3639272250

3639272250 - Lecture Notes ME4906 Mechanical Engineering...

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1 Lecture Notes ME4906, Mechanical Engineering, PolyU 2008/2009 Chapter 5 Partial Differential Equations and Finite Difference Methods 5.1 Finite Difference Taylor’s Expansion (Revisit) ) ( ) ( ) ( ! ) ( ) ( ! 2 ) ( ) )( ( ) ( ) ( 1 2 1 1 1 i i i n n i i i i i i i i i i x x at dx df x f where x f n x x x x x f x x x f x f x f = = + + + + + = + + + + L Denoting ) ( 1 i i x x x = + + + + + + = + ) ( ! ) ( ) ( ! 2 ) ( ) )( ( ) ( ) ( 2 1 i n n i i i i x f n x x x f x x f x f x f L Denoting ) ( 1 i i x x x = + + + + = ) ( ! ) ( ) 1 ( ) ( ! 2 ) ( ) )( ( ) ( ) ( 2 1 i n n n i i i i x f n x x x f x x f x f x f L When terms up to nth-order derivative are retained in the series, the error is given by ) ( )! 1 ( ) ( 1 1 i n n n x f n x R + + + = , or ) ) (( 1 + = n n x O R , which means the error is of order 1 ) ( + n x

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2 First Derivatives . .......... ) ( ! 3 ) ( ! 2 ) ( 3 2 1 + + + = i i i i i f h f h f h f f where ) ( 1 = = i i x x x h and . .......... ) ( ! 3 ) ( ! 2 ) ( 3 2 1 + + + + = + i i i i i f h f h f h f f where ) ( 1 i i x x x h = = + y x x i x i-1 X i+1 ) ( ' x f y = ) ( x f y = h h Forward difference Backward difference Central difference
3 Difference Equations Re-arranging difference Backward ......... ) ( 2 difference Forward .......... ) ( 2 1 1 1 1 = + + = = + = + + h f f f or f h h f f f and h f f f or f h h f f f i i i i i i i i i i i i i i The error is h E 1 with the order of O(h) and difference Central = + = + + h x f x f x f or x f h h x f x f x f i i i i i i i 2 ) ( ) ( ) ( ...

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3639272250 - Lecture Notes ME4906 Mechanical Engineering...

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