Talk:Differential algebraic equation

From Wikipedia, the free encyclopedia
WikiProject Mathematics (Rated Stub-class, Low-importance)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
Stub Class
Low Importance
 Field:  Analysis

First equation

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

Modelica is not a software

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 (talk) 20:32, 11 September 2007 (UTC)

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

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)


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 (talkcontribs) 20:23, 29 December 2008 (UTC)

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)


This article says:

In mathematics, differential algebraic equations (DAEs) are a general form of differential equation, given in implicit form. They can be written
  • , a vector in , are variables for which derivatives are present (differential variables),
  • , a vector in , are variables for which no derivatives are present (algebraic variables),
  • , 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)

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)
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
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)

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)

Yes, but that is not very helpful, since the solution theory of ODEs is much simpler. One can think of DAEs as implicit ODEs where the part of the Jacobian of related to the variables does not have a full rank, and moreover has a nullspace 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 where is decreasing the dimension, that is, m < n.--LutzL (talk) 08:58, 22 October 2009 (UTC)

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)

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 containing coordinates and their time derivatives, . Unlike an ODE system, a DAE system is not (uniquely) solvable for the first derivatives .
  • 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. . This system does not contain but is usually supposed to be (locally) solvable for .
  • Even more explicit is the form , 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: Solution contains , it requires at least the second derivative of the equations.
  • Example pendulum in Laplace formulation 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 , with improper initial values leads to nonsensical paths.(?)
--LutzL (talk) 14:01, 26 October 2009 (UTC)
LutzL, having read your explanation, I have added a clarification to the article's lead. Charvest (talk) 21:11, 26 October 2009 (UTC)

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)

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. (talk) 05:46, 14 November 2009 (UTC)
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)

It might be a good idea to add an external link to scholarpedia which have an excellent article on the subject. — Preceding unsigned comment added by (talk) 12:35, 8 July 2011 (UTC)


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. (talk) 21:35, 16 December 2014 (UTC)

External links modified

Hello fellow Wikipedians,

I have just modified one external link on Differential algebraic equation. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

  • Added archive to

When you have finished reviewing my changes, please set the checked parameter below to true or failed to let others know (documentation at {{Sourcecheck}}).

You may set the |checked=, on this template, to true or failed to let other editors know you reviewed the change. If you find any errors, please use the tools below to fix them or call an editor by setting |needhelp= to your help request.

  • If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
  • If you found an error with any archives or the URLs themselves, you can fix them with this tool.

If you are unable to use these tools, you may set |needhelp=<your help request> on this template to request help from an experienced user. Please include details about your problem, to help other editors.

Cheers.—InternetArchiveBot (Report bug) 00:04, 13 December 2016 (UTC)

Retrieved from ""
This content was retrieved from Wikipedia :
This page is based on the copyrighted Wikipedia article "Talk:Differential algebraic equation"; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA