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Inmathematics, adifferential-algebraic system of equations (DAE) is asystem of equations that either containsdifferential equations andalgebraic equations, or is equivalent to such a system.

The set of the solutions of such a system is adifferential algebraic variety, and corresponds to anideal in adifferential algebra ofdifferential polynomials.

In theunivariate case, a DAE in the variablet can be written as a single equation of the form

${\displaystyle F(\dot{x},x,t)=0,}$

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

They are distinct fromordinary differential equation (ODE) in that a DAE is not completely solvable for the derivatives of all components of the functionx 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 theJacobian matrix${\displaystyle \frac{\mathrm{\partial }F(\dot{x},x,t)}{\mathrm{\partial }\dot{x}}}$ is asingular matrix for a DAE system.^[1] This distinction between ODEs and DAEs is made because DAEs have different characteristics and are generally more difficult to solve.^[2]

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;^[3] this issue is commonly encountered innonlinear systems withhysteresis,^[4] such as theSchmitt trigger.^[5]

This difference is more clearly visible if the system may be rewritten so that instead ofx we consider a pair${\displaystyle (x,y)}$ of vectors of dependent variables and the DAE has the form

where${\displaystyle x(t)\in {\mathbb{R}}^n}$,${\displaystyle y(t)\in {\mathbb{R}}^m}$,${\displaystyle f:{\mathbb{R}}^{n+m+1}\to {\mathbb{R}}^n}$ and${\displaystyle g:{\mathbb{R}}^{n+m+1}\to {\mathbb{R}}^m.}$

A DAE system of this form is calledsemi-explicit.^[1] Every solution of the second halfg of the equation defines a unique direction forx via the first halff of the equations, while the direction fory is arbitrary. But not every point(x,y,t) is a solution ofg. The variables inx and the first halff of the equations get the attributedifferential. The components ofy and the second halfg of the equations are called thealgebraic variables or equations of the system. [The termalgebraic in the context of DAEs only meansfree 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 for this process is called thedifferentiation index. The equations derived in computing the index and consistent initial values may also be of use in the computation of the trajectory. A semi-explicit DAE system can be converted to an implicit one by decreasing the differentiation index by one, and vice versa.^[6]

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${\displaystyle (x,y)}$ and the system of differential equations of the DAE appears in the form

${\displaystyle F\left(\dot{x},x,y,t\right)=0}$

where

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

As a whole, the set of DAEs is a function

${\displaystyle F:{\mathbb{R}}^{(2n+m+1)}\to {\mathbb{R}}^{(n+m)}.}$

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

${\displaystyle F\left(\dot{x}(t_0),x(t_0),y(t_0),t_0\right)=0.}$

The behaviour of apendulum of lengthL with center in (0,0) in Cartesian coordinates (x,y) is described by theEuler–Lagrange equations

where${\displaystyle \lambda }$ is aLagrange multiplier. The momentum variablesu andv 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${\displaystyle L^2\dot{\lambda }-3gv=0}$ which implies the conservation of energy since after integration the constant${\displaystyle E=\frac{3}{2}gy-\frac{1}{2}{L}^2\lambda =\frac{1}{2}(u^2+v^2)+gy}$ 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${\displaystyle (x_0,u_0)}$ and a sign fory are given, the other variables are determined via${\displaystyle y=\pm \sqrt{L^2-x^2}}$, and if${\displaystyle y\ne 0}$ then${\displaystyle v=-ux/y}$ and${\displaystyle \lambda =(gy-u^2-v^2)/L^2}$. To proceed to the next point it is sufficient to get the derivatives ofx andu, 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${\displaystyle (y_0,v_0)}$ and a sign forx.

DAEs also naturally occur in the modelling of circuits with non-linear devices.Modified nodal analysis employing DAEs is used for example in the ubiquitousSPICE family of numeric circuit simulators.^[7] Similarly,Fraunhofer'sAnalog InsydesMathematica package can be used to derive DAEs from anetlist and then simplify or even solve the equations symbolically in some cases.^[8][9] It is worth noting that the index of a DAE (of a circuit) can be made arbitrarily high by cascading/coupling via capacitorsoperational amplifiers withpositive feedback.^[4]

DAE of the form

are called semi-explicit. The index-1 property requires thatg issolvable fory. In other words, the differentiation index is 1 if by differentiation of the algebraic equations fort an implicit ODE system results,

which is solvable for${\displaystyle (\dot{x},\dot{y})}$ if${\displaystyle det\left({\mathrm{\partial }}_{y}g(x,y,t)\right)\ne 0.}$

Every sufficiently smooth DAE is almost everywhere reducible to this semi-explicit index-1 form.

Two major problems in solving DAEs areindex reduction andconsistent initial conditions. Most numerical solvers requireordinary differential equations andalgebraic equations of the form

It is a non-trivial task to convert arbitrary DAE systems into ODEs for solution by pure ODE solvers. Techniques which can be employed includePantelides algorithm anddummy derivative index reduction method. Alternatively, a direct solution of high-index DAEs with inconsistent initial conditions is also possible. This solution approach involves a transformation of the derivative elements throughorthogonal collocation on finite elements ordirect transcription into algebraic expressions. This allows DAEs of any index to be solved without rearrangement in the open equation form

Once the model has been converted to algebraic equation form, it is solvable by large-scale nonlinear programming solvers (seeAPMonitor).

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Several measures of DAEs tractability in terms of numerical methods have developed, such asdifferentiation index,perturbation index,tractability index,geometric index, and theKronecker index.^[10][11]

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

For the pendulum DAE above, the variables are${\displaystyle (x_1,x_2,x_3,x_4,x_5)=(x,y,u,v,\lambda )}$. The corresponding signature matrix is

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

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

• Numerical analysis
• Differential calculus

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