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]

Filter by
Sorted by
Tagged with
1 vote
0 answers
126 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 ...
user avatar
  • 997
25 votes
2 answers
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 ...
user avatar
  • 997
6 votes
2 answers
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 ...
user avatar
  • 997
7 votes
1 answer
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 ...
user avatar
  • 997
0 votes
1 answer
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 ...
user avatar
  • 997
1 vote
1 answer
94 views

Does $x(t) = \exp\left(\frac{t}{t-1}\right)\cdot\theta(1-t)$ solve $\dot{x}\cdot(1-t)^2+x=0,\,\,x(0)=1$?

Does $x(t) = \exp\left(\frac{t}{t-1}\right)\cdot\theta(1-t)$ solve $\dot{x}\cdot(1-t)^2+x=0,\,\,x(0)=1$ with $\theta(t)$ the standard unitary step/Heaviside function $$\theta(t) := \begin{cases} 0 &...
user avatar
  • 997
1 vote
1 answer
87 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 ...
user avatar
  • 997
5 votes
0 answers
163 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 ...
user avatar
  • 997
0 votes
1 answer
91 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 ...
user avatar
  • 997
1 vote
2 answers
188 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 ...
user avatar
  • 997
2 votes
2 answers
315 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 ...
user avatar
  • 997
5 votes
0 answers
194 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? (...
user avatar
  • 997
4 votes
0 answers
271 views

It is possible to find a solution to $y''+\sqrt{|y|}\operatorname{sgn}(y)+\sqrt{|y'|}\operatorname{sgn}(y')=0,$ $\,y'(0)=0,\,y(0)= 1/4$?

It is possible to find an exact solution (hopefully in "close form") to $$y''+\sqrt{|y|}\operatorname{sgn}(y)+\sqrt{|y'|}\operatorname{sgn}(y')=0, \,y'(0)=0,\,y(0)= 1/4$$? How?... There ...
user avatar
  • 997
0 votes
1 answer
81 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 ...
user avatar
  • 997
1 vote
1 answer
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 ...
user avatar
  • 997
0 votes
1 answer
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 ...
user avatar
  • 997
2 votes
0 answers
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 ...
user avatar
  • 997
2 votes
2 answers
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 ...
user avatar
  • 997
2 votes
1 answer
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 ...
user avatar
  • 997
0 votes
1 answer
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 ...
user avatar
  • 997
3 votes
1 answer
125 views

It is possible for a scalar finite-duration continuous system to achieve an infinite speed (in finite-time)? How if true? Why not if false?

It is possible for a scalar finite-duration continuous system to achieve an infinite speed (in finite-time)? How if it true? Why not if it false? (Please read first the restrictions of the system I am ...
user avatar
  • 997
2 votes
1 answer
139 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 ...
user avatar
  • 127