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8 votes

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

Computation by hand of the Fourier transform First, recall that (with your convention for the Fourier transform), $$ \mathcal F[\theta(1-t^2)](w) = \mathcal F \unicode{x1D7D9}_{[-1,1]}(w)= 2 \...
Calvin Khor's user avatar
6 votes
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Closed-form solutions to $x''+\frac{k}{m}\ x+\mu\ g\ \text{sgn}(x')=0$

Let's define the natural frequency $\omega = \sqrt{k/m}$ and the frictional length scale $L = \mu g/\omega^2$, then normalize to $\bar{x} = x/L$ and $\bar{t} = \omega t$. The differential equation ...
eyeballfrog's user avatar
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5 votes
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Confusion with the Fourier Transform and Complex Differentiability: example with compact-supported function

Computing directly, we can find the Fourier transform of your function to be $$\int_{-1}^1e^{-i\omega t}\:dt = 2\operatorname{sinc}(\omega)$$ $$ \implies \int_{-1}^1(1-4t^2+6t^4-4t^6+t^8)e^{-i\omega t}...
Ninad Munshi's user avatar
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5 votes
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Can smooth ODE converge to its equilibrium in finite time?

Let $y(t) = x^*$ and note that $y$ is a solution to the system $\dot{y} = -f(y)$ with $y(0) = x^*$. Suppose $x$ is a solution with $x(T) = x^*$ and $x(0) = x_0$. Then since $x(T)=y(T)$ we must have $x(...
copper.hat's user avatar
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4 votes
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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?

You are confusing compactly supported and finite duration - these do not mean the same thing in the context of PDEs that distinguish between time and spatial variables. Not many people would ...
Ninad Munshi's user avatar
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3 votes

Closed-form solutions to $x''+\frac{k}{m}\ x+\mu\ g\ \text{sgn}(x')=0$

Let us for convenience go to dimensionless barred coordinates for position, velocity and time, $$\begin{align} x ~=~& \frac{\mu g}{\omega^2} \bar{x}, \cr v ~=~& \frac{\mu g}{\omega} \bar{v},\...
Qmechanic's user avatar
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3 votes

Closed-form solutions to $x''+\frac{k}{m}\ x+\mu\ g\ \text{sgn}(x')=0$

The following phase-space analysis confirms many of the results of eyeball frog using simple geometrical principles. (Note however that in contrast to eyeball frog's analysis, I do not assume that ALL ...
MathFont's user avatar
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3 votes
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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

Your $\hat f$ is correct. The graphing programs are not able to give accurate graphs of that $\hat f$ function near $x=0$, at least not from that formula. The most common form of representing real ...
aschepler's user avatar
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3 votes
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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.?

Your question is too vague, unclear and does not respect this site standards. However my flag did not do anything, as you made some efforts in the direction I suggested in comment and explained that ...
blamethelag's user avatar
  • 2,017
2 votes
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Does $x(t) = \exp\left(\frac{t}{t-1}\right)\cdot\theta(1-t)$ solve $\dot{x}\cdot(1-t)^2+x=0,\,\,x(0)=1$?

Your equation is not an ODE at $t=1$, so as ODE it only has solutions for $t<1$ and for $t>1$. However, as all these solutions have limit $0$ at $t=1$, you can combine any solution left of $t=1$ ...
Lutz Lehmann's user avatar
2 votes
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How to "formally" prove that $x(t)=\frac{1}{4}(2-t)^2\theta(2-t)$ solves $\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$?

Since $x(0)=1>0$, then one can see the solution is decreasing to zero and then stays there when it reaches this value. So, the differential equations simplifies to $\dot{x}=-\sqrt{x}$. This can be ...
KBS's user avatar
  • 7,224
2 votes

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

An illustrative example of a system with a finite-duration solution would be Norton’s dome: If you roll a ball with the right speed up this dome, it will come to rest at the top after a finite time (...
Wrzlprmft's user avatar
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2 votes

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

Introduction In what follows, we'll only consider first-order Ordinary differential equations (ODE) defined as $$y^\prime(t) = f(t, y(t))$$ where $f : U \to \mathbb R$ is continuous and $U\subseteq \...
mathcounterexamples.net's user avatar
2 votes
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Are these equations "properly" defined differential equations? (finite duration solutions to diff. eqs.)

1) Are all these differential equations with terms $(T-t)$ "well-defined" as differential equations? Their well-posenedness will depend on the multiplicity of the singularities. You will ...
KBS's user avatar
  • 7,224
2 votes
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Proving that these solutions are formally solving these differential equations: $x'' = -\text{sgn}(x')$ and $y'' = \sqrt{|y'|}$

Perhaps I don't understand the problem, but I don't think that it's difficult to prove within distribution theory that those are solutions to the differential equations (I skip the initial conditions ...
md2perpe's user avatar
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2 votes

Confusion with the Fourier Transform and Complex Differentiability: example with compact-supported function

There are two parts to your difficulty. 1. How did you end up with a Bessel function? Mathematica uses specific, very general families to represent common special functions internally, because those ...
Jacob Manaker's user avatar
2 votes

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?

$$\frac{d^2 x}{dt^2}+x(t)+\text{sgn}\left(\frac{dx}{dt}\right)\sqrt{\Bigg|\frac{dx}{dt} \Bigg|}=0$$ This second order ODE of autonomous kind can be reduced to a first order ODE thanks to the change of ...
JJacquelin's user avatar
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1 vote
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Solving $x'=-\text{sgn}(x)\sqrt{|x|}$: Uniqueness of solutions of finite duration

IVPs for $x'(t)= -{\rm sgn}(x(t))\sqrt{|x(t)|}$ are uniquely solvable to the right, since $x \mapsto -{\rm sgn}(x)\sqrt{|x|}$ is monotone decreasing on $\mathbb{R}$. More general, let $f:\mathbb{R} \...
Gerd's user avatar
  • 7,349
1 vote

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?

Is there a closed form solution? Likely not. Write it as follows: $$x''+x+\frac{x'}{\sqrt{|x'|}}=0.$$ Now, assuming one has found a solution $w$, or a convenient $w$, the other solution is obtained as ...
Alexander Conrad's user avatar
1 vote

Closed-form solutions to $x''+\frac{k}{m}\ x+\mu\ g\ \text{sgn}(x')=0$

$\color{green}{\textbf{The task standing.}}$ Let $\;P=\dot x = \dfrac{\text dx}{\text dt}= P(x(t)),\;$ then $$\ddot x=\dot P=\dfrac{\text dP}{\text dt}=\dfrac{\text dP}{\text dx} \dfrac{\text dx}{\...
Yuri Negometyanov's user avatar
1 vote
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Does Lipschitz-kind "non-scalar ODEs" and PDEs could stand having finite-duration solutions?

Let's recall precisely Picard Lindelöf theorem where I pick up here the definition used in Wikipedia: Let $D\subseteq \mathbb {R} \times \mathbb {R} ^{n}$ be a closed rectangle with $(t_{0},y_{0})\in ...
mathcounterexamples.net's user avatar
1 vote
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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)

So, yes, the solution that you provide is the solution to the dynamical system you are are considering under the assumption that the the initial conditions of the system are $r(0)=T^4/144$ and $\dot{r}...
KBS's user avatar
  • 7,224
1 vote

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.?

The only points where smoothness is not clear are $\{-1,1\}$. However, the singularities as a result of your compactly supported function are of the form $|1-x|)^{-R}$ for some $R>0$. However, ...
Diffusion's user avatar
  • 5,591
1 vote

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

You seem to be confused by the following two statements: The Laplace transform of a time-limited function that is absolutely integrable converges everywhere. The Laplace transform of a causal ...
Matt L.'s user avatar
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1 vote
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Are there any continuous time-limited Linear and Time-Invariant (LIT) functions with unbounded derivative?

I have realized later that the questions is ill-posed, but since other person bookmarked it, before deleting it I will give what know I think is the answer. I have found the following paper named &...
Joako's user avatar
  • 1,416
1 vote

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?

Maybe look into the Rayleigh-Plesset equation for the radius of a collapsing bubble within a fluid subject to an external acoustic field. I'm not too sure but I think one form of the equation is $\...
Pranav Chandrashekar's user avatar

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