Differential algebraic equation

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In mathematics, differential algebraic equations (DAEs) are a general form of differential equation, given in implicit form. They can be written

$f\left(\frac{dx}{dt}, x, y, t\right) = 0$

where

$x$ , a vector in $R n$ , are variables for which derivatives are present (differential variables),
$y$ , a vector in $R m$ , are variables for which no derivatives are present (algebraic variables),
$t$ , a scalar (usually time) is an independent variable.

The set of DAEs is

$f: R^{(2n+m+1)} \rightarrow R^{(n+m)}$

Initial conditions be a solution of the system of equations of the form

$f\left(\frac{dx}{dt}\Big\vert_{t=0}, x(0), y(0), 0 \right) = 0.$

Physical systems are often readily specified in terms of DAEs, and software can be used to attempt to solve these problems. Such software includes Modelica, ABACUSS, EMSO, Sim42 and others.

A major problem in the solution of DAEs is the problem of index reduction. Most numerical solvers require ordinary differential equations of the form

$\left[\frac{dx}{dt}, \frac{dy}{dt}\right]^T = g(x,y,t) = 0.$

However it is a non-trivial task to convert arbitrary DAE systems into ODEs. Techniques which can be employed include Pantelides algorithm and dummy variable substitution.