The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS 183
Each curve will be plotted in its own style: the first column of data (A) as a thin,
dashed, red line; the second (B) as a thick blue dashed line; the third (X) as a thick
green line; and the forth (e) as a thin purple line:
The figure shows how all of the substrate (A, thin, red, dashed curve) is eventually
converted to the product (B, thick green). The concentration of enzyme dips, and
the concentration of intermediate (X, thick blue dashes) while the conversion is
taking place, but as the process reaches completion they return to their steady state
values.
Unfortunately, the simple process of writing
0 = F
(
t n , y n , y )
n − y n−1
h
(9.164)
(or using any other discretization method besides backwards Euler) does not always
work. For some equations, it is even possible to find any discretization formula at
all. Consider, for example, the index-2 differential-algebraic equation
The exact solution is
x ′ − ty ′ = 0 (9.165)
x − ty = f(t) (9.166)
x = f(t) − tf ′ (t) (9.167)
y = −f ′ (t) (9.168)
which may be verified by substitution. Backward Euler gives the formulas
x n − x n−1
h
− t n
y n − y n−1
h
= 0 (9.169)
x n − t n y n = f(t n ) (9.170)
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge