In mathematics, a differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system.

The set of the solutions of such a system is a differential algebraic variety, and corresponds to an ideal in a differential algebra of differential polynomials.

In the univariate case, a DAE in the variable t can be written as a single equation of the form

:<math>F(\dot x, x, t)=0,</math>

where <math>x(t)</math> is a vector of unknown functions and the overdot denotes the time derivative, i.e., <math>\dot x = \frac{dx}{dt}</math>.

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 because these may not all appear (i.e. some equations are algebraic); technically the distinction between an implicit ODE system [that may be rendered explicit] and a DAE system is that the Jacobian matrix <math>\frac{\partial F(\dot x, x, t)}{\partial \dot x}</math> is a singular matrix for a DAE system. This distinction between ODEs and DAEs is made because DAEs have different characteristics and are generally more difficult to solve.

In practical terms, the distinction between DAEs and ODEs is often that the solution of a DAE system depends on the derivatives of the input signal and not just the signal itself as in the case of ODEs; this issue is commonly encountered in nonlinear systems with hysteresis, such as the Schmitt trigger.

This difference is more clearly visible if the system may be rewritten so that instead of x we consider a pair <math>(x,y)</math> of vectors of dependent variables and the DAE has the form

::<math>\begin{align}\dot x(t)&=f(x(t),y(t),t),\\0&=g(x(t),y(t),t).\end{align}</math>

:where <math>x(t)\in\R^n</math>, <math>y(t)\in\R^m</math>, <math>f:\R^{n+m+1}\to\R^n</math> and <math>g:\R^{n+m+1}\to\R^m.</math>

A DAE system of this form is called semi-explicit.

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 <math>(x,y)</math> and the system of differential equations of the DAE appears in the form

::<math> F\left(\dot x, x, y, t\right) = 0 </math>

where

  • <math>x</math>, a vector in <math>\R^n</math>, are dependent variables for which derivatives are present (differential variables),
  • <math>y</math>, a vector in <math>\R^m</math>, are dependent variables for which no derivatives are present (algebraic variables),
  • <math>t</math>, a scalar (usually time) is an independent variable.
  • <math>F</math> is a vector of <math>n+m</math> functions that involve subsets of these <math>n+m+1</math> variables and <math>n</math> derivatives.

As a whole, the set of DAEs is a function

::<math> F: \R^{(2n+m+1)} \to \R^{(n+m)}. </math>

Initial conditions must be a solution of the system of equations of the form

::<math> F\left(\dot x(t_0),\, x(t_0), y(t_0), t_0 \right) = 0. </math>

Examples

The behaviour of a pendulum of length L with center in (0,0) in Cartesian coordinates (x,y) is described by the Euler–Lagrange equations

::<math>\begin{align}

\dot x&=u,&\dot y&=v,\\

\dot u&=\lambda x,&\dot v&=\lambda y-g,\\

x^2+y^2&=L^2,

\end{align}</math>

where <math>\lambda</math> 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

::<math>\begin{align}

&&\dot x\,x+\dot y\,y&=0\\

\Rightarrow&& u\,x+v\,y&=0,

\end{align}</math>

restricting the direction of motion to the tangent of the circle. The next derivative of this equation implies

::<math>\begin{align}

&&\dot u\,x+\dot v\,y+u\,\dot x+v\,\dot y&=0,\\

\Rightarrow&& \lambda(x^2+y^2)-gy+u^2+v^2&=0,\\

\Rightarrow&& L^2\,\lambda-gy+u^2+v^2&=0,

\end{align}</math>

and the derivative of that last identity simplifies to <math>L^2\dot\lambda-3gv=0</math> which implies the conservation of energy since after integration the constant <math>E=\tfrac32gy-\tfrac12L^2\lambda=\frac12(u^2+v^2)+gy</math> 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 <math>(x_0,u_0)</math> and a sign for y are given, the other variables are determined via <math>y=\pm\sqrt{L^2-x^2}</math>, and if <math>y\ne0</math> then <math>v=-ux/y</math> and <math>\lambda=(gy-u^2-v^2)/L^2</math>. 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

:: <math>\begin{align}

\dot x&=u,\\

\dot u&=\lambda x,\\[0.3em]

0&=x^2+y^2-L^2,\\

0&=ux+vy,\\

0&=u^2-gy+v^2+L^2\,\lambda.

\end{align}</math>

This is a semi-explicit DAE of index 1. Another set of similar equations may be obtained starting from <math>(y_0,v_0)</math> and a sign for x.

DAEs also naturally occur in the modelling of circuits with non-linear devices. Modified nodal analysis employing DAEs is used for example in the ubiquitous SPICE family of numeric circuit simulators. Similarly, Fraunhofer's Analog Insydes Mathematica package can be used to derive DAEs from a netlist and then simplify or even solve the equations symbolically in some cases. It is worth noting that the index of a DAE (of a circuit) can be made arbitrarily high by cascading/coupling via capacitors operational amplifiers with positive feedback.

Structural analysis for DAEs

We use the <math>\Sigma</math>-method to analyze a DAE. We construct for the DAE a signature matrix <math>\Sigma=(\sigma_{i,j})</math>, where each row corresponds to each equation <math>f_i</math> and each column corresponds to each variable <math>x_j</math>. The entry in position <math>(i,j)</math> is <math>\sigma_{i,j}</math>, which denotes the highest order of derivative to which <math>x_j</math> occurs in <math>f_i</math>, or <math>-\infty</math> if <math>x_j</math> does not occur in <math>f_i</math>.

For the pendulum DAE above, the variables are <math>(x_1,x_2,x_3,x_4,x_5)=(x,y,u,v,\lambda)</math>. The corresponding signature matrix is

:<math>\Sigma =

\begin{bmatrix}

1 & - & 0^\bullet & - & - \\

- & 1^\bullet & - & 0 & - \\

0 & - & 1 & - & 0^\bullet \\

- & 0 & - & 1^\bullet & 0 \\

0^\bullet & 0 & - & - & -

\end{bmatrix}

</math>

See also

  • Algebraic differential equation, a different concept despite the similar name
  • Delay differential equation
  • Partial differential algebraic equation
  • Modelica Language

References

Further reading

Books

  • (Covers the structural approach to computing the DAE index.)

Various papers

  • http://www.scholarpedia.org/article/Differential-algebraic_equations