N10 - ME – 510 NUMERICAL METHODS FOR ME II Fall 2007...

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Unformatted text preview: ME – 510 NUMERICAL METHODS FOR ME II g51g85g82g73g17g3g39g85g17g3g41g68g85g88g78g3g36g85g213g81g111 Fall 2007 ORTHOGONAL FUNCTIONS Two functions f (x) and g (x) are orthogonal with respect to a weight(ing) function w (x) on an interval [a , b] if b a w(x) f(x) g(x) dx = 0 g179 Example: sin(n g652 x) and sin(m g652 x), m g143g3g81g15g3g68g85g72g3g82g87g75g82g74g82g81g68g79g3g90g76g87g75g3g85g72g86g83g72g70g87g3g87g82 w(x) = 1 over the interval [0 , 1]. 1 , m n sin(n x) sin(m x) dx = 0 g83 g83 g122 g179 ME – 510 NUMERICAL METHODS FOR ME II g51g85g82g73g17g3g39g85g17g3g41g68g85g88g78g3g36g85g213g81g111 Fall 2007 ORTHOGONAL FUNCTIONS Theorem: A function f(x) can be expressed as a linear combination of a complete set of mutually orthogonal functions If Q n (x), n = 1, 2, 3, ..., form a complete orthogonal set of functions, then 1 1 2 2 f(x) = a Q (x) + a Q (x) + ... Where the coefficients can be determined by g11 g12 b n a n b 2 n a w(x) Q (x) f(x) dx a = w(x) Q (x) dx g179 g179 ME – 510...
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