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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A brick sliding in an horizontal plane after an initial push (under Coulomb's dry friction) - closed form solutions validation?
A brick sliding in an horizontal plane after an initial push (under Coulomb's dry friction) - closed form solutions validation?
Posted later after comments: In summary, I am trying to understand what ...
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How to force my differential equations give a bounded solution?
I have modeled the interaction of two physical quantities, $S$ and $B$, by the following differential equations (the second one is a delay differential equation):
$$S'(t) = 0.31 S(t) \Big( 1 - \frac{S(...
user1264437
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Solving $x'=-\text{sgn}(x)\sqrt{|x|}$: Uniqueness of solutions of finite duration
Show that $$x(t) = \frac{\text{sgn}(x(0))}{4}\left(2\sqrt{|x(0)|}-t\right)^2\cdot\theta\!\left(2\sqrt{|x(0)|}-t\right)$$ is a solution to$$x'=-\text{sgn}(x)\sqrt{|x|}.$$
Is the solution unique?
This ...
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Solving $x''+x+\text{sgn}(x')\sqrt{|x'|} = 0\ $ Does it have closed form solutions? Does it stop moving? Could it stop at a different place than zero?
Solving $x''+x+\text{sgn}(x')\sqrt{|x'|} = 0\ $
Does it have closed form solutions? Please show how you got them.
Does it stop moving? There exists a finite extinction time $|T|<\infty$ such $x'(t)...
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Closed-form solutions to $x''+\frac{k}{m}\ x+\mu\ g\ \text{sgn}(x')=0$
Closed-form solutions to $x''+\frac{k}{m}\ x+\mu\ g\ \text{sgn}(x')=0$
Introduction______________________
I am looking for simple mechanics models that could have closed-form solutions that achieves ...
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Proving that these solutions are formally solving these differential equations: $x'' = -\text{sgn}(x')$ and $y'' = \sqrt{|y'|}$
Please take a look also to the comments section, here, and in other people answers, since there are extended what are my apprehensions about the validity of the found answers.
I have found these two ...
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Confusion with the Fourier Transform and Complex Differentiability: example with compact-supported function
I have a misconception when applying the Fourier Transform to a compacted-supported function and the characteristics of the function obtained.
Intro
I am going to list what I believe is true so you ...
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Find the extinction time $T$ for the solutions $\ddot{y}+a\,\text{sgn}(\dot{y})\sqrt{|\dot{y}|}(1+|\dot{y}|^{3/2})+b\,\sin(y)=0,\,y(0)>0,\,y'(0)=0$?
For the differential equation with two real-valued positive parameters $\{a,\,b\}>0$:
$$\ddot{y}+a\cdot\text{sgn}(\dot{y})\sqrt{|\dot{y}|}\left(1+|\dot{y}|^{\frac{3}{2}}\right)+b\cdot\sin(y)=0,\,y(...
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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 ...
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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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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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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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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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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 &...
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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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