Page 166 - Numerical Methods for Chemical Engineering

ODE-IVP 155

ODEs and nonlinear algebraic equations of the general form
M ˙x = f (x) x(t 0 ) = x [0] (4.5)

M is a matrix, in general itself a function of x and t, and singular, as it contains a row of
all zeros for each algebraic equation. Finally, we present a robust method, based upon IVP
solvers, to study how the solution to a set of nonlinear algebraic equations depends upon
its parameters, parametric continuation.

Initial value problems of ordinary differential equations (ODE-IVPs)

IVPs arise when we study the dynamics of a system governed by a set of first-order ODEs,
such as the batch reactor kinetics for the network of two elementary reactions
A + B → C r R1 = k 1 c A c B
C + B → D r R2 = k 2 c C c B (4.6)
At t 0 = 0, we start with the initial concentrations
c C (t 0 ) = c D (t 0 ) = 0 (4.7)
c A (t 0 ) = c A0 c B (t 0 ) = c B0
The time evolution of the system follows the set of first-order ODEs
dc A dc B
=−r R1 =−r R1 − r R2
dt dt
(4.8)
dc C dc D
= r R1 − r R2 = r R2
dt dt
We wish to use a general notation system for IVPs, and so define a state vector, x, that
completely describes the state of the system at any time sufficiently well to predict its future
behavior; here,
T
x = c A c B c C c D (4.9)
We then write the ODE system, substituting for the reaction rates, as
˙ x 1 =−k 1 x 1 x 2 = f 1 (x; k 1 , k 2 ) ˙ x 2 =−k 1 x 1 x 2 − k 2 x 3 x 2 = f 2 (x; k 1 , k 2 )
(4.10)
˙ x 3 = k 1 x 1 x 2 − k 2 x 3 x 2 = f 3 (x; k 1 , k 2 ) ˙ x 4 = k 2 x 3 x 2 = f 4 (x; k 1 , k 2 )
We collect the parameters of the system into a parameter vector

Θ = [k 1 k 2 ] T (4.11)
and write (4.10) in the standard ODE-IVP form

˙ x = f (x; Θ) x(t 0 ) = x [0] (4.12)
We next show that this problem formulation is quite general by considering the following:
How do we express the system in the form of (4.12) if the function vector is itself time-
dependent?
What if we have ODEs of higher order than one?