Talk:Differential-algebraic system of equations

This Article Should be Refactored[edit]

So this article is clearly directed at engineers and the style in which it is written is guiding engineers away from a body of literature available to them in mathematics (namely foliation theory and differential algebra). These theorems and constructions could potentially be helpful in their investigations.

In particular a DAE in the sense of this article is just a collection of differential polynomials as polynomials are naturally included in the term ``differential polynomial. From here there are two natural ways to construct objects that people are interested in: via differential algebraic varieties, and via foliations (or more precisely exterior differential systems).

I'm not sure what to do but there is a language problem. There shouldn't exist distinct article from algebraic differential equation and differential algebraic equation because it only serves to separate and confuse people who are searching for information that could be useful to them. — Preceding unsigned comment added by DupuyTaylor (talk • contribs) 01:45, 9 December 2022 (UTC)[reply]

First equation[edit]

Istn't the first equation not a simple implicit ODE? — Preceding unsigned comment added by 85.178.223.187 (talk) 11:22, 27 January 2014 (UTC)[reply]

Modelica is not a software[edit]

Modelica is not some software but a modeling language. There are some programs which use this language such as SimulationX or Dymola. —Preceding unsigned comment added by 84.179.9.10 (talk) 20:32, 11 September 2007 (UTC)[reply]

Could someone please explain what an index is of a DAE system? —Preceding unsigned comment added by 152.1.164.114 (talk) 19:05, 30 May 2008 (UTC)[reply]

It's essentially the number of times you have to differentiate some of the equations in the system until you can extract an ODE. This is the differentiation index and most of the times the perturbation index. For the numerical treatment, also index 1 systems with explicit separation of algebraic and differental equations are fine. This is what the strangeness index and tractability index are about.--LutzL (talk) 18:47, 31 May 2008 (UTC)[reply]

Mathematica[edit]

Shouldn't Mathematica be included with the other programs? I am fairly certain it can easily manipulate DAEs. Either way, just a suggestion. —Preceding unsigned comment added by PromisedProgress (talk • contribs) 20:23, 29 December 2008 (UTC)[reply]

Could you please explain what capabilities you had in mind? Surely there will be numerical routines for index 1 and 2 systems. For general higher index systems, even the experts aren't exactly sure what a sensible symbolic manipulation of DAEs should consist of. And one needs some symbolic, index-reducing pre-processing, since numerics only works well for index 1 systems.--LutzL (talk) 14:32, 30 December 2008 (UTC)[reply]

Huh?[edit]

This article says:

In mathematics, differential algebraic equations (DAEs) are a general form of differential equation, given in implicit form. They can be written

f\left({\frac {dx}{dt}},x,y,t\right)=0

where

$x$ , a vector in $R^{n}$ , are variables for which derivatives are present (differential variables),
$y$ , a vector in $R^{m}$ , are variables for which no derivatives are present (algebraic variables),
$t$ , a scalar (usually time) is an independent variable.

I was expecting next to see the most important part:

ƒ is [.....]

My guess was that ƒ was to be a polynomial function and that justified the word "algebraic". Or maybe ƒ is a function defined implicitly by polynomial relations. Or something.

Michael Hardy (talk) 04:02, 20 October 2009 (UTC)[reply]

No, in this context, algebraic refers to not containing derivatives. Seems to be a historical numerics thing.--LutzL (talk) 07:39, 20 October 2009 (UTC)[reply]

I have no idea how to modify the article in a small way. The german version has more contents, but leans to the so called tractability index, leaving out, for example, the strangeness and perturbation index.---The most simple DAE has the semi-explicit form

{\begin{aligned}{\dot {x}}&=f(x,y)\\0&=g(x,y)\end{aligned}}

with g solvable for y. Then f is the "differential" part and g is the "algebraic" part. Every sufficiently smooth DAE is almost everywhere reducible to this form. The reduction may include differentiation of some equations and introduction of additional variables for the higher derivatives.--LutzL (talk) 07:48, 20 October 2009 (UTC)[reply]

If the article is correct as it stands, then isn't every differential equation a "differential algebraic equation"? Michael Hardy (talk) 21:33, 21 October 2009 (UTC)[reply]

Yes, but that is not very helpful, since the solution theory of ODEs is much simpler. One can think of DAEs as implicit ODEs

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

where the part of the Jacobian of

F_{p}(p,x,t)={\tfrac {\partial F(p,x,t)}{\partial p}}

related to the variables

p

does not have a full rank, and moreover has a nullspace

{\text{kern}}F_{p}(p,x,t)

that is at least continuous in some open set (that is, there is a continuous basis of the nullspaces). One typical occurrence of DAEs in nonlinear circuits takes the form

0=F({\tfrac {d}{dt}}D(x,t),\,x,t)

where

D:\mathbb {R} ^{n}\to \mathbb {R} ^{m}

is decreasing the dimension, that is, m < n.--LutzL (talk) 08:58, 22 October 2009 (UTC)[reply]

How could the solution theory of ODEs be simpler than that of DAEs if every ODE is a DAE? That doesn't make any sense at all. Why don't you say what you really mean, whatever that is? I'm asking what the difference is, if any between an ODE and a DAE. And you haven't attempted to say that.

If not having full rank is the essence of the matter, then why is that omitted from the article entirely?

Please don't be abusive toward your readers. Either answer the question or refrain from pretending to answer it. Michael Hardy (talk) 06:27, 24 October 2009 (UTC)[reply]

Sorry, I don't own this article. Please don't be so accusative towards voluntary editors.---To the matter at hand, the confusion is hopefully restricted to the talk page, and that is its purpose, to work out such misunderstandings. My error was to use ODE in two different meanings. When saying "DAEs are implicit ODEs with rank deficient jacobian", then I meant ODE as opposed to PDE, an equation system for a path containing coordinates and their time derivatives as inputs. Later I used ODE (of order 1) as a system of equations that defines, explicitly or implicitly, a velocity field. As such, an ODE would be a trivial DAE of index 0. So there are two inverse directions of specialization. For this article, the second variant would be more appropriate, and where the first is needed, it should be expressed as in the reformulation. A much longer introduction could be:

Like an ODE system, a system of DAEs consists of equations for a differentiable path $x:[a,b]\to \mathbb {R} ^{n}$ containing coordinates and their time derivatives, $F({\dot {x}},x,t)$ . Unlike an ODE system, a DAE system is not (uniquely) solvable for the first derivatives ${\dot {x}}$ .
A more explicit form of a DAE separates the variables in those that occur with their first derivatives, the "differential" variables x, and the derivative free ones, the "algebraic" variables y. $F({\dot {x}},x,y,t)$ . This system does not contain ${\dot {y}}$ but is usually supposed to be (locally) solvable for ${\dot {x}}$ .
Even more explicit is the form ${\dot {x}}=f(x,y,t),\quad 0=g(x,y,t)$ , where the equations with f are called "differential", with g "algebraic". If g is locally solvable for y, it is called semi-explicit of index 1. This is the most friendly form for numerical solvers.
Example for difference of DAEs to ODEs: $x_{1}+{\dot {x}}_{2}=f_{1}(t),\;x_{2}+{\dot {x}}_{3}=f_{2}(t),\;x_{3}=f_{3}(t).$ Solution contains $x_{1}(t)=f_{1}(t)-{\dot {f}}_{2}(t)+{\ddot {f}}_{3}(t)$ , it requires at least the second derivative of the equations.
Example pendulum in Laplace formulation $m{\ddot {x}}=-\lambda x,\;m{\ddot {y}}=-\lambda y-g,\;x^{2}+y^{2}=L^{2}$ contains no equation for the first derivatives, but they are constraint by conservation of energy. The conservation law follows from the second derivative of the last equation. Brute force discretisation of the original system, as in $F({\tfrac {x_{n+1}-x_{n}}{h}},\,{\tfrac {x_{n+1}+x_{n}}{2}},t+{\tfrac {h}{2}})$ , with improper initial values leads to nonsensical paths.(?)

--LutzL (talk) 14:01, 26 October 2009 (UTC)[reply]

LutzL, having read your explanation, I have added a clarification to the article's lead. Charvest (talk) 21:11, 26 October 2009 (UTC)[reply]

Seems that after all I own the article for the moment. I started with the above points for the introduction. I'm not sure if the more special forms belong into the introduction or to sections on their theoretical or numerical treatment. I'll try to incorporate information on numerical methods from an overview talk by Linda Petzold (1995) and other resources such as this tech report.--LutzL (talk) 16:13, 13 November 2009 (UTC)[reply]

The key thing about DAEs is that, compared to systems of ODEs, they tightly incorporate purely algebraic variables in their midst, and even some equations which may involve no derivatives at all. This is particularly relevant to the definition of numerical methods used to solve such systems, because efficiency gains are made by converging the solution for the algebraic and differential variables at the same time, using a single common process. 124.168.195.132 (talk) 05:46, 14 November 2009 (UTC)[reply]

Yeah, but that was one point of discussion above: what, in this context, is meant with "purely algebraic"? That is, is there a better explanation than "no derivatives"? And in the end, ODEs are DAEs of index 0, so the distinction is not that sharp.---In a separated system one can of course solve the algebraic equations to any desired precision. But how do you "converge" the differential part? Especially considering hidden constraints? Which numerical methods should occur in the article in greater length, BDF, DASSL,...,RADAU? Pryce? Mehrmann/Kunkel, März/etal.?--LutzL (talk) 10:25, 14 November 2009 (UTC)[reply]

It might be a good idea to add an external link to scholarpedia which have an excellent article on the subject.http://www.scholarpedia.org/article/Differential-algebraic_equations — Preceding unsigned comment added by 130.237.43.79 (talk) 12:35, 8 July 2011 (UTC)[reply]

Title[edit]

This isn't worth fretting much about, but the more common name in sources appears to use a dash as in "differential-algebraic equations". Also the title should probably use the plural "equations" because in order for the definition to be interesting you need to have a mixture of algebraic and differential eq., i.e. have at least two equations in the system. 86.121.137.150 (talk) 21:35, 16 December 2014 (UTC)[reply]

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Examples[edit]

In the pendulum example, the last equation (algebraic constraint) is derived 3 times to get an explicit expression for each dependent variable (i.e., if I understood correctly, an ODEs system - please somebody clarify this in the Introduction, when talking about differences between ODEs and DAEs). Hence, a differentiation index of 3 is obtained. However, right below, it is written that the DAE has index 1. Can somebody please clarify this part? Are possibly those separate indexes (not properly defined in that case)? — Preceding unsigned comment added by 131.175.28.194 (talk) 13:29, 17 May 2018 (UTC)[reply]

Structural analysis for DAEs[edit]

A method called Sigma-method is presented to analyze DAEs. However, no conclusions are drawn from this analysis. How is the matrix Sigma useful/informative? Also, the balck "dot" which appears on top right of some numbers is not a well-defined symbol, and somebody should please clarify its meaning. — Preceding unsigned comment added by 131.175.28.194 (talk) 13:28, 18 May 2018 (UTC)[reply]

Shortening lead section[edit]

The article currently has a warning suggesting that the lead section is too long. Perhaps the discussion about the distinction between DAEs and ODEs could be moved to its own section under the heading "Distinction between DAEs and ODEs"? Zor Quatre (talk) 20:01, 9 June 2023 (UTC)[reply]