Differential algebra and differential-algebraic equations

Question

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?

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EDIT:

About differential algebra I found also this link.

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

Answer 1

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?

Answer 2

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 $t$.

Answer 3

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.