Chemical Engineering Hand Written_Notes_Part_70

Chemical Engineering Hand Written_Notes_Part_70 - 142 4....

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142 4. ODE-IVPS AND RELATED NUMERICAL SCHEMES ODEs. Thus, we can de f ne auxiliary variables (3.4) x 1 ( t )= y ( t ) x 2 ( t )= dy dt ....... ....... x m ( t )= d m 1 y dt m 1 Using these variables, the original nth order ODE can be converted to n f rst order ODE’s as, (3.5) dx 1 dt = x 2 dx 2 dt = x 3 ....... dx m 1 dt = x m dx m dt = f [ x 1 ,x 2 ,x 3 , ....... , x m ,t ] De f ning function vector (3.6) F ( x )= x 2 ..... x m f [ x 1 ,x 2 ,x 3 , ....... , x m ,t ] we can write the above set of d x dt = F ( x ,t ) (3.7) x (0) = y (0) dy dt (0) ....... d m 1 y dt m 1 (0) ¸ T (3.8) Thus, it is su cient to study only the solution methods for solving n f rst order ODE’s. Any set of higher order ODEs can be reduced to a set of f rst order ODEs. Also, as forced systems (non-homogeneous systems) can be looked upon as unforced systems (homogenous systems) with variable parameters, it is su cient to study the solution methods for homogenous set of equations of the
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This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

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Chemical Engineering Hand Written_Notes_Part_70 - 142 4....

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