Introduction to Differential-Algebraic Equations (DAEs)
The systems of equations that govern certain phenomena (in electrical circuits, chemical kinetics, etc.) contain a combination of differential equations and algebraic equations. The differential equations are responsible for the dynamical evolution of the system, while the algebraic equations serve to constrain the solutions to certain manifolds. It is therefore of some interest to study the solutions of such
differential-algebraic equations (DAEs).
Here is a simple example of a DAE. The first equation is an ODE for the function
, while the second equation constrains the functions
and
to lie in a submanifold (a straight line) in
-
space.
These tutorials are restricted to
linear DAEs, which are defined as systems of equations of the following type.
Here
and
are matrix functions of the independent variable
,
is a vector function of
, and
is the vector of unknowns. If the matrix
is nonsingular (that is, invertible) then this is a system of ODEs. Thus, the system is a DAE if the matrix
is
singular.
If
, then the system is said to be
homogeneous. As for ODEs, the general solution to a DAE is composed of the general solution to the corresponding homogeneous problem and a particular solution to the inhomogeneous system.
DSolve can find the solutions to all DAEs in which the entries of the matrices
and
are constants. Such DAEs are said to have constant coefficients. The algorithm used by
DSolve is based on decomposing both
and
into a nonsingular and nilpotent part. This decomposition is used to calculate a generalized inverse for
and
, which effectively reduces the problem to solving a system of ODEs.
It is important to realize that the initial values for a DAE must be prescribed carefully to guarantee a solution for the problem. This can be seen by considering the following system of equations.
Hence the only solution is
But this solution is inconsistent with the initial condition
.
DSolve can solve DAEs with constant coefficients; see
"Examples of DAEs".
