hw 1 tips

# Hw 1 tips - 6 2 R u n g e K u t t a M e t h o d s R K M A 2 n d O r d e r R K M o r I m p r o v e d E u l e r M e t h o d F a i l u r e o f E u l e

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Unformatted text preview: 6 . 2 : R u n g e K u t t a M e t h o d s ( R K M ) ( A ) 2 n d O r d e r R K M ( o r I m p r o v e d E u l e r M e t h o d ) F a i l u r e o f E u l e r M e t h o d : O n l y s l o p e o n l e f t e n d o f i n t e r v a l [ t , t + h ] i s u s e d . I m p r o v e m e n t : G i v e n t , y ( t ) , • c o m p u t e s l o p e a t t s l = f ( t , y ( t ) ) • fi n d s l o p e a t t + h v i a E M y E = y ( t ) + h s l s r = f ( t + h , y E ) • a p p r o x i m a t e y ( t + h ) v i a a v e r a g e s l o p e y ( t + h ) ≈ y ( t ) + h ( s l + s r ) / 2 I t e r a t i o n S c h e m e S t a r t : y , t F o r k = t o k = N : t k + 1 = t k + h s l = f ( t k , y k ) s r = f ( t k + 1 , y k + h s l ) y k + 1 = y k + h ( s l + s r ) / 2 1 E x . A p p r o x i m a t e t h e s o l u t i o n t o y ′ = t − y , y ( ) = . 5 i n ≤ t ≤ 1 u s i n g h = . 2 5 . S t a r t : y = . 5 , t = t 1 = . 2 5 s l = t − y = − . 5 s r = t 1 − ( y + h s l ) = − . 1 2 5 y 1 = y + h ( s l + s r ) / 2 = . 4 2 1 9 t 2 = . 5 s l = t 1 − y 1 = − . 1 7 1 9 s r = t 2 − ( y 1 + h s l ) = . 1 2 1 1 y 2 = y 1 + h ( s l + s r ) / 2 = . 4 1 5 5 t 3 = . 7 5 s l = t 2 − y 2 = . 8 4 5 s r = t 3 − ( y 2 + h s l ) = . 3 1 3 4 y 3 = y 2 + h ( s l + s r ) / 2 = . 4 6 5 3 t 4 = 1 s l = t 3 − y 3 = . 8 4 5 s r = t 4 − ( y 3 + h s l )...
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## This note was uploaded on 02/19/2012 for the course ENGR 361 taught by Professor Drexel during the Spring '12 term at Bloomsburg.

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Hw 1 tips - 6 2 R u n g e K u t t a M e t h o d s R K M A 2 n d O r d e r R K M o r I m p r o v e d E u l e r M e t h o d F a i l u r e o f E u l e

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