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lecture15 - Summary from last lecture Starting from the...

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Summary from last lecture - Starting from the Taylor’s series expansion, we derive: - Euler’s method - Improvement of Euler’s method - Heun’s method - Midpoint method - Higher order Taylor method
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Higher order Taylor method Euler method is only a first order approximation (use only two terms in Taylor’s series with truncation error O(h 2 )). It is obvious that if we want a better solution, we should use more terms in the Taylor series in order to obtain a higher order truncation error. Suppose the solution y(t) to the initial-value problem: ) , ( y t f dt dy = for a t b y(t o ) = y o has (n+1) continuous derivatives. If we expand the solution, y(t) in therms of its nth Taylor polynomial about t: 1 ) 1 ( 1 ) ( 2 1 ) ( )! 1 ( ) ( ! ) ( ! 2 ) ( ) ( ) ( + + + + + + + + + = i i i n n i n n i i i i t t y n h t y n h t y h h t y t y t y ξ ξ K
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Higher order Taylor method 1 ) ( 1 ) 1 ( 2 1 ) , ( )! 1 ( ) , ( ! ) , ( ! 2 ) , ( ) ( ) ( + + + + + + + + = i i i i n n i i n n i i i i i i t t y f n h y t f n h y t f h y t f h t y t y ξ ξ K Successive differentiation of the solution y(t) gives: )) ( , ( ) ( )) ( , ( ) ( )) ( , ( ) ( ) 1 ( ) ( t y t f t y t y t f t y t y t f t y n n = = = M Substitute: ( ) ( ) ( ) + + + = = + i i n n i i i i i i o o w t f n h w t f h w t f h w w y w , ! , 2 , ) 1 ( 1 1 K
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