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Unformatted text preview: Problem Set 3 January 24, 2007 Due February 2, 2007 ACM 95b/100b 3pm in Firestone 303 Niles A. Pierce (2 pts) Include grading section number 1. Consider the initial value problem y = f ( t,y ) , y (0) = y . and the trapezoidal rule y n +1 = y n + Δ t 2 ( f n +1 + f n ) , where f n ≡ f ( t n ,y n ) and Δ t = t n +1- t n . a) (15 pts) Show that the method is second order with truncation error T n =- 1 12 Δ t 2 y 000 ( ξ n ) , for some ξ n ∈ ( t n ,t n +1 ) , where y is the unknown analytical solution to the IVP. b) (10 pts) Suppose that | y 000 ( t ) | ≤ M for some positive constant independent of t and that f satisfies the Lipschitz condition | f ( t,y 1 )- f ( t,y 2 ) | ≤ L | y 1- y 2 | for all real t , y 1 , y 2 , where L is a positive constant independent of t . Show that the solution error e n ≡ y ( t n )- y n satisfies | e n +1 | ≤ | e n | + 1 2 Δ tL ( | e n +1 | + | e n | ) + 1 12 Δ t 3 M....
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This note was uploaded on 11/23/2009 for the course ACM 95b taught by Professor Nilesa.pierce during the Winter '09 term at Caltech.
- Winter '09