Differential algebraic equation
In mathematics, differentialalgebraic equations (DAEs) are a general form of (systems of) differential equations 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 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 is a singular 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 in 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 of vectors of dependent variables and the DAE has the form

 where , , and
A DAE system of this form is called semiexplicit.^{[1]} 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 for 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. A semiexplicit DAE system can be converted to an implicit one by decreasing the differentiation index by one, and vice versa.^{[6]}
Contents
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 equations
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 semiexplicit 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 nonlinear 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.^{[4]}
Semiexplicit DAE of index 1
DAE of the form
are called semiexplicit. The index1 property requires that g is solvable 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 semiexplicit index1 form.
Numerical treatment of DAE and applications
Two major problems in the solution of DAEs are index reduction and consistent initial conditions. Most numerical solvers require ordinary differential equations and algebraic equations of the form
It is a nontrivial 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 highindex 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 largescale nonlinear programming solvers (see APMonitor).
Tractability
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Several measures of DAEs tractability in terms of numerical methods have developed, such as differentiation index, perturbation index, tractability index, geometric index, and the Kronecker index.^{[10]}^{[11]}
Structural analysis for DAEs
We use the method to analyze a DAE. We construct for the DAE a signature matrix , where each row corresponds to each equation and each column corresponds to each variable . The entry in position is , which denotes the highest order of derivative to which occurs in , or if does not occur in .
For the pendulum DAE above, the variables are . The corresponding signature matrix is
See also
 Algebraic differential equation, a different concept despite the similar name
 Delay differential equation
 Partial differential algebraic equation
 Modelica Language
References
 ^ ^{a} ^{b} Uri M. Ascher; Linda R. Petzold (1998). Computer Methods for Ordinary Differential Equations and DifferentialAlgebraic Equations. SIAM. p. 12. ISBN 9781611971392.
 ^ Achim Ilchmann; Timo Reis (2014). Surveys in DifferentialAlgebraic Equations II. Springer. pp. 104–105. ISBN 9783319110509.
 ^ Renate Merker and Wolfgang Schwarz, eds. (2001). System Design Automation: Fundamentals, Principles, Methods, Examples. Springer Science & Business Media. p. 221. ISBN 9780792373131.
 ^ ^{a} ^{b} K. E. Brenan; S. L. Campbell; L. R. Petzold (1996). Numerical Solution of Initialvalue Problems in Differentialalgebraic Equations. SIAM. pp. 173–177. ISBN 9781611971224.
 ^ Günther, M.; Feldmann, U.; Ter Maten, J. (2005). "Modelling and Discretization of Circuit Problems". Numerical Methods in Electromagnetics. Handbook of Numerical Analysis. 13. p. 523. doi:10.1016/S15708659(04)130068. ISBN 9780444513755., pp. 529531
 ^ Ascher and Petzold, p. 234
 ^ Ricardo Riaza (2013). "DAEs in Circuit Modelling: A Survey". In Achim Ilchmann and Timo Reis. Surveys in DifferentialAlgebraic Equations I. Springer Science & Business Media. ISBN 9783642349287.
 ^ Platte, D.; Jing, S.; Sommer, R.; Barke, E. (2007). "Improving Efficiency and Robustness of Analog Behavioral Models". Advances in Design and Specification Languages for Embedded Systems. p. 53. doi:10.1007/9781402061493_4. ISBN 9781402061479.
 ^ Hauser, M.; Salzig, C.; Dreyer, A. (2011). "Fast and Robust Symbolic Model Order Reduction with Analog Insydes". Computer Algebra in Scientific Computing. Lecture Notes in Computer Science. 6885. p. 215. doi:10.1007/9783642235689_17. ISBN 9783642235672.
 ^ Ricardo Riaza (2008). Differentialalgebraic Systems: Analytical Aspects and Circuit Applications. World Scientific. pp. 5–8. ISBN 9789812791818.
 ^ http://www.ise.chuou.ac.jp/iselabs/takamatsulab/takamatsu/metr/METR0810.pdf
Further reading
Books
 Hairer, E.; Wanner, G. (1996). Solving Ordinary Differential Equations II: Stiff and DifferentialAlgebraic Problems (2nd revised ed.). Berlin: SpringerVerlag.
 Ascher, Uri M.; Petzold, Linda R. (1998). Computer Methods for Ordinary Differential equations and DifferentialAlgebraic equations. Philadelphia: SIAM. ISBN 9780898714128.
 Kunkel, Peter; Mehrmann, Volker Ludwig (2006). Differentialalgebraic equations: analysis and numerical solution. Zürich, Switzerland: European Mathematical Society. ISBN 9783037190173.
 Kazuo Murota (2009). Matrices and Matroids for Systems Analysis. Springer Science & Business Media. ISBN 9783642039942. (Covers the structural approach to computing the DAE index.)
 Matthias Gerdts (2012). Optimal Control of ODEs and DAEs. Walter de Gruyter. ISBN 9783110249996.
 Lamour, René; März, Roswitha; Tischendorf, Caren (2013). DifferentialAlgebraic equations: a Projector based analysis. Heidelberg: Springer. ISBN 9783642275548.
Various papers
 G. Fábián; D.A. van Beek; J.E. Rooda (2001). "Index Reduction and Discontinuity Handling using Substitute Equations" (PDF). Mathematical and Computer Modelling of Dynamical Systems. 7 (2): 173–187. doi:10.1076/mcmd.7.2.173.3646. Archived from the original (PDF) on 20050426.
 Ilie, Silvana; Corless, Robert M.; Reid, Greg (2006). "Numerical Solutions of Differential Algebraic Equations of Index −1 Can Be Computed in Polynomial Time". Numerical Algorithms. 41 (2): 161–171. doi:10.1007/s1107500590071.
 Nedialkov, Ned S.; Pryce, John D. (2005). "Solving DifferentialAlgebraic Equations by Taylor Series (I): Computing Taylor Coefficients" (PDF). BIT. 45 (3): 561–591. doi:10.1007/s105430050019y.
 Nedialkov, Ned S.; Pryce, John D. (2005). "Solving DifferentialAlgebraic Equations by Taylor Series (II): Computing the System Jacobian" (PDF). BIT. 47: 121–135. doi:10.1007/s1054300601068.
 Nedialkov, Ned S.; Pryce, John D. (2007). "Solving DifferentialAlgebraic Equations by Taylor Series (III): the DAETS Code" (PDF). Journal of Numerical Analysis, Industrial and Applied Mathematics (JNAIAM). 1 (1): 1–30. ISSN 17908140.
 Nedialkov, Ned S.; Pryce, John D.; Tan, Guangning (2014). "DAESA — a Matlab Tool for Structural Analysis of DifferentialAlgebraic Equations: Software" (PDF). ACM Transactions on Mathematical Software.
 Pryce, John D.; Nedialkov, Ned S.; Tan, Guangning (2014). "DAESA — a Matlab Tool for Structural Analysis of DifferentialAlgebraic Equations: Algorithm" (PDF). ACM Transactions on Mathematical Software.
 Roubíček, T.; Valášek, M. (2002). "Optimal control of causal differential algebraic systems". J. Math. Anal. Appl. 269 (2): 616–641. doi:10.1016/s0022247x(02)000409.
External links
 http://www.scholarpedia.org/article/Differentialalgebraic_equations