All Questions
Tagged with convolution laplace-transform
187
questions
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58
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Solving 1st order PDE including convolution
I'm studying Van Kampen's "Stochastic processes in physics and chemistry" and stuck to some exercise (p.78):
That is, solving
\begin{equation}
\frac{\partial P(y, t)}{\partial t}=\int_{-\...
- 1
2
votes
0
answers
50
views
General solution for linear Volterra-like integral equation?
A linear Volterra integral equation looks like this (see the wiki)
\begin{align}
x(t) = f(t) + \int_0^t K(t, s)x(s)~\mathrm{d}s.
\end{align}
If the Kernel function $K$ is of the form $K(t, s) = K(...
- 138
1
vote
0
answers
22
views
Prove that the convolution of the signals and its time reversal is an odd signal.
Suppose signal $g(t)$ is obtained by time reversal of signal $f(t)$ for all times $t$. Prove that the convolution of the signals $f$ and $g$ is an odd signal.
My attempt at proof
Given: $g(t)=f(-t)\...
- 1,210
5
votes
2
answers
248
views
Calculate $\sum_{n=1}^\infty (1-\alpha) \alpha^nP^{*n}(x)$
$$
\mbox{Let}\quad
P'(x)=\sum_{j=1}^n a_j\frac{a^jx^{j-1}}{(j-1)!}e^{-ax},x\geq 0
$$
be the density function of a mixture of Erlangs and let $\alpha\in(0,1)$:
Is is possible to determine an analytic ...
- 91
2
votes
0
answers
48
views
Is there a good method to recover the original functions from their convolution?
Occasionally, I failed to detect the function which is just the convolution of two simple functions. For instance
$$
2a\sin{at}*\sin{at}=\sin{at}-at\cos{at}
$$
One possible way is to observe the ...
- 1,329
0
votes
0
answers
76
views
Solving the integro-differential equation (Cauchy problem) using Laplace transform
I am stuck trying to solve this equation using Laplace transform:
$$ y''(t)+2y(t)=\displaystyle 3\int_0^t (t−u)y(u)du\\ y(0)=2 , y'(0)=3 $$
I got stuck after performing the Laplace transformation ...
4
votes
1
answer
179
views
defective renewal equation
I am reading this paper of Lin and Willmot. I dont understand how they come up with formula $2.7$ and why $\tilde{\phi}(s)=\frac{\tilde{H}(s)}{1+\beta-\tilde{g}(s)}$. Can someone help me?
So i want to ...
- 351
1
vote
0
answers
97
views
Convolution equation equals a constant
How would you solve this?
I did the standard way of solving questions like these - I took the Laplace of both sides and used the convolution identity. But the solution I got, $f(t)=3+t^2/2$, does not ...
1
vote
1
answer
44
views
Solve using Convolution theorem (Inverse Laplace)
Using Convolution theorem, find:
$$L^{-1} [\frac{s^2}{(s^2+1)^3}]$$
Note: This was an exam question and was worth "3" marks only, so it should not be so "long" I suppose.
So my ...
- 395
1
vote
0
answers
28
views
Using convolution to derive first passage probability
I'm reading through Sidney Redner's lectures on first passage processes (lectures at this link) and I've been stuck on the convolution steps he takes to go from equation (3.1) to (3.2). He starts by ...
- 11
2
votes
0
answers
119
views
Is there a 3D version of the convolution and cross-correlation theorems for the Laplace transform?
The Laplace-transform version of the convolution and cross-correlation theorems is essentially the same as the "usual" (Fourier-transform) version: if $\mathcal{L}[f(t)]$ is the Laplace ...
- 125
0
votes
1
answer
58
views
Convolution theorem of Laplace transform; Schiff
I'm reading Schiff's The Laplace Transform and I have some questions about the convolution theorem he proves on page 92 to 93.
Theorem and proof
Theorem 2.39 (Convolution Theorem). If $f$ and $g$ are ...
- 813
3
votes
0
answers
105
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Are there functions with a Fourier transform but no Laplace transform?
Let
$$A = \text{the collection of functions with a Fourier transform}$$
and
$$B = \text{the collection of functions with a Laplace transform.}$$
What is the relationship between $A$ and $B$? Based on ...
- 75
2
votes
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answers
188
views
Finding the relation between Laplace and the CTFT
Let us take a system where the input is $V_{i}(t)$ and output is $V_{o}(t)$ and the impulse response of the system be $I(t)$ where $t$ represents time domain and $w$ be frequency. we get $\tag{1}V_{o}(...
- 21
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59
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Laplace transform of translated convolution
I need to calculate the Laplace transform $\mathcal{L_t}$ of the function
$$ h(t)=\int_0^{\tau-t}f(\tau-t-u)g(u)du, $$
in terms of $\mathcal{L_t}[(f\ast g)(t)]$. Here, I do not know $g$ and $\tau>t&...
- 349