Chemical Engineering Hand Written_Notes_Part_73

Chemical Engineering Hand Written_Notes_Part_73 - 148 4....

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148 4. ODE-IVPS AND RELATED NUMERICAL SCHEMES Modi f ed Euler-Cauchy Method: Set b = 1. (3.42) x ( n +1)= x ( n )+ hf t n + h 2 ,x ( n )+ h 2 f ( n ) ¸ It must be emphasized that 3.40. and 3.30 do not give identical results. However, if we start from the same x ( n ) ,then x ( n +1) given by 3.30 and 3.40 would di f er only by O ( h 3 ) . 3.4.2. Multivariate Case. Even though the above derivation has been worked for one dependent variable case, the method can be easily extended to multi- variable case. For example, the most commonly used fourth order R-K method for one variable can be stated as (3.43) x ( n +1)= x ( n )+ h 6 ( k 1 +2 k 2 +2 k 3 ++ k 4 ) k 1 = f ( t n ,x ( n )) = f ( n ) (3.44) k 2 = f μ t n + h 2 ,x ( n )+ h 2 k 1 (3.45) k 3 = f μ t n + h 2 ,x ( n )+ h 2 k 2 (3.46) k 4 = f ( t n + h, x ( n )+ hk 3 ) (3.47) Now,supposewewanttousethismethodforsolving m simultaneous ODE-IVPs d x dt = F ( x ,t ) (3.48) x (0) = x 0 (3.49) where x R
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Chemical Engineering Hand Written_Notes_Part_73 - 148 4....

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