# 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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### 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 ...
560 views

### 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 ...
198 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 ...
177 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 ...
67 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 ...
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### It is possible for a (electromagnetic) wave equation to have as a solution a finite-duration/compact-supported function? Any closed-form examples?

A) It is possible for a wave equation to have as a solution a finite-duration function? Any closed-form example? (please share the specific wave equation with its finite-duration solution, showing how ...
1 vote
65 views

### 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 ...
46 views

### It is possible to (continuous-time) finite-duration continuous systems to be linear?

It is possible to (continuous-time) finite-duration continuous systems to be linear? I am trying to understand which effects are introduced in continuous-time systems described by continuous functions ...
57 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 ...
58 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 ...
33 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 ...
42 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 ...
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 ...