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 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

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 DASSL/DASPK, Modelica, ABACUSS, EMSO, APMonitor, Sim42, and others. Matlab solves Index-1 DAEs with its stiff ODE solver.

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).

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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