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lec10_09292006

# lec10_09292006 - 10.34 Numerical Methods Applied to...

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10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #10: Function Space. Functional Approximation (Variables are scalar in this example) ) ( ) ( ) ( 0 x x c x f N n n n Δ + = φ Figuring out Δ (x) is similar to solving whole problem Increase N until function converges { φ n (x)} favorite set of functions length M {v n } favorite set of vectors = N n n n v c w 0 N<M v n { } m Basis : e l = = N n n n l v d 0 , = = = n l n n l l l l l N n n n approx v d a e a v c w , , 0 e l ·e j = δ jl Î orthonormal c = a T D We want to do the same with functions. How do you take dot product? Define “ φ n · φ m ” = “works”: < φ x of range g interestin * ) ( ) ( ) ( x x x g dx m n φ φ m | φ n > = δ mn weighting function g(x) = k x: 0 Æ 2 π φ m = e imx = cos(mx) + i·sin(mx) ) cos( 2 mx e e imx imx = + g(x) = 1 x: -1 Æ +1 Legendre polynomials 2 ) ( x e x g = x: - Æ + Hermite polynomials 2 2 ) ( x x g 1 = π x: -1 Æ +1 Chebyshev polynomials 1) We chose a basis { φ n (x)} and an inner product orthonormal: < φ m | φ n > = δ mn

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lec10_09292006 - 10.34 Numerical Methods Applied to...

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