Table of Contents

Definitions
An Initial Value Problem (IVP) is defined as
(1)If the Jacobian $F_{\dot y}$ is nonsingular, (1) represents an Ordinary Differential Equation (ODE). In principle, one could solve for $\dot y$ as
(2)to obtain an ODE in socalled explicit form.
On the other hand, if $F_{\dot y}$ is singular, (1) represents a Differential Algebraic Equation (DAE). DAEs are best classified using various concepts of their index:
 index of nilpotency (for linear constant coefficient DAE): measure of the numerical difficulty in solving the DAE;
 differentiation index: measure of the "departure" from an ODE;
 perturbation index: measure of the sensitivity of the solutions with respect to perturbations;
 etc.
The differentiation index is defined as follows [Gear and Petzold, 1983]:
Equation (1) has differentiation index $m$ if $m$ is the minimal number of analytical differentiation
(3)such that, by algebraic manipulations, we can extract an explicit ODE $\dot y = \phi(t,y)$ (called an underlying ODE)
Note that, an ODE has therefore index0.
Rheinboldt (1984) advocated looking at DAEs as differential equations on manifolds:
DAE examples
index1
If
(5)and both $f_{\dot u}$ and $g_v$ are square and nonsingular, then $F(t,y,\dot y) = 0$ has index1. Indeed, one can solve $g=0$ for $v$, substitute in $f=0$, and solve for $\dot u$. Note that there are more general forms of index1 DAEs.
Hessenberg index1
(6)is an index1 DAE in Hessenberg form if $g_z$ is nonsingular.
Example
single perturbation problems (e.g. Robertson's chemical kinetics problem):
Hessenberg index2
(8)Example
modeling of incompressible fluid flow by NavierStokes:
with appropriate spatial discretization.
Hessenberg index3
(10)is an index3 DAE in Hessenberg form if $h_y g_x f_z$ is nonsingular.
Example
modeling of constrained mechanical systems with holonomic constraints:
If the constraints $\Phi$ are linearly independen, then $\Phi_qM^{1}\Phi_q^T$ is nonsingular.