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
where
- x, a vector in Rn, are variables for which derivatives are present (differential variables),
- y, a vector in Rm, are variables for which no derivatives are present (algebraic variables),
- t, a scalar (usually time) is an independent variable.
The set of DAEs is
Initial conditions be a solution of the system of equations of the form
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.](../../../math/3/f/8/3f8bee5e473f8341bcb3f8c6a6af2947.png)
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.
