# Questions tagged [finite-duration]

This tag is for questions of Finite-Duration Solutions to Differential Equations, which after an ending time by itself becomes zero forever after. For ordinary functions which have a starting and ending time, see [tag:piecewise-continuity], and if time is not the involved variable, search for [tag:compact-support]. Finite-Duration solution cannot be solutions of Linear ODE, since they fail uniqueness. Synonyms: [tag:finite-time], [tag:time-limited]

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### Are these equations "properly" defined differential equations? (finite duration solutions to diff. eqs.)

Are these equations properly defined differential equations? Modifications were made to a deleted question to re-focus it. I am trying to find out if there exists any exact/accurate/non-approximated ...
1 vote
143 views

### Does the Differential Topology/Geometry frameworks being able to model solutions to diff. eqs. that are Non-Smooth?

I don't have much knowledge about Differential Topology neither Differential Geometry, but working on this another question about solutions to differential equations, and someone recommend me to ...
191 views

### Is the solution to $\theta''+0.021\,\text{sgn}(\theta')\sqrt{|\theta'|}+0.02\sin(\theta)=0,\,\theta_0=\pi/2,\,\theta'_0=0$ of finite duration?

Is the solution to $\ddot{\theta}+0.021\,\text{sgn}(\dot{\theta})\sqrt{|\dot{\theta}|}+0.02\sin(\theta)=0,\,\,\theta(0)=\frac{\pi}{2},\,\dot{\theta}(0) = 0 \quad\text{(Eq. 1)}$ of finite duration? I ...
265 views

### Does Lipschitz-kind "non-scalar ODEs" and PDEs could stand having finite-duration solutions?

Does Lipschitz-kind "non-scalar ODEs" and PDEs could stand having finite-duration solutions? Intro Recently I have found on these papers by Vardia T. Haimo (1985) Finite Time Controllers ...
331 views

### Issues with the Fourier Transform of $f(t)=(1-t^2)^4$ on $[-1,\,1]$, should be analytical but looks like having a singularity with noise-like rippling

Issues with the Fourier Transform of $f(t)=(1-t^2)^4$ on $[-1,\,1]$, should be analytical but looks like having a singularity with noise-like rippling Intro I was trying to made a compact-supported ...
91 views

### Is $r(t) = \frac{1}{144}(T-t)^4\theta(T-t)$ a valid solution to $\ddot{r}=\sqrt{r},\,r(0)=\frac{T^4}{144}>0$? (with $\theta(t)$ the Heaviside step fn)

Is $r(t) = \frac{1}{144}(T-t)^4\theta(T-t)$ a valid solution to $\ddot{r}=\sqrt{r},\,r(0)=\frac{T^4}{144}>0$? (with $\theta(t)$ the Heaviside unitary step function) I am looking here for examples ...
1 vote
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1 vote
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### Non-smooth continuous and compact-supported 1-D functions with Fourier transform: Could they be defined through differential equations? Any examples?

Non-smooth continuous and compact-supported 1-D functions with Fourier transform: Could they be defined through differential equations? Any examples? Motivation I have learned recently here that non ...
441 views

### Can smooth ODE converge to its equilibrium in finite time?

Consider the following nonliner system: \begin{align} \dot{x}=f(x) \end{align} where $x\in\mathbb{R}^n$ and $f(x)\in\mathbb{R}^n$ is sufficiently smooth and Lipschitz in $x$. Then the system is ...
1 vote
519 views

### Examples of Finite-Duration solutions to Autonomous Ordinary Differential Equations ODEs?

Examples of Finite-Duration solutions to Autonomous Ordinary Differential Equations ODEs? Examples of the scalar versions: 1st order: $\dot{x} = F(x)$ 2nd order: $\ddot{x} = F(x,\dot{x})$ I have ...
125 views

### Could a continuous time-limited and absolute integrable function be the output of a causal continuous-time LTI system (Linear and Time-Invariant)??

Could a continuous time-limited and absolute integrable function be the output of a causal continuous-time Linear and Time-Invariant system (CT-LTI)?? I am trying to understand if there exists any ...
50 views

### Could non-smooth time-limited functions been Analytical?

Could non-smooth time-limited functions been Analytical? Please read the scenarios first I was reading about analytic functions definitions on Wiki and looks like some of its properties where ...
54 views

### Are there any continuous time-limited Linear and Time-Invariant (LIT) functions with unbounded derivative?

Are there any continuous time-limited Linear and Time-Invariant (LIT) functions with unbounded derivative? Let think about a continuous and time-limited function $q(t)$ that is representing a ...
141 views

### Is the rate of change of Finite-Duration Solutions always bounded?

Is the rate of change of Finite-Duration Solutions always bounded? (between times $[t_0;\,t_F]$) I have found recently a paper Finite Time Differential Equations (V. T. Haimo - 1985), where its proved ...
518 views

### Is the function $f(x)=\frac{\left(1-x^2+\sqrt{\left(1-x^2\right)^2}\right)}{2}e^{-\frac{x^2}{1-x^2}}$ a bump-function $\in C_c^\infty$? Diff. Eq.?

Is the function $f(x)=\frac{\left(1-x^2+\sqrt{\left(1-x^2\right)^2}\right)}{2}e^{-\frac{x^2}{1-x^2}}$ a bump-function $\in C_c^\infty$? Which autonomous differential equation it fulfill? (note it is ...
220 views

### What are the solutions for $y(t)\cdot\left(y'(t) + a\right)=-b\sin(t)$?

What are the solutions for $y(t)\cdot\left(y'(t) + a\right)=-b\sin(t)$? It could be proben that there exists some solutions? Are these solutions unique? and obviously, which are these solutions? (...
67 views

### For finite-duration continuous $f(t)$ with $\|f'(t)\|_\infty < \infty$: It is true $\|f'(t)\|_\infty \leq \frac{2\pi \|f'(t)\|_2^2}{\|f'(t)\|_1}$?

For finite-duration continuous $f(t)$ with bounded derivative $\|f'(t)\|_\infty < \infty$: It is true that $\|f'(t)\|_\infty \leq \frac{2\pi \|f'(t)\|_2^2}{\|f'(t)\|_1}$? I am looking for an upper ...