Chemical Engineering Hand Written_Notes_Part_72

Chemical Engineering Hand Written_Notes_Part_72 - 146 4....

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146 4. ODE-IVPS AND RELATED NUMERICAL SCHEMES is a function of time. Thus, for sti f systems it is better to use variable step size methods or special algorithms for sti f systems. 3.3. Taylor’s series based methods. Consider a simple scalar case (3.24) dx dt = f ( x, t ); x R Suppose we know the exact solution x ( t ) at time t n , i.e. x ( n ) ,thenwecan compute x ( n +1) using Taylor series as follows: (3.25) x ( n +1)= x + h dx ( t n ) dt + 1 2! h 2 d 2 x ( t n ) dt 2 + ....... The various derivatives in the above series can be calculated using the di f erential equation, as follows: dx ( t n ) dt = f [ t n ,x ( n )] (3.26) d 2 x ( t n ) dt 2 = ∂f ∂x ¸ ( x ( n ) ,t n ) f [ x ( n ) ,t n ]+ ∂f [ x ( n ) ,t n ] ∂t (3.27) and so on. Let us now suppose that instead of actual solution x ( n ) ,wehave available an approximation to x ( n ) , denoted as x ( n ) .W i
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Chemical Engineering Hand Written_Notes_Part_72 - 146 4....

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