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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 + t 2 ( f n +1 + f n ) , where f n f ( t n ,y n ) and t = t n +1 t n . a) (15 pts) Show that the method is second order with truncation error T n = 1 12 t 2 y 000 ( n ) , for some 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 tL (  e n +1  +  e n  ) + 1 12 t 3 M....
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 Winter '09
 NilesA.Pierce

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