NationMaster - Encyclopedia: Differential algebraic equation

Differential algabraic equations (DAEs) are a generalised form of Differential equation. They can be written In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

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 a independent variable

The set of DAEs is

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

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. The object-oriented modeling language Modelica is designed to allow convenient, component-oriented modeling of complex systems, , systems containing mechanical, electrical, electronic, hydraulic, thermal, control, electric power or process-oriented subcomponents. ... EMSO is the existential fragment of monadic second-order logic (MSO). ...

A major problem in the solution of DAEs is the problem of index reduction. Most numerical solvers require ordinary differential equations of the form: In mathematics, a differential equation is an equation that describes a prescribed relationship between a set of unknowns which are to be regarded as an unknown function and its (ordinary or partial) derivatives. ...

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.

For more information, try

Results from FactBites:

Differential equation - Wikipedia, the free encyclopedia (779 words)
	In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables.
	A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process.
	The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates.

How do I solve systems of differential and algebraic equations? (383 words)
	A set of equations that includes both differential equations and algebraic equations is called a differential-algebraic equation, or DAE.
	Differential-algebraic equations are usually solved either by first solving the algebraic equations and using the solution to construct a set of purely differential equations, or by solving the differential equations and using the solution to construct a set of purely algebraic equations.
	If neither the algebraic equations nor the differential equations can be solved symbolically, then it may be useful to consider a purely numerical solution.

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