Questions tagged [differential-algebraic-equations]

A differential-algebraic system of equations (DAEs) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system.

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a question on visualizing a state of DAE and a question from a continous time nonlinear dynamics

Could someone explain me what is the fundamental difference between the dynamical system of the kind $\dot x = f(x)$ and $E \dot x= f(x)$ where $E$ is a singular matrix with real entries. For the ...
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Constants of universal derivation on an $R$-algebra $A$ under localization ($d: S^{-1}A \to S^{-1}\Omega_{A/R}$)

Suppose that $A$ is an $R$-algebra, with both $R, A$ integral domains. Let $B = S^{-1}A$ be a localization of $A$. Let $$\partial_A: A \to M$$ $$\partial_B: B \to S^{-1}M$$ be an $R$-derivation on $A$ ...
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Index reduction of a DAE from a PDE system

I have a system of 2 non-linear, coupled PDEs that I would like to transform to a stiff ODE system to solve them using the method of lines. The 2 equations: $$\begin{align} \frac{\partial \phi}{\...
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Is there a stability theorem akin to Lyapunov's direct or indirect method for differential algebraic equations (DAE)?

Is there a stability theorem akin to Lyapunov's direct or indirect method for differential algebraic equations (DAE)? Specifically, is there a stability theorem that does not require transforming the ...
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Differential Algebraic Equation Question

This isn't a homework question, just something I'm trying to learn. Consider the differential equation $\dot{y}=f(y)$ with known invariants $g(y)=$ Const , and assume that $g^{\prime}(y)$ has full ...
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Classifying and solving a nonlinear differential algebraic equation (DAE)

Suppose that $x,y$ are functions of time and I'm considering the dynamics in a problem with equations as below.$$ \alpha\ddot{z} - k_{1}z - k_2y=0 \, \text{ (differential equation)}$$ $$ k_1 y + k_3 \...
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Implicit Euler integration scheme for DAEs

Consider following DAE \begin{equation} \begin{bmatrix} a_{11} & a_{12} & a_{13} & 0\\ a_{21} & a_{22} & a_{23} & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}\frac{\mathrm{d}}{\...
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(Approximate) Explicit solutions for mass matrix DAEs with nil-potent linear dynamics

I am writing a package to solve linear differential equations of the form $\frac{d\vec{x}}{dt} = A\vec{x}$ where $A$ is nil potent. The knowledge of the nil potency guarantees that the matrix ...
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The relation between $\ker(E)$ and the differential-algebraic equation $E\mathbf x'(t)=Ax(t)+f(t)$

A differential-algebraic equation is, loosely speaking, an equation of the form $$E\mathbf x'(t)=Ax(t)+f(t)$$ where $E,A\in\mathbb C^{n\times n}$, $t\in[0,\infty)$ and $f:[0,\infty)\to\mathbb C^n$, ...
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On an example for consistent initial values in differential-algebraic equations

I am studying differential-algebraic equations, that is, loosely speaking, equations of the form $$E\mathbf x'(t)=Ax(t)+f(t)$$ where $E,A\in\mathbb C^{n\times n}$, $t\in[0,\infty)$ and $f:[0,\infty)\...
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Confusion about Big O-Notation in convergence proof for differential algebraic equations

I understand almost all of this proof. But the part I can't find an explanation for is how they get the result for $r$ with the sums over the $\mathcal{O}(h^{\nu})$ all the way at the end of the proof....
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Properties of blocks in blockmatrix A if the matrix pair (E,A) is regular

I am working on a problem from the book "Differential-Algebraic Equations: Analysis and Numerical Solution" by Kunkel, Mehrmann. It is the Exercise 3 from Page 53 concerning matrix pairs $(E,A)$ and ...
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The differential equation $u'\cdot v + A\cdot u = w$

Let $A\colon \mathbb R^n\to\mathbb R^{n\times n}$ be a matrix valued function, and $v, w\colon \mathbb R^n\to\mathbb R^n$ vector fields. I want to find a function/vector field $u\colon\mathbb R^n\to\...
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Matlab's DAE example

I am a newcomer to working with differential algebraic equations and I am trying to work my way through Matlab's pendulum DAE system example: https://www.mathworks.com/help/symbolic/solve-differential-...
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