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

CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS 169
Since x 2 + y 2 = 1,
Differentiating a third time gives an equation for T ′ ,
T = u 2 + v 2 + yg (9.27)
T ′ = 2uu ′ + 2vv ′ + gy ′ (9.28)
= −2uT x + 2vg − 2vT y + gv (9.29)
= −2T (ux + vy) + 3gv (9.30)
= 3gv (9.31)
Since the DAE can be reduced to an explicit ODE in 3 differentiations, it is an
index-3 DAE.
Equation 9.1 is sometimes referred to as a fully implicit DAE. It becomes a
semi-implicit DAE if we can express the constraint explicitly:
F(t, y, y ′ ) = 0 (9.32)
g(t, y, y ′ ) = 0 (9.33)
where the Jacobian F y ′ is non-singular. If we can express all the derivatives explicitly
then we have a semi-explicit DAE:
A linear time-varying DAE can be expressed as
y ′ = f(t, y, y ′ ) (9.34)
0 = g(t, y, y ′ ) (9.35)
A(t)y ′ (t) + B(t)y(t) = f(t) (9.36)
where the matrix A(t) is singular for all t (if it were not singular the equation
would reduce to linear system of ordinary differential equations). If the matrices are
composed of constants we call the system a Linear constant coefficient DAE,
and write it as
Ay ′ (t) + By(t) = f(t) (9.37)
It is convenient to write the linear equations in block form, e.g., as
( ) ( ) ( ) ( ) ( )
A11 A 12 x
′ B11 B
A 21 A 22 y ′ +
12 x f(t)
=
B 22 y g(t)
where the A ij and B ij are matrices, and the A 2j are singular. Expanding,
B 21
The system is semi-explicit if it has the form
(9.38)
A 11 x ′ + A 12 y ′ + B 11 x + B 12 y = f(t) (9.39)
A 21 x ′ + A 22 y ′ + B 21 x + B 22 y = g(t) (9.40)
A 11 x ′ + B 11 x + B 12 y = f(t) (9.41)
B 21 x + B 22 y = g(t) (9.42)
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge