Differential-algebraic system of equations

Short description: System of equations in mathematics

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Differential equations

Navier–Stokes differential equations used to simulate airflow around an obstruction.

Scope

* Natural sciences
* Engineering

* Astronomy
* Physics
* Chemistry

Biology
* Geology

Applied mathematics

* Continuum mechanics
* Chaos theory
* Dynamical systems

Social sciences

* Economics
* Population dynamics

Classification

Types

* Ordinary
* Partial
* Differential-algebraic
* Integro-differential
* Fractional
* Linear
* Non-linear

By variable type

* Dependent and independent variables

* Autonomous
* Complex
* Coupled / Decoupled
* Exact
* Homogeneous / Nonhomogeneous

Features

* Order
* Operator

* Notation

Relation to processes

* Difference (discrete analogue)

* Stochastic
* Stochastic partial
* Delay

Solution

General topics

* Picard–Lindelöf theorem
* Wronskian

* Phase portrait
* Phase space

* Lyapunov / Asymptotic / Exponential stability
* Rate of convergence
* Series / Integral solutions

* Numerical integration
* Dirac delta function

Solution methods

* Inspection
* Method of characteristics

Euler
* Exponential response formula
* Finite difference (Crank–Nicolson)
* Finite element
* Infinite element
* Finite volume
* Galerkin
* Petrov–Galerkin
* Integrating factor
* Integral transforms
* Perturbation theory
* Runge–Kutta

* Separation of variables
* Undetermined coefficients
* Variation of parameters

People

* Isaac Newton
* Gottfried Leibniz
* Leonhard Euler
* Émile Picard
* Józef Maria Hoene-Wroński
* Ernst Lindelöf
* Rudolf Lipschitz
* Augustin-Louis Cauchy
* John Crank
* Phyllis Nicolson
* Carl David Tolmé Runge
* Martin Kutta

* v
* t
* e

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

F(x˙,x,t)=0,

where x(t) is a vector of unknown functions and the overdot denotes the time derivative, i.e., x˙=dxdt.

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 ∂F(x˙,x,t)∂x˙ 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 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 (x,y) of vectors of dependent variables and the DAE has the form

x˙(t)=f(x(t),y(t),t),0=g(x(t),y(t),t).
where x(t)∈ℝn, y(t)∈ℝm, f:ℝn+m+1->ℝn and g:ℝn+m+1->ℝm.

A DAE system of this form is called semi-explicit.[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 semi-explicit DAE system can be converted to an implicit one by decreasing the differentiation index by one, and vice versa.[6]

## Contents

* 1 Other forms of DAEs
* 2 Examples
* 3 Semi-explicit DAE of index 1
* 4 Numerical treatment of DAE and applications
* 4.1 Tractability
* 5 Structural analysis for DAEs
* 6 See also
* 7 References
* 8 Further reading
* 8.1 Books
* 8.2 Various papers
* 9 External links

## 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 (x,y) and the system of differential equations of the DAE appears in the form

F(x˙,x,y,t)=0

where

* x, a vector in ℝn, are dependent variables for which derivatives are present (differential variables),
* y, a vector in ℝm, are dependent variables for which no derivatives are present (algebraic variables),
* t, a scalar (usually time) is an independent variable.
* F is a vector of n+m functions that involve subsets of these n+m+1 variables and n derivatives.

As a whole, the set of DAEs is a function

F:ℝ(2n+m+1)->ℝ(n+m).

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

F(x˙(t0),x(t0),y(t0),t0)=0.

## 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

x˙=u,y˙=v,u˙=λx,v˙=λy−g,x2+y2=L2,

where λ 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

x˙x+y˙y=0⇒ux+vy=0,

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

u˙x+v˙y+ux˙+vy˙=0,⇒λ(x2+y2)−gy+u2+v2=0,⇒L2λ−gy+u2+v2=0,

and the derivative of that last identity simplifies to L2λ˙−3gv=0 which implies the conservation of energy since after integration the constant E=32gy−12L2λ=12(u2+v2)+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 (x0,u0) and a sign for y are given, the other variables are determined via y=±L2−x2, and if y≠0 then v=−ux/y and λ=(gy−u2−v2)/L2. 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

x˙=u,u˙=λx,0=x2+y2−L2,0=ux+vy,0=u2−gy+v2+L2λ.

This is a semi-explicit DAE of index 1. Another set of similar equations may be obtained starting from (y0,v0) 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.[4]

## Semi-explicit DAE of index 1

DAE of the form

x˙=f(x,y,t),0=g(x,y,t).

are called semi-explicit. The index-1 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,

x˙=f(x,y,t)0=∂xg(x,y,t)x˙+∂yg(x,y,t)y˙+∂tg(x,y,t),

which is solvable for (x˙,y˙) if det⁡(∂yg(x,y,t))≠0.

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

## Numerical treatment of DAE and applications

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

dxdt=f(x,y,t),0=g(x,y,t).

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

0=f(dxdt,x,y,t),0=g(x,y,t).

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

### Tractability

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 Σ-method to analyze a DAE. We construct for the DAE a signature matrix Σ=(σi,j), where each row corresponds to each equation fi and each column corresponds to each variable xj. The entry in position (i,j) is σi,j, which denotes the highest order of derivative to which xj occurs in fi, or −∞ if xj does not occur in fi.

For the pendulum DAE above, the variables are (x1,x2,x3,x4,x5)=(x,y,u,v,λ). The corresponding signature matrix is

Σ=[1−0∙−−−1∙−0−0−1−0∙−0−1∙00∙0−−−]

## See also

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

## References

1. ↑ 1.0 1.1 Uri M. Ascher; Linda R. Petzold (1998). Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM. p. 12. ISBN 978-1-61197-139-2.
2. ↑ Achim Ilchmann; Timo Reis (2014). Surveys in Differential-Algebraic Equations II. Springer. pp. 104–105. ISBN 978-3-319-11050-9.
3. ↑ Renate Merker, ed (2001). System Design Automation: Fundamentals, Principles, Methods, Examples. Springer Science & Business Media. p. 221. ISBN 978-0-7923-7313-1. https://archive.org/details/systemdesignauto00teic.
4. ↑ 4.0 4.1 K. E. Brenan; S. L. Campbell; L. R. Petzold (1996). Numerical Solution of Initial-value Problems in Differential-algebraic Equations. SIAM. pp. 173–177. doi:10.1137/1.9781611971224. ISBN 978-1-61197-122-4.
5. ↑ Günther, M.; Feldmann, U.; Ter Maten, J. (2005). "Modelling and Discretization of Circuit Problems". Numerical Methods in Electromagnetics. Handbook of Numerical Analysis. 13. pp. 523. doi:10.1016/S1570-8659(04)13006-8. ISBN 978-0-444-51375-5. https://research.tue.nl/nl/publications/7e8db823-0664-4b2d-b03d-7c5bd5edb0b5. , pp. 529-531
6. ↑ Ascher and Petzold, p. 234
7. ↑ Ricardo Riaza (2013). "DAEs in Circuit Modelling: A Survey". in Achim Ilchmann. Surveys in Differential-Algebraic Equations I. Springer Science & Business Media. ISBN 978-3-642-34928-7.
8. ↑ 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. pp. 53. doi:10.1007/978-1-4020-6149-3_4. ISBN 978-1-4020-6147-9. https://infoscience.epfl.ch/record/108820/files/maehne_fdl06_book_modellib_chapter.pdf.
9. ↑ 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. pp. 215. doi:10.1007/978-3-642-23568-9_17. ISBN 978-3-642-23567-2.
10. ↑ Ricardo Riaza (2008). Differential-algebraic Systems: Analytical Aspects and Circuit Applications. World Scientific. pp. 5–8. ISBN 978-981-279-181-8. https://archive.org/details/differentialalge00riaz.
11. ↑ Takamatsu, Mizuyo; Iwata, Satoru (2008). "Index characterization of differential-algebraic equations in hybrid analysis for circuit simulation" (in en). International Journal of Circuit Theory and Applications 38 (4): 419–440. doi:10.1002/cta.577. http://www.ise.chuo-u.ac.jp/ise-labs/takamatsu-lab/takamatsu/metr/METR08-10.pdf. Retrieved 9 November 2022.

## Further reading

### Books

* K. E. Brenan; S. L. Campbell; L. R. Petzold (1996). Numerical Solution of Initial-value Problems in Differential-algebraic Equations. SIAM. pp. 173–177. doi:10.1137/1.9781611971224. ISBN 978-1-61197-122-4.
* Hairer, E.; Wanner, G. (1996). Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (2nd revised ed.). Berlin: Springer-Verlag.
* Ascher, Uri M. (1998). Computer Methods for Ordinary Differential equations and Differential-Algebraic equations. Philadelphia: SIAM. ISBN 978-0-89871-412-8.
* Kunkel, Peter; Mehrmann, Volker Ludwig (2006). Differential-algebraic equations: analysis and numerical solution. Zürich, Switzerland: European Mathematical Society. ISBN 978-3-03719-017-3. https://books.google.com/books?id=iRZPqCwkI_IC.
* Kazuo Murota (2009). Matrices and Matroids for Systems Analysis. Springer Science & Business Media. ISBN 978-3-642-03994-2. (Covers the structural approach to computing the DAE index.)
* Matthias Gerdts (2012). Optimal Control of ODEs and DAEs. Walter de Gruyter. ISBN 978-3-11-024999-6.
* Lamour, René (2013). Differential-Algebraic equations: a Projector based analysis. Heidelberg: Springer. ISBN 978-3-642-27554-8.

### Various papers

* G. Fábián; D.A. van Beek; J.E. Rooda (2001). "Index Reduction and Discontinuity Handling using Substitute Equations". Mathematical and Computer Modelling of Dynamical Systems 7 (2): 173–187. doi:10.1076/mcmd.7.2.173.3646. http://se.wtb.tue.nl/~vanbeek/pub/mcmds01.pdf.
* 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/s11075-005-9007-1.
* Nedialkov, Ned S.; Pryce, John D. (2005). "Solving Differential-Algebraic Equations by Taylor Series (I): Computing Taylor Coefficients". BIT 45 (3): 561–591. doi:10.1007/s10543-005-0019-y. http://www.cas.mcmaster.ca/~nedialk/PAPERS/DAEs/taylcoeff_I/taylorcoeffsdae.pdf.
* Nedialkov, Ned S.; Pryce, John D. (2005). "Solving Differential-Algebraic Equations by Taylor Series (II): Computing the System Jacobian". BIT 47: 121–135. doi:10.1007/s10543-006-0106-8. http://www.cas.mcmaster.ca/~nedialk/PAPERS/DAEs/taylcoeff_II/sysjac.pdf.
* Nedialkov, Ned S.; Pryce, John D. (2007). "Solving Differential-Algebraic Equations by Taylor Series (III): the DAETS Code". Journal of Numerical Analysis, Industrial and Applied Mathematics 1 (1): 1–30. ISSN 1790-8140. http://www.cas.mcmaster.ca/~nedialk/PAPERS/DAEs/daets/daets_color.pdf.
* Nedialkov, Ned S.; Pryce, John D.; Tan, Guangning (2014). "DAESA — a Matlab Tool for Structural Analysis of Differential-Algebraic Equations: Software". ACM Transactions on Mathematical Software 41 (2): 1–14. doi:10.1145/2700586. http://www.cas.mcmaster.ca/~nedialk/PAPERS/DAEs/daesa_software/daesaSoftware.pdf.
* Pryce, John D.; Nedialkov, Ned S.; Tan, Guangning (2014). "DAESA — a Matlab Tool for Structural Analysis of Differential-Algebraic Equations: Algorithm". ACM Transactions on Mathematical Software 41 (2): 1–20. doi:10.1145/2689664. http://www.cas.mcmaster.ca/~nedialk/PAPERS/DAEs/daesa_theory/daesaTheory.pdf.
* Roubíček, T.; Valášek, M. (2002). "Optimal control of causal differential algebraic systems". J. Math. Anal. Appl. 269 (2): 616–641. doi:10.1016/s0022-247x(02)00040-9.
* Estévez Schwarz, D.; März, R.; Lamour, R. (2025). "The Common Ground of DAE Approaches: An overview of diverse DAE frameworks emphasizing their commonalities". DAE Panel 3: 1-110. doi:10.52825/dae-p.v3i.2547. https://www.tib-op.org/ojs/index.php/dae-p/article/view/2547.

## External links

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

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* e

Differential equations

Classification|

| Operations|

* Differential operator
* Notation for differentiation
* Ordinary
* Partial
* Differential-algebraic
* Integro-differential
* Fractional
* Linear
* Non-linear
* Holonomic

|

Attributes of variables|

* Dependent and independent variables
* Homogeneous
* Nonhomogeneous
* Coupled
* Decoupled
* Order
* Degree
* Autonomous
* Exact differential equation
* Complex differential equation

Relation to processes|

* Difference (discrete analogue)
* Stochastic
* Stochastic partial
* Delay

Solutions|

| Solution topics|

* Picard–Lindelöf theorem (existence and uniqueness)
* Wronskian
* Phase portrait
* Phase space
* Lyapunov stability
* Asymptotic stability
* Exponential stability
* Rate of convergence
* Series solutions
* Integral solutions
* Numerical integration
* Dirac delta function

|

Solution methods|

* Inspection
* Separation of variables
* Method of undetermined coefficients
* Variation of parameters
* Integrating factor
* Integral transforms
* Euler method
* Finite difference method
* Crank–Nicolson method
* Runge–Kutta methods
* Finite element method
* Finite volume method
* Galerkin method
* Perturbation theory

Applications|

* List of named differential equations

Mathematicians|

* Isaac Newton
* Gottfried Wilhelm Leibniz
* Leonhard Euler
* Jacob Bernoulli
* Émile Picard
* Józef Maria Hoene-Wroński
* Ernst Lindelöf
* Rudolf Lipschitz
* Joseph-Louis Lagrange
* Augustin-Louis Cauchy
* John Crank
* Phyllis Nicolson
* Carl David Tolmé Runge
* Martin Kutta

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