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# 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 ...
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1 vote
1 answer
141 views

### How to "formally" prove that $x(t)=\frac{1}{4}(2-t)^2\theta(2-t)$ solves $\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$?

How to "formally" prove that $x(t)=\frac{1}{4}(2-t)^2\theta(2-t)$ solves $\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$? (with $\theta(t)$ the standard unitary step function). I have found the ...
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2 votes
2 answers
519 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 ...
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1 vote
2 answers
520 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 ...
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0 votes
1 answer
201 views

### 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 ...
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0 votes
1 answer
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### 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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6 votes
1 answer
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 ...
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6 votes
2 answers
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 ...
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1 vote
1 answer
130 views

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1 vote
0 answers
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 ...
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0 votes
1 answer
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 ...
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