All Questions
Tagged with convolution ordinary-differential-equations
96
questions
0
votes
2
answers
46
views
How to linearize time-variant ordinary differential equation (ODE), in particular, if there is a convolution term?
Letting $(\ * \ )$ denote convolution, I have an ODE of the form
\begin{align}
\dot{r} &= -7.4r -1.6f - 8.8 (f(t)*e^{-t}) - 10.4(f(t)*(te^{-t})) \\
\dot{f} &= 0.25r
\end{align}
with initial ...
- 286
0
votes
0
answers
35
views
Exact Successor State Distribution for a Pendulum
I want to solve the following problem. Suppose we have a simple pendulum, which follows the differential equation
\begin{equation}
\dot{x} = f(x) = [x_2, -\sin(x_1)]^T, \text{with } x=[x_1, x_2]^T.
\...
- 101
2
votes
0
answers
67
views
Method to solve a first order non linear ODE with a convolution inside
Working on thermodynamics, I arrived to an equation like this in spatial and frequency domain:
$$ \partial_z G(z,\omega) + \left( G(z,\omega) F(\omega) \right) \ast G(z,\omega) = 0 $$
with $0 < ...
- 191
7
votes
1
answer
129
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How to approach the following differential equation
I have a differential equation of the form
$$
\frac{\mathrm{d}g}{\mathrm{d}x} = f(x) + \int_0^x g(y)f(x-y)\mathrm{d}y + \alpha g(x).
$$
$f$ is a monotonous decreasing function, satisfying $\int_0^\...
- 1,119
1
vote
0
answers
62
views
Method to solve a first order ODE with convolution exponential
In thermodynamics, I have an equation like this in frequency domain $\omega$ and spatial domain $z$:
$$ \partial_z g(z,\omega) - f(\omega) \ast g(z,\omega) = h(z,\omega) $$
with the boundary ...
- 191
0
votes
0
answers
121
views
Spectrum of a convolution operator on $L^2(\mathbb{R})$
For context, this question is related to a previous one of mine.
Consider the operator $H$ acting on $L^2(\mathbb{R})$ and of the form $$Hv:=\phi\ast v,$$ where $\phi=e^{-|x|}/2$ and $\ast$ denotes ...
- 1,139
3
votes
0
answers
203
views
Find all functions whose convolution is the same as their square.
Find all functions who's self-convolution is the same as their square.
To make it explicit, find all $f:\mathbb{R} \rightarrow \mathbb{R}$ such that:
$$
\int_{-\infty}^{\infty} f(t - \tau)f(\tau) d\...
- 59
0
votes
0
answers
161
views
How to solve a recursive convolution equation
In the most general case, I ask how to solve (either analytically or numerically) this equation for $x(t)$
$$x(t) = \int_{-\infty}^t g(t-\tau) f\big(x(\tau)\big) d\tau$$
where $f, g$ are functions ...
- 251
1
vote
1
answer
56
views
How to calculate the convolution product of $(H_0 e^{\alpha t}) * (H_0 e^{-\alpha t})$?
everyone!
I am doing an exercise concerning the resolution of differential eaqtion in the sense of distributions.
Let $\alpha \in \mathbb{R}_+^*$. Let be the differential equation, for $f \in \mathrm{...
- 25
3
votes
1
answer
243
views
Solving $ y'' + a y = f(x) $ with zero initial conditions
Given the following initial value problem (IVP) $$ y'' + a y = f(x), \qquad y(0)= y'(0) = 0$$ show that $$y = \frac1a \int_{0}^{x}f(t)\sin(ax-at)\ {\rm d}t$$
My attempt
To solve the given initial ...
- 484
1
vote
2
answers
408
views
ODE solution using convolution and (inverse) Fourier transform
I'm studying PDEs from old course material which includes answers to most of the exercises. Given a ODE
$-u''(x)+au(x)=f(x)$
and a function $g(x)=e^{-|x|}$, we should find $u(x)$ by calculating the ...
- 261
1
vote
1
answer
180
views
Solve Integral Equation With Convolution and a constant added
$7t + 8/5 $$\int_0^t \cos (a(t - \tau)) y(\tau) d\tau$$ = y(t)$ , for $ a>10^{50}$
Im sort of confused by working with the constant $a$.
i used convolution theorem and applied laplace, giving it:
$...
- 13
5
votes
1
answer
332
views
How "practical" is the Laplace transform method for constant coefficient ODE?
I just finished teaching a chapter on using Laplace transform to solve constant coefficient second order linear differential equations. I touted how amazing the method was because it incorporates the ...
- 5,239
4
votes
1
answer
209
views
How to prove that if $g*g=0$, then $g=0$?
Given a continuous function $g$, if the convolution $g*g(t)$ (defined as: $\int^t_0(g(r)g(t−r))dr$ equals $0$, $\forall t\geq 0$, then $g=0$.
My attempt was to use Laplace transformation, but such ...
- 166
2
votes
2
answers
4k
views
Solving $y'' - 2y' + y = \delta(t-2)$ for y(0) = 0, y'(0) = 0 by using Laplace Transforms. Need help finishing the problem.
So I'm working on this question and I've taken the Laplcace transforms for everything and separated it as $Y(s) = F(s)G(s)$, and then I've put it into the convolution equation to get $y(t)$. My ...
- 367