Questions tagged [laplace-transform]
The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform.
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$\ln(x)$ Does Have a Laplace Transform?
My introductory Laplace transform textbook says the following:
The improper integral is defined in the obvious way by taking the limit:
$$\lim_{R \to \infty} \int_a^R F(x) \ dx = \int_a^\infty ...
4
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0answers
38 views
Find the solution of $x''-x=f(t)$ using Heaviside Functions
Find the solution of $$x''-x=f(t), \ \ t\geq 0, \ \ x(0)=1, \ \ x'(0)=0,$$ where $$f(t)=\begin{cases}
0 & 0\leq t\leq 1 \\
t-1 & t>1. \\
\end{cases}
$$
First, I note that $...
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1answer
18 views
Solving system of first order differential equations using eigenvalue /eigenvector and matrix exponential approaches
I am attempting to solve the following first order linear system of differential equations using the eigenvalue /eigenvector approach and then by the matrix exponential approach.
$$ \mathbf{x}^{\prime}...
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How to prove that $ \mathscr{L}^{-1}\left\{\frac{-As}{B^2+s^2}\right\}=Ae^{-B|t|}. $ [on hold]
How to prove that
$$
\mathscr{L}^{-1}\left\{\frac{-As}{B^2+s^2}\right\}=Ae^{-B|t|}\, ?
$$
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0answers
26 views
Laplace transformation of $\frac{1-e^{-t}}{t}$
Can anybody help me with the Laplace transformation of $$\mathcal{L}\left(\frac{1-e^{-t}}{t}\right)?$$
This was on old exam question, but I can not find a way to solve it.
2
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1answer
30 views
Finding a difficult inverse Laplacetransform
I'm trying to solve the following problem:
$$\mathcal{L}_\text{s}^{-1}\left[\frac{\text{F}(\text{s})+\text{G}(\text{s})}{1-\exp\left(-\frac{\text{s}}{4x}\right)}\right]_{\left(t\right)}\tag1$$
Where:...
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0answers
11 views
Definition of H(0) when inversing Laplace transform results in heaviside step function
We have: $ L^{-1}\{e^{-cs}F(s)\} = H(t - c)f(t-c) $, with $ H $ is a heaviside function.
In many documents, $ H(t - c) $ is defined as:
$ H(t - c) = \left\{\begin{matrix}
0 &, t < c \\
1 &...
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0answers
21 views
What does this piece-wise function look like?
So I have $f(t) = t$ for $0\le t<1$, and $f(t + 1) = f(t)$ for all $t \ge 0$, i.e., $f$ is a periodic
function with period $T = 1$. I am wondering what this function actually looks like. I know ...
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0answers
17 views
Does the Laplace Transform time shift property hold for t<0?
Here's the example that caused the question
$$ y(t) = x_1(t-3) \circledast x_2(-t+2) \quad x_1(t) = e^{-4t}u(t) \quad x_2(t) = e^{-2t}u(t)$$
For $x_1$, the $t-3$ factor can be adjusted to be (time ...
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1answer
20 views
Is there an inverse Laplace Transform of $-a\cdot e^{-b\cdot x^c}$
Is there an inverse Laplace Transform of $-a\cdot e^{-b\cdot x^c}$,
where $a,b$ and $c$ are constants, in my case its:
$a=-0.9898;
b=0.3511;
c=0.2553$
The property, that the function tends to zero ...
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votes
1answer
14 views
What is the proof for time shifting property of Laplace Transform? Can anyone explain or has an external link for it? [closed]
what is time shifting property? and what does it mean in Laplace Transform and signal processing? Im so bamboozled . I have had 20 lectures ,i dont understand anything yet. Also please write the proof ...
2
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1answer
55 views
Laplace transform of $\frac{2\sin(t) \sinh(t)}{t}$
Laplace transform of $$\frac{2\sin(t) \sinh(t)}{t}$$
Using the property:
$$\int_0^\infty \frac{f(t)}{t}dt = \int_0^\infty L(f(t)) du$$
where laplace transform is written as function of $u$, we get:
...
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1answer
53 views
Calculate inverse Laplace transform of $\exp(\sqrt{s^2-r^2})$
I have some trouble with the calculation of the inverse Laplace transform $e^{-k\sqrt{s^2-r^2}}$ , $k\geq0$, $r$ is known. ##
##
And I believe it has some relation with the inverse Laplace transform ...
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1answer
55 views
Evaluate $\int_0^\infty 2 u \sin u\, \exp(-su^2)\mathrm du$
Evaluate $\int_0^\infty 2u \sin u \,e^{-su^2} \mathrm du$.
This integral emerged while finding Laplace Transform of $\sin(\sqrt u)$
As a start, I used:
\begin{align*}
I &= 2\int_{u=0}^{\infty} ...
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1answer
30 views
Solving Heat Transfer PDE using Laplace Transform
I have the following PDE in the time domain:
$$\frac{r}{\alpha}\frac{\partial T}{\partial t} =\frac{\partial T}{\partial r}+r\frac{\partial ^2T}{\partial r^2} $$
Where temperature ($T$) is a ...
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0answers
16 views
How would I get the Heaviside function to be period in my solution?
so I am trying to solve an RC circuit problem that represents the square wave. Which I have already done on paper and now I am trying to plot it. So for this problem, we solved it using two methods, ...
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1answer
20 views
Inverse Laplace with a irreducible quadratic in the denominator and a 1 in the numerator
Please help me find the Inverse Laplace transform of:
$$F(s) = \frac{1}{s(s^2+8s+4)}$$
After completing the square, I obtained
$$F(s) = \frac{1}{s((s+4)^2-12)}$$
Thank you.
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2answers
31 views
Can 3D co-ordinates be transferred into 2D co-ordinates?
Is it possible to transform co-ordinates $(a,b,c)$ into $(x,y) $ such that $(x,y)$ is unique for each $(a,b,c)$ ? $a, b, c, x, y$ are in $\Bbb{R}$ .
2
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1answer
21 views
How to rearrange this fraction so it matches a Laplace Transform table identity?
I have the fraction:
$$\frac{s}{s^2+2s+2}$$
I want to rearrange the fraction so that I can solve find the inverse Laplace of it using the following identities from a Laplace Transform table:
$$f(t)....
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1answer
39 views
How to evaluate the Laplace transform of the square root using Residue theory?
My lecturer mentioned that it is possible to evaluate the Laplace integral transform (definition below) of $\sqrt{t}$ using complex analysis. How is that possible?
$$\hat f (s)=\int^{\infty} _0 {\...
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1answer
40 views
Laplace transform coming out to infinity
Trying to find the Laplace transform of $$\frac{\cos t}{t}.$$
It comes out as infinity, but that doesn't make any sense.
Does this mean that this function doesn't have a Laplace transform or is ...
1
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0answers
31 views
Identity a Laplace Transform
I am looking for a function on the positive real line whose Laplace transform, with parameter $s$, is
$$\left(\frac{\lambda}{1+\lambda}\right)^s,$$
where $s$ and $\lambda$ are greater than $0$.
The ...
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0answers
16 views
Density of hitting time of absolute value of a Brownian motion
I am interested in the probability density of
$$
\tau =\inf\{t\geq 0: \vert W(t)\vert = 1\}
$$
where $W(t)$ is a standard Wiener process. I have two approaches in mind:
First approach: I could ...
0
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0answers
34 views
Laplacian from spherical to rectangular
Given:
$$x = r\sin\theta\cos\phi$$
$$y = r\sin\theta\sin\phi$$
$$z = r\cos\theta$$
$${\nabla_{spherical}}^2={1\over r^2}{\partial\over\partial r}\left(r^2{\partial\over{\partial r}}\right)+{1\over{...
14
votes
2answers
384 views
“Continuized” Taylor Series? $\sin(x)=\sum \frac{(-1)^nx^{2n+1}}{(2n+1)!}=\int_{-1}^\infty \frac{\cos(\pi n) x^{2n+1}}{G(2n+1)}dn$?
~~not trying to reinvent the Laplace transform, but just an exploration into these particular series and integrals~~
Current answers don't fully address the 5 questions, so any new ideas or ...
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1answer
54 views
Use the Laplace transform to solve $ u_{tt}(x, t) − c^2u_{xx}(x, t) = 0$
Use the Laplace transform to solve the following initial boundary value problem for the wave equation $ u_{tt}(x, t) − c^2u_{xx}(x, t) = 0$
$u(x, 0) = 0$,
$u_{t}(x, 0) = 0 ∀x > 0$ , and
$u(0, t) ...
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0answers
25 views
Inverting a Laplace transform (for a Lévy process)
Let $\psi(\theta) = c\theta + \frac{\sigma^{2}}{2}\theta^{2} - \frac{\lambda\theta}{\alpha + \theta}.$
For those who are wondering where this function comes from, $\psi$ is the Laplace exponent for a ...
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1answer
25 views
Laplace Transform of an integral function of a convolution
Making suitable assumptions wherever necessary, what is the Laplace Transform $\mathcal{L}(S(t))$ where
$S(t)=\int_{0}^{t}\int_{0}^{t}f(t-s_1,t-s_2)g(s_1)h(s_2)ds_1ds_2$.
I tried using the Double ...
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0answers
36 views
Initial condition Impulse Answer of a first order system
Given the following first order system
$\tau y'(t) + y(t) = ku(t)$
The LaPlace transform yields
$\tau [s y(s) - y(t=0)] + y(s) = ku(s)$
Given $y(t=0) = 0$, this simplifies to
$y(s)[\tau s + 1] = ...
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0answers
26 views
Laplace Transform of Riesz Fractional Derivative
I wanted to calculate the Laplace Transform of Riesz Fractional Derivative. But got some troubles in the middle.
The Riesz Fractional Derivative is given by:
$R^{\alpha}f(t)=-\frac{1}{2\pi} \int_{-\...
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0answers
26 views
Popular papers on the Laplace Transform
What are some popular papers or articles introducing "new" Laplace transform based methods or techniques for solving ordinary and partial differential equations?
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0answers
29 views
Laplace transform methods for ODEs and PDEs
I am doing a research project on "Laplace transform based methods for differential and partial differential equations". I'm having a hard time finding a few of these "methods". Can someone name a few ...
0
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1answer
15 views
Does the Laplace transform of a probability distribution function (pdf) also integrates to 1?
I have a pdf $p(t)$
$$\int_{0}^{\infty}p(t)dt=1$$ now I take its laplace transform as:
$P(s)$ should
$$\int_{0}^{\infty}P(s)ds=1?$$
Intuitively, I think it should integrate to $1$.
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0answers
24 views
Inverse Laplace Transform of $F(s)=\frac{1}{\sqrt{s} \coth(\sqrt{s})-1}$.
I try to find the inverse laplace transform of the function
$\displaystyle F(s)=\frac{1}{\sqrt{s} \coth(\sqrt{s})-1}$. I check numerically that this function has no root in the right half complex ...
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1answer
83 views
Filtering property of Dirac $\delta$ function
Evaluate $$\int_0^\infty f(x)\delta(x-1)dx$$where$$f(x)=\begin{cases}x^2,&0\le x<1\\\sin 2,&x=1\\x,&x>1\end{cases}$$
Attempt
Since the function is discontinuous at $1$, I couldn't ...
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0answers
23 views
Find the transfer function G(s)
I have a set of equations from which I have to find the transfer function $G\left(s\right)=\frac{y\left(s\right)}{u\left(s\right)}$
I am given:
$0.15\cdot \frac{d^2\left(x+l\theta \right)}{dt^2}+1....
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0answers
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Laplace transform of inverse power law: $t^{-(1+\beta)}$ for $t > 0$ and $0 < \beta < 1$
I came across a paper writing about continuous-time random walk, which derived the number of distinct sites visited by a random walker. It says that given the waiting time distribution $\psi(t) \sim t^...
2
votes
2answers
71 views
Showing that integral is related to sine function in elementary means
So I am trying to prove the reflection formula for the gamma function by showing that $$\int_{0}^{\infty} \frac{v^{s-1}}{1+v}dv=\frac{\pi}{\sin(\pi s)}$$
for $0 < \Re(s) < 1$ , as these two ...
1
vote
1answer
46 views
$\lim_{n \to \infty} E[e^{-X*\frac{t}{n}}]^n \approx \lim_{n \to \infty} E(1-\frac{E[X] t}{n})^n = e^{-E[X]*t}$
X is a random variable.
I don't understand this passage. Please someone can explain it to me each equality at a time? Thank you!
$\lim_{n \to \infty} E[e^{-X\frac{t}{n}}]^n \approx \lim_{n \to \...
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0answers
38 views
Discrete gradient operator
I am working on implementing a heat diffusion paper and I'm a bit stuck. Would be glad if someone can share their thoughts!
I have a pointcloud $P \subset \mathbb{R}^n$. Each point has a value $\...
1
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1answer
65 views
Laplace Transform of $ te^{2t}$ using unit step function
I was wondering if I could get some help with this question:
Consider the function:
$$f(t) = \begin{cases} te^{2t} \quad \,0 \leq t < 3\\
0 \quad \, 3 \leq t
\end{cases}$$
(a) ...
2
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0answers
41 views
Laplace transform of heat conduction PDE in cylindrical coordinates.
I'm trying apply the Laplace transformation to solve the non-dimensional heat conduction PDE for a hollow cylinder with convection boundary conditions and a non-homogenous initial condition.
$$\frac{...
1
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1answer
70 views
Solving second order PDE using Laplace transforms. $u_{t}(x,t)-u_{x}(x,t)=u_{xx}(x,t)$ with $u(x,0)=\cos(2x)$
I am asked to use the Laplace transform to solve the initial value problem for the advection-diffusion equation:
$$u_{t}(x,t)-u_{x}(x,t)=u_{xx}(x,t)$$ with $u(x,0)=\cos(2x)$
Taking the Laplace ...
3
votes
3answers
70 views
Using Laplace transform to solve differential equation $y'' -4y' = -4te^{2t}$
$y'' -4y' = -4te^{2t}, y(0)=0, y'(0)=1$
If you take laplace Transform of all terms, isolate L(y), I got
$$L(y) = \frac{1}{(p-2)^2} + \frac{-2}{(p-2)^2 -4}$$
Then, taking inverse Laplace, you get
$$y(...
2
votes
3answers
63 views
If $f(t)=\frac{e^{bt}-e^{at}}{t}$ how to compute $\mathcal{L} \{ f(t) \}$
I have a problem trying to solve
$$
\int_0^{+\infty}\frac{e^{(b-s)t}}{t}
$$
I know that I can use the Ei (exponential integral function) but after that I don't know what's exactly this means.
I ...
2
votes
0answers
36 views
Two dimensional Laplace transform of convolutions
On a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, let us consider the joint law of two random variables $X$ and $Y$
$$\nu(B_1\times B_2):=\mathbb{P}(X\in B_1, Y \in B_2)
$$
where $B_i\in\...
4
votes
2answers
61 views
Laplace transforms with Fresnel(?) integrals
I've come into contact with this two part question, and the latter I'm not too sure how to go about; at least to me upon researching, I can't find anything remotely similar to what I've been asked.
...
0
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0answers
22 views
Doubt about Laplace Transform existence for a given $f(t)$
I have some doubts about how to practically check if a function admits a Laplace Transform; I write what I think I understood, could you please tell me what I got wrong?
A sufficient condition for a ...
0
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0answers
14 views
Consider the following problem for Laplace's equation, I wonder how to use mathematica to solve it.
Consider the following problem for Laplace's equation, I wonder how to use mathematica to solve it.
enter image description here
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1answer
26 views
Calculating inverse Laplace transform of $\exp(-s c)/s$?
One can look up in this table that the inverse Laplace transform of $\exp(-c s)/s$ with $c\in\mathbb{R}$ is given by:
$$\frac{1}{2\pi i}\lim_{T\to \infty}\int_{\gamma-i T}^{\gamma+iT}ds\frac{e^{s(t-c)...
