Math128_hw11

# Math128_hw11 - Math 128A, Spring 2007 Homework 11 Solution...

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Math 128A, Spring 2007 Homework 11 Solution 1. Compute the local truncation error for the 3-Point Adams-Bashforth method. Solution: Recall that we obtained AB-3 by ﬁnding constants A , B , and C so that the approximation y ( t n +1 ) - y ( t n ) = Z t n +1 t n y 0 ( t ) dt Ay 0 ( t n ) + By 0 ( t n - 1 ) + Cy 0 ( t n - 2 ) was exact for y 0 ( t ) a polynomial of degree 2. We can use this to compute the local truncation error: If p 2 ( t ) is the interpolation of y 0 ( t ) at t n , t n - 1 , and t n - 2 , then we may write y 0 ( t ) = p 2 ( t ) + e 2 ( t ) and thus Z t n +1 t n y 0 ( t ) dt = Ay 0 ( t n ) + By 0 ( t n - 1 ) + Cy 0 ( t n - 2 ) + Z t n +1 t n e 2 ( t ) dt Using the Newton form for the error term, namely, e 2 ( t ) = y 0 [ t n - 2 , t n - 1 , t n , t ]( t - t n - 2 )( t - t n - 1 )( t - t n ) We can apply the mean value theorem and integrate to get Z t n +1 t n e 2 ( t ) dt = y 0 [ t n - 2 , t n - 1 , t n , ¯ ξ ] h 4 4 and then apply a theorem from class to get Z t n +1 t n e 2 ( t ) dt = y 5 ( ξ ) 4! h

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## This note was uploaded on 05/07/2008 for the course MATH 128A taught by Professor Rieffel during the Spring '08 term at Berkeley.

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Math128_hw11 - Math 128A, Spring 2007 Homework 11 Solution...

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