Chemical Engineering Hand Written_Notes_Part_112

Chemical Engineering Hand Written_Notes_Part_112 - 226 5...

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226 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES To begin with, let us check whether the operator L de f ned by equations (10.18-10.20) is self-adjoint. h v,Lu i = 1 Z 0 v ( z )( d 2 u/dz 2 ) dz (10.22) = v ( z ) du dz ¸ 1 0 + 1 Z 0 dv dz du dz dz = v ( z ) du dz ¸ 1 0 + dv dz u ( z ) ¸ 1 0 + 1 Z 0 μ d 2 v dz 2 u ( z ) dz Using the boundary conditions u (0) = u (1) = 0 ,wehave (10.23) dv dz u ( z ) ¸ 1 0 = dv dz u (1) dv dz u (0) = 0 If we set B 1 : v (0) = 0 (10.24) B 2 : v (1) = 0 (10.25) then (10.26) du dz v ( z ) ¸ 1 0 =0 and we have (10.27) h v,Lu i = 1 Z 0 μ d 2 v dz 2 u ( z ) dz = h L v,u i In fact, it is easy to see that the operator L is self adjoint as L = L , B 1 = B 1 and B 2 = B 2 . In addition to the self-adjointness of L, we have h u, Lu i = u ( z ) du dz ¸ 1 0 + 1 Z 0 μ du dz 2 dz (10.28) = 1 Z 0 μ du dz 2 dz > 0 for all u ( z ) (10.29) when u ( z ) isanon -zerovectorin C (2) [0 , 1] .
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Chemical Engineering Hand Written_Notes_Part_112 - 226 5...

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