Differential algebraic equation
Encyclopedia
In mathematics
, differential algebraic equations (DAEs) are a general form of (systems of) differential equation
s for vector–valued functions x in one independent variable t,
where
is a vector of dependent variables 
and the system has as many equations, 
.
They are distinct from ordinary differential equation
(ODE) in that a DAE is not completely solvable for the derivatives of all components of the function x.
This difference is more clearly visible if the system may be rewritten so that instead of x we consider a pair
of vectors of dependent variables and the DAE has the form
Every solution of the second half g of the equation defines a unique direction for x via the first half f of the equations, while the direction for y is arbitrary. But not every point (x,y,t) is a solution of g. The variables in x and the first half f of the equations get the attribute differential. The components of y and the second half g of the equations are called the algebraic variables or equations of the system. The term algebraic in the context of DAEs only means free of derivatives and is not related to (abstract) algebra.
The solution of a DAE consists of two parts, first the search for consistent initial values and second the computation of a trajectory. To find consistent initial values it is often necessary to consider the derivatives of some of the component functions of the DAE. The highest order of a derivative that is necessary in this process is called the differentiation index. The equations derived in computing the index and consistent initial values may also be of use in the computation of the trajectory.
and the system of differential equations of the DAE appears in the form
where
As a whole, the set of DAEs is a function
Initial conditions must be a solution of the system of equations of the form
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, differential algebraic equations (DAEs) are a general form of (systems of) differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s for vector–valued functions x in one independent variable t,
where
is a vector of dependent variables and the system has as many equations, .They are distinct from ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
(ODE) in that a DAE is not completely solvable for the derivatives of all components of the function x.
This difference is more clearly visible if the system may be rewritten so that instead of x we consider a pair
of vectors of dependent variables and the DAE has the form
-
- where
, 
, 
and 
.
Every solution of the second half g of the equation defines a unique direction for x via the first half f of the equations, while the direction for y is arbitrary. But not every point (x,y,t) is a solution of g. The variables in x and the first half f of the equations get the attribute differential. The components of y and the second half g of the equations are called the algebraic variables or equations of the system. The term algebraic in the context of DAEs only means free of derivatives and is not related to (abstract) algebra.
The solution of a DAE consists of two parts, first the search for consistent initial values and second the computation of a trajectory. To find consistent initial values it is often necessary to consider the derivatives of some of the component functions of the DAE. The highest order of a derivative that is necessary in this process is called the differentiation index. The equations derived in computing the index and consistent initial values may also be of use in the computation of the trajectory.
Other forms of DAEs
The distinction of DAEs to ODEs becomes apparent if some of the dependent variables occur without their derivatives. The vector of dependent variables may then be written as pair and the system of differential equations of the DAE appears in the form
where
-
, a vector in 
, are dependent variables for which derivatives are present (differential variables), -
, a vector in 
, are dependent variables for which no derivatives are present (algebraic variables), -
, a scalar (usually time) is an independent variable. -
is a vector of 
functions that involve subsets these 
variables and 
derivatives.
As a whole, the set of DAEs is a function
Initial conditions must be a solution of the system of equations of the form
Examples
The pendulum in Cartesian coordinates (x,y) with center in (0,0) and length L has the Euler-Lagrange equationsEuler-Lagrange equation
In calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation, is a differential equation whose solutions are the functions for which a given functional is stationary...
-
where
is a Lagrange multiplier. The momentum variables u and v should be constrained by the law of conservation of energy and their direction should point along the circle. Neither condition is explicit in those equations. Differentiation of the last equation leads to
-
restricting the direction of motion to the tangent of the circle. The next derivative of this equation implies-
and the derivative of that last identity simplifies to
which implicitly implies the conservation of energy since after integration the constant 
is the sum of kinetic and potential energy.
To obtain unique derivative values for all dependent variables the last equation was three times differentiated. This gives a differentiation index of 3, which is typical for constrained mechanical systems.
If initial values
and a sign for y are given, the other variables are determined via 
, and if 
then 
and 
. To proceed to the next point it is sufficient to get the derivatives of x and u, that is, the system to solve is now
-
-
This is a semi-explicit DAE of index 1. Another set of similar equations may be obtained starting from
and a sign for x.
Semi-explicit DAE of index 1
DAE of the form
are called semi-explicit. The index-1 property requires that g is solvableImplicit function theoremIn multivariable calculus, the implicit function theorem is a tool which allows relations to be converted to functions. It does this by representing the relation as the graph of a function. There may not be a single function whose graph is the entire relation, but there may be such a function on...
for y. In other words, the differentiation index is 1 if by differentiation of the algebraic equations for t an implicit ODE system results,-
which is solvable for
if 
.
Every sufficiently smooth DAE is almost everywhere reducible to this semi-explicit index-1 form.
Numerical treatment of DAE and applications
A major problem in the solution of DAEs is the problem of index reduction. Most numerical solvers require ordinary differential equations of the form
However it is a non-trivial task to convert arbitrary DAE systems into ODEs. Techniques which can be employed include Pantelides algorithmPantelides algorithmPantelides algorithm gives a systematic method for reducing high-index systems of differential-algebraic equations to lower index, by selectively adding differentiated forms of the equations already present in the system...
and dummy variable substitution.
Numerical solution of DAEs
Many physical systems are naturally described by a set of DAEs. Software can be used to attempt to solve these problems. The table below lists a number of software packages for the numerical solution of DAEs.
Name Brief info ACADO Automatic Control and Dynamic Optimization (C++ and Matlab) APMonitor APMonitorAPMonitor, or "Advanced Process Monitor", is a modeling language for differential and algebraic equations. It is used for describing and solving representations of physical systems in the form of implicit DAE models. APMonitor is suited for large-scale problems and allows solutions of dynamic...Any index DAEs (Index 0 to 3 Pendulum example) DASSL/DASPK Up to index-1 DAEs (and a special form of index-2), sequential solution approach EMSO EMSO simulatorEMSO simulator is an equation-oriented process simulator with a graphical interface for modeling complex dynamic or steady-state processes. It is CAPE-OPEN compliant. EMSO stands for Environment for Modeling, Simulation, and Optimization...Jacobian (software) MATLAB MATLABMATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...Up to index-1 DAEs with ode15s, sequential solution approach Modelica ModelicaModelica is an object-oriented, declarative, multi-domain modeling language for component-oriented modeling of complex systems, e.g., systems containing mechanical, electrical, electronic, hydraulic, thermal, control, electric power or process-oriented subcomponents.The free Modelica languageis...Applications in the auto industry OdePkg Used with Octave DAETS Use Structural Analysis and Taylor methods
See also
- Delay differential equationDelay differential equationIn mathematics, delay differential equations are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times....
- Algebraic differential equationAlgebraic differential equationIn mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used....
, a different concept despite the similar name
- Delay differential equation
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