In the univariate case, a DAE in the variable t can be written as a single equation of the form where
is a vector of unknown functions and the overdot denotes the time derivative, i.e.,
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
[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 in nonlinear systems with hysteresis,[4] such as the Schmitt trigger.
[5] This difference is more clearly visible if the system may be rewritten so that instead of x we consider a pair
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.
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 the differentiation index.
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
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 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 where
The momentum variables u and v should be constrained by the law of conservation of energy and their direction should point along the circle.
Differentiation of the last equation leads to restricting the direction of motion to the tangent of the circle.
which implies the conservation of energy since after integration the constant
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.
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. 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.
[7] 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.
[8][9] 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.
Every sufficiently smooth DAE is almost everywhere reducible to this semi-explicit index-1 form.
Two major problems in solving DAEs are index reduction and consistent initial conditions.
Most numerical solvers require ordinary differential equations and algebraic 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 include Pantelides algorithm and dummy 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 through orthogonal collocation on finite elements or direct 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 (see APMonitor).