Chemical Engineering Hand Written_Notes_Part_77

Chemical Engineering Hand Written_Notes_Part_77 - 156 4....

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156 4. ODE-IVPS AND RELATED NUMERICAL SCHEMES 3.5.4. Multivariate Case. Even though the above derivations have been worked for one dependent variable case, these methods can be easily extended to multi- variable case (3.117) dx dt = F ( x ,t ); x R n where F ( x ,t ) is a n × 1 function vector. In the multivariable extension, the scalar function f ( x, t ) is replaced by the function vector F ( x ,t ) ,i .e . (3.118) x ( n +1) = α 0 x ( n )+ α 1 x ( n 1) + ....... + α p x ( n p ) + h £ β 1 F ( n +1)+ β 0 F ( n )+ β 1 F ( n 1) + .... + β p F ( n p ) ¤ where F ( n i ) F [ x ( t n ih ) , ( t n ih )] (3.119) i = 1 , 0 , 1 , ...p and the scalar coe cients © α 0 ....α p 1 0 1 , ...... β p ª are identical with the coe cients derived for the scalar case as described in the above section. The main advantages and limitations of multi-step methods can be summa- rized as follows
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Chemical Engineering Hand Written_Notes_Part_77 - 156 4....

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