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

CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS 177
Solution. Write the pendulum equations from the earlier example as
p ′ = u (9.107)
q ′ = v (9.108)
u ′ = −T p (9.109)
v ′ = g − T q (9.110)
0 = p 2 + q 2 − 1 (9.111)
and make the associations x = (u, v) T , y = (p, q) T , and z = (T ). Then
( ) −zy1
f(t, x, y, z) =
(9.112)
g − zy 2
( )
x1
g(t, x, y, z) =
(9.113)
x 2
h(t, y) = y1 2 + y2 2 − 1 (9.114)
and
( ) ( )
∂h ∂g ∂f
1 0
∂y ∂x ∂z = (2y −y1
1, 2y 2 )
0 1 −y 2
(9.115)
= −2(y 2 1 + y 2 2) = −2(p 2 + q 2 ) = −2 ≠ 0 (9.116)
9.5 Implementation: A Detailed Example
Implementing a general solver for the differential algebraic equation F(t, y, y ′ ) = 0
can be quite complicated. If we use a method that approximates y ′ with
y ′ n = φ n (. . . , y n+1 , y n , y n−1 , . . . ) (9.117)
then we end up with a system of m nonlinear equations for the y n , where m = dim y:
F (t n , y n , φ n (. . . , y n , . . . )) = 0 (9.118)
For, example, with the Backward Euler Method we have
φ n = 1 h (y n − y n−1 ) (9.119)
which gives us the nonlinear system
(
F t n , y n , 1 )
h (y n − y n−1 ) = 0 (9.120)
Let us consider an implementation of the index-1 system that describes enzymatic
conversion according to the system of chemical reactions
A + E α → X (9.121)
X β → A + E (9.122)
X γ → B + E (9.123)
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge