESM4A - Numerical Methods Homework Problems 13.1. Consider Poisson’s equation 2 u ∂x 2 ( x, y ) + 2 u ∂y 2 ( x, y ) = xe y , for 0 < x < 2 and 0 < y < 1 with the boundary conditions u (0 , y ) = 0 , u (2 , y ) = 2 e y , for 0 y 1 u ( x, 0) = x, u ( x, 1) = e x , for 0 x 2. Use the Finite-Difference method to approximate the exact solution u ( x, y ) = xe y with n = 6 and m = 5. 13.2. Use the Crank-Nicolson method to approximate the solution of the following problem with the choices m = 10, h = 0 . 1, k = 0 . 01, and λ = 1: ∂u ∂t ( x, t ) - 2 u ∂x 2 ( x, t ) = 0, for 0
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Unformatted text preview: &lt; x &lt; 1 and 0 &lt; t with the boundary conditions u (0 , t ) = u (1 , t ) = 0, for 0 &lt; t and u ( x, 0) = sin πx , for 0 ≤ x ≤ 1. 13.3. (a) Find the natural cubic spline function whose knots are-1 , , 1 and that takes the values S (-1) = 13, S (0) = 7, and S (1) = 9. (b) Prove that: sup-∞ &lt;x&lt; ∞ | ∞ X i =-∞ c i B k i ( x ) | ≤ sup-∞ &lt;i&lt; ∞ | c i | (c) If S is the natural cubic spline interpolating f ( x ) = sin x at the knots 0 = t &lt; t 1 &lt; . . . &lt; t n = 1, then prove that Z 1 ( S 00 ( x )) 2 dx ≤ 1 2 (1-cos 2)....
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