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

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

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

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

Initial conditions be a solution of the system of equations, such that

$f\backslash left(\backslash frac\{dx\}\{dt\}\backslash Big\backslash vert\_\{t=0\},\; x(0),\; y(0),\; 0\; \backslash right)\; =\; 0$

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:

$\backslash left\backslash frac\{dy\}\{dt\}\backslash right^T\; =\; g(x,y,t)\; =\; 0$

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.

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Differential equations | Numerical analysis