Chemical Engineering Hand Written_Notes_Part_117

Chemical Engineering Hand Written_Notes_Part_117 - 236 5...

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236 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES Again, the matrix A is symmetric and positive de f nite matrix and this guar- antees that stationary point of Φ ( u ) is the minimum. At the minimum, we have (10.84) Φ /∂ b u = A b u b =0 The matrix A will also be a sparse matrix. The main limitation of Relay-Ritz method is that it works only when the operator L is symmetric or self adjoint. 10.3.2. Gelarkin’s method. The Gelarkin’s method can be applied for any problem where di f erential operator is not self adjoint or symmetric. Instead of minimizing Φ ( b u ) ,weso lvefor ­ v ( i ) ( z ) ,L b u ( z ) ® = ­ v ( i ) ( z ) ,f ( z ) ® (10.85) i =0 , 2 , .... n where b u ( z ) is chosen as f nite dimensional approximation to u ( z ) (10.86) b u ( z )= b u 0 v (0) ( z )+ ...... + b u n v ( n ) ( z ) Rearranging above equations as (10.87) ­ v ( i ) ( z ) , ( L b u ( z ) f ( z )) ® =0; ( i =0 , 1 , 2 , .... n ) we can observe that parameters u 0 , ..... c n
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Chemical Engineering Hand Written_Notes_Part_117 - 236 5...

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