lec10_09292006 - 10.34, Numerical Methods Applied to...

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Unformatted text preview: 10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #10: Function Space. Functional Approximation (Variables are scalar in this example) ) ( ) ( ) ( x x c x f N n n n + = Figuring out (x) is similar to solving whole problem Increase N until function converges { n (x)} favorite set of functions length M {v n } favorite set of vectors = N n n n v c w N<M v n { } m Basis : e l = = N n n n l v d , = = = n l n n l l l l l N n n n approx v d a e a v c w , , e l e j = jl orthonormal c = a T D We want to do the same with functions. How do you take dot product? Define n m = works: < x of range g interestin * ) ( ) ( ) ( x x x g dx m n m | n > = mn w e i g h t i n g f u n c t i o n g(x) = k x: 0 2 m = e imx = cos(mx) + isin(mx) ) cos( 2 mx e e imx imx = + g(x) = 1 x: -1 +1 Legendre polynomials 2 ) ( x e x g = x: - + Hermite polynomials...
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This note was uploaded on 11/27/2011 for the course CHEMICAL E 10.302 taught by Professor Clarkcolton during the Fall '04 term at MIT.

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lec10_09292006 - 10.34, Numerical Methods Applied to...

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