Chemical Engineering Hand Written_Notes_Part_74

Chemical Engineering Hand Written_Notes_Part_74 - 150 4....

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150 4. ODE-IVPS AND RELATED NUMERICAL SCHEMES Now, there are several ways we could go about estimating the unknown para- meters of the polynomial. Explicit algorithm: Let us use only the current and the past infor- mation of state and derivatives, which will lead to an explicit algorithm. f ( n 1) = a 1 ,n 2 a 2 ,n h f ( n )= a 1 ,n (3.60) (3.61) x ( n )= a 0 ,n Solving above equations simultaneously, we get coe cients (3.62) a 0 ,n = x ( n ); a 1 ,n = f ( n ); a 2 ,n = f ( n ) f ( n 1) 2 h which can be used to extrapolate the value of x ( t ) at t = t n +1 as x ( n +1) = a 0 ,n + a 1 ,n h + a 2 ,n h 2 (3.63) = x ( n )+ f ( n ) h + f ( n ) f ( n 1) 2 h ¸ h 2 = x ( n )+ h 3 2 f ( n ) 1 2 f ( n 1) ¸ Implicit algorithm: Alternatively, we can choose to estimate x ( n +1) based on derivative at t n +1 , i.e. f ( n +1) = a 1 ,n +2 a 2 ,n h (3.64) f ( n )= a 1 ,n (3.65) (3.66) x ( n )= a 0 ,n These equations yield following set of coe
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Chemical Engineering Hand Written_Notes_Part_74 - 150 4....

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