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

Chapter 9
Differential Algebraic Equations
9.1 Concepts
A differential algebraic equation 1 (DAE) is a system of differential equations and
algebraic constraints. The most general form is
F(t, y, y ′ ) = 0 (9.1)
where the Jacobian matrix F y ′ may be singular. 2 For example, the DAE with
y = (u, v, w) T described by the system of scalar equations
has
The Jacobian is
u ′ = v (9.2)
v ′ = w (9.3)
0 = ue w + we u (9.4)
⎛
u ′ ⎞
− v
F(t, y, y ′ ) = ⎝ v ′ − w ⎠ (9.5)
ue w + we u
⎛ ⎞
1 0 0
∂F
∂y ′ = ⎝0 1 0⎠ (9.6)
0 0 0
Under certain conditions any DAE can be reduced to a system of ordinary differential
equations by repeatedly differentiating the constraints. The idea is to get an explicit
equation for each of the derivatives, of the form y ′ = f(t, y). For example, if we
differentiate 9.4 we get
0 = u ′ e w + uw ′ e w + w ′ e u + wu ′ e u (9.7)
= u ′ (e w + we u ) + w ′ (ue w + e u ) (9.8)
w ′ = − u′ (e w + we u )
ue w + e u = − v(ew + we u )
ue w + e u (9.9)
1 The material in this and the following sections is based on [5].
2 In general, throughout this chapter we will be informal with notation and may omit the boldface
even though the referenced variable may be a vector or matrix as in the Jacobian F y ′
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