Differential algebra and differential-algebraic equations

Could you give me some information about differential algebra? What is it about?

Differential-algebraic equations (DAEs) are polynomials with complex coefficients and the unknown variables are $z, x, x'$.

Is this correct?

What is the difference between them and the ODEs?

Two possible solutions of DAEs $C^{\infty}$ functions and formal power series, right?



EDIT:

Does this mean that Differential algebra is about differential equations over a field or a ring?

These are two distinct and not very related concepts (though not completely unrelated).

Differential algebra is the study of differential rings and fields and related structures. Let me briefly mention some things about differential fields to give you some idea about what differential algebra is about.

A differential field is a field $\mathbb F$ together with a function $\partial : \mathbb F \to \mathbb F$, called a derivation, that satisfies the product rule: $$\partial(xy) = \partial(x)y + x\partial(y)$$ An element $x \in \mathbb F$ is called a constant of the differential field if $$\partial(x) = 0.$$ The set of all constants form a subfield of the differential field.

An example of a differential ring is $\mathbb R(t)$, the field of rational functions in $t$ over $\mathbb R$, with the derivation $\frac{d}{dt}$, differentiation with respect to $t$. The constants of $\mathbb R(t)$ is $\mathbb R$.

Elements of differential algebra are used in e.g. differential Galois theory and symbolic integration.

A differential algebraic system of equations is a system of equations where some equations are algebraic equations and some are differential equations. The equations need not be polynomial. I say system of equations, because if it is not a system of equations, i.e. there is only one equation, it will either be purely algebraic or differential.

An example of a DAE system is the equations describing the motions of a planar pendulum, having position $(x,y)$, velocity $(u,v)$, all functions of time $t$, with length $L$: \begin{align} \dot x &= u \\ \dot y &= v \\ \dot u &= \lambda x \\ \dot v &= \lambda y - g \\ L^2 &= x^2 + y^2 \end{align} as you can see, the last equation is algebraic and not differential.

I have explained the difference between an ODE and a DAE here: What is the difference between an implicit ordinary differential equation and a differential algebraic equation?

As an aside, your description of DAEs ("polynomials with complex coefficients and the unknown variables are $z,x,x'$") got me thinking of holonomic functions, and while they are not exactly what you described, they do come close. A holonomic function $y(t)$ is a function that satisfies $$a_r(t) y^{(r)}(t) + a_{r-1}(t) y^{(r-1)}(t) + \dots a_1(t)y'(t) + a_0(t)y(t) = 0$$ where each $a_i(t)$ is a polynomial in $t$.

For a little on differential algebra, see https://en.wikipedia.org/wiki/Differential_algebra ... probably for a full understanding, you need to consult one of the books cited.

Differential algebraic equation (a term I had not heard before) is here https://en.wikipedia.org/wiki/Differential_algebraic_equation ...

Are there some points on these pages that are confusing to you?

• About differential algebra I found also this link: mathworld.wolfram.com/DifferentialAlgebra.html Does this mean that Differential algebra is about differential equations over a field or a ring? – Mary Star Aug 26 '15 at 20:28
• Differential algebra uses an algebraic structure that has not only the usuall operations, but also a "derivative" operation. (Or even more than one.) So it will model a situation where you have a space of $C^\infty$ functions with the usual operations of addition, multiplication, and differentiation. One common topic is something like Galois theory, where you want to study whether solution of a DE is possible in terms of certain simpler solutions (such as quadratures, or elementary functions). – GEdgar Aug 27 '15 at 15:16

Differential-algebraic equations are important for mathematical modeling and scientific computation. If you write down the mathematical laws for some chemical, electrical, or physical system, you often will just end up with a system of equations involving parameters, various partial derivatives and purely algebraic quantities. Maybe you also get some equations involving integrals.

Now equations involving partial derivatives quickly get challenging for the available theory and numerical methods, and general integral equations are also not exactly easy to solve (both for the available theory and numerical methods). Those complicated (partial/integral) equations do arise all the time, and one can often still solve, simulate or optimize them, but not fully automatic.

But if no partial derivatives and no integrals are there, then one is in a situation where theory and numerical methods are available. The equation systems are still slightly more complicated than systems of ordinary differential equations, but the theory and the numerical methods are able to cope with this. The basic way to imagine this complication over an ordinary differential equation is to imagine that some purely algebraic equations describe a path that the solution must follow, and some purely algebraic quantities serve as control signals whose time evolution has to ensure that the solution follows the path described by the purely algebraic equations.

• Thomas Klimpel, am I reading your profile correctly if I read that you have done research on the relation between the structural properties of DAEs and symmetries? That sounds fascinating, any tips where I can read more about it? – Calle Sep 1 '15 at 19:58
• @Calle If you speak German, then section 3.2.1 in mod_geom_betr_klas_diffop.pdf gives a good idea for a relation between symmtries and structural properties, i.e. coupling or non-zero structure. Here are visualizations of non-zero structure of DAEs in integral form and non-zero structure of DAEs in semi-explicit form. See "Matrices and Matroids for System Analysis" – Thomas Klimpel Sep 1 '15 at 22:10
• thank you. It's been a while since I've used my German, but since it is mathematics I think I will manage. – Calle Sep 3 '15 at 18:34

as a PhD of DAE I want you to read these 3 books in this order:

1. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations (Classics in Applied Mathematics) 2nd Edition by K. E. Brenan (Author), S. L. Campbell (Author), L. R. Petzold (Author)

1. Singular Systems of Differential Equations Vol. I and II, by S. L. Campbell (Author)
• Although it is just the beginning. find me here: linkedin.com/in/mohammad-golchian-0a4146a8 Good Luck – Mohammad Golchian Aug 8 '19 at 22:55
• Welcome to MSE, and thanks for providing an answer. I think that perhaps you meant to say "As a PhD in DAE, I suggest you read these three books in order." As an answer to an MSE question, that's asking a great deal of the person asking the question, who was hoping to read a few paragraphs, not hundreds of pages. But it may be useful to some later reader of this question who really wants a pointer to a more thorough set of references. One last thought: I found myself wondering "where's the 3rd book?" until I saw "Vol 1 and II". You might have labelled this one "2. and 3." to make it clear. – John Hughes Aug 8 '19 at 23:38