Differential Algebraic Equations

Differential Algebraic Equations If we had a mass moving under the influence of an external force, viscous friction and windage, we might use

These are explicit ordinary differential equations but if we decide that is negligbile, we have

and this is an implicit ODE

Flow through an orifice is often modeled using the formula

where Cq is constant. This is an example of an explicit algebraic equation. If on the other hand

we have an implicit algebraic equation and the value of Q must be determined iteratively.

If the governing system of equations includes any sort of implicit ordinary differential equation or implicit algebraic equation, they are described as differential algebraic equations. The most general way of writing a system of DAEs is

Note that there is no distinction between the displacement x in the first example and the flow rate Q in the second. Both are regarded as state variables. Hence initial values are required for the implicit algebraic variables as well as the traditional state variables. The objective in solving these equations is to force the two following residuals (Ɛ1,Ɛ2) to zero