Chemical Engineering Hand Written_Notes_Part_113

Chemical Engineering Hand Written_Notes_Part_113 - 228 5...

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228 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES As 1 Z 0 ( du/dz ) 2 dz is positive and symmetric , we are guaranteed to f nd the minimum. The main di culty in performing the search is that, unlike the previous case where we were working in R n , the search space is in f nite dimensional as u ( z ) C (2) [0 , 1] .One remedy to alleviate this di culty is to reduce the in f nite dimensional search problem to a f nite dimensional search space by constructing an approximate solution using n trial functions. Let v (1) ( z ) , ..... , v ( n ) ( z ) represent trial function. The approximate solution is constructed as (10.36) b u ( z )= c 0 v (0) ( z )+ ..... + c n v ( n ) ( z ) where v ( i ) ( z ) represents trial functions . Using this approximation, we convert the in f nite dimensional optimization problem to a f nite dimensional optimiza- tion problem as follows Min C b Φ ( c )= 1 / 2 1 Z 0 ( d b u/dz ) 2 dz 1 Z 0 b uf ( z ) dz (10.37) =1 / 2 1
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Chemical Engineering Hand Written_Notes_Part_113 - 228 5...

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