Questions tagged [inverse-laplace]
This tag is for questions regarding to "Inverse Laplace Transform" which is the transformation of a Laplace transform into a function of time.
324
questions
0
votes
1
answer
47
views
Laplace of air resistance
I have the next linear equation to get its laplace transform:
$$\frac{dv}{dt}+cv=g$$
and its values are:
$g = 32ft/sec^2$
$c = 0.25/sec$
My teacher tell me that I need to do a laplace transform to ...
- 13
1
vote
0
answers
21
views
The distribution function of a running supremum of a diffusion process in a particular limit.
Let $x>0$, $\theta \le 1$, $\mu_0 \in {\mathbb R}$ and $\nu:= -{\bar \mu_0}/(1-\theta)$ and ${\bar \mu_0} := \mu_0 - 1/2 $. Now, consider a non-linear diffusion model $d X_t = \mu_0 X_t^{2 \theta-1}...
- 9,810
1
vote
0
answers
33
views
A simple question for numerical inverse Laplace Transform
I recently met a problem when I tried to apply Laplace transform to a simple complex-valued function, for example
\begin{equation}
f(t) = \exp\left(-t(a+bi)\right)
\end{equation}
where $a,b,t>0$. ...
- 11
1
vote
3
answers
85
views
Inverse Laplace transform of $\frac{6s^2-2}{\left(s^2+1\right)^3}$
I would like to determine the inverse Laplace transform of $$\frac{6s^2-2}{\left(s^2+1\right)^3}.$$
What's the easiest way to do this? (Easy in the sense that it follows easily from some Laplace ...
- 473
0
votes
0
answers
28
views
Computing matrix exponential using the inverse Laplace transform
Trying to compute matrix exponential, I ran into a problem. My solution:
\begin{equation*}
B= \begin{pmatrix}
1 & 2 \\
0 & 1 \\
\end{pmatrix}
\end{equation*}
$(sE-A)= \begin{pmatrix}
...
- 1
0
votes
0
answers
24
views
Demonstrate the resolution of laplace inverses
demonstrate the resolution of laplace inverses, I got a slightly scary result, but I wanted to clear up a doubt.
$\mathcal{L}^{-1}\left\{ \dfrac{1}{s^2(s+2)} \right\} - \mathcal{L}^{-1}\left\{ \dfrac{...
1
vote
0
answers
42
views
Inverse Laplace with square root of s in quotient
I am trying to find the inverse Laplace transform of expressions of the form $$\frac{a + b \sqrt{s}}{cs + d\sqrt{s} +e}$$ Even an instance of it like $$\frac{\sqrt{s}}{s + \sqrt{s} +1}$$ perhaps? ...
- 23
0
votes
0
answers
14
views
Information about Laplace transform and its derivatives - What does that tell us about the function itself?
Let $\alpha \in\left( \frac12, 1 \right)$. We set
\begin{equation}
\mu:\mathbb N_0 \rightarrow \mathbb R_{+}, \qquad 0\mapsto 0,\, k\mapsto k^{-\alpha}-(k+1)^{-\alpha}.
\end{equation}
And define the ...
- 317
3
votes
1
answer
252
views
Solve $y''-y'-6y=e^{3t}+5$
Solve the following initial value problem $$y'' - y' - 6y = e^{3t} + 5, \quad y(0) = 0, \quad y'(0) = 0 $$
I used the Laplace transform and got
$$ Y(s) \left(s^2-s-6\right)=\dfrac 1{s-3}+\dfrac 5s $$
...
- 167
0
votes
0
answers
19
views
Can Inverse Laplace Transform reveal reveal adequate information about intensities obscured by other confounding signal intensities in FTIR and Raman?
I’m using MATLAB for my platform and I want to be able to do signal processing for spectra that we have recorded in FTIR and Raman. For now, the focus is on FTIR. I have done analyses unrelated to ...
4
votes
0
answers
62
views
I need help solving this differential delay equation (inverse Laplace transform problem)
Let us consider a differential delay equation (DDE) with $a,b\in\mathbb{R}$:
$$
\frac{d}{dt}y(t)=ay(t)+bH(t-1)y(t-1),~0\le t<\infty,
$$
where $H(t)=\int_{-\infty}^t\delta(t')dt'$ is the Heaviside ...
- 41
1
vote
1
answer
51
views
Finding the inverse transform of a Laplace Transform
I'm trying to solve an inverse laplace function. The equation is quite simple actually but has a lot of small constants in the initial expression.
The initial expression:
$$\frac{1+5*10^{-3}+100*10^{-...
- 11
0
votes
0
answers
23
views
Can the inverse Laplace transformation of these functions be found by using the residue theorem?
While solving a partial differential equation, I came across the following four functions:
$$ F_1(s) = e^{ k \sqrt{ q^2 + cs^2 } } \tag 1$$
$$ F_2(s) = e^{ -k \sqrt{ q^2 + cs^2 } } \tag 2$$
$$ F_3(s) ...
- 461
1
vote
0
answers
32
views
Inverse laplace transform of $\frac{s}{s+a}$
There is a missing entry in my table of Laplace transforms. I want to make a transform of the general form of the fraction
$$\frac{s}{s+a}$$
Using the definition:
$$\mathscr{L}^{-1}\{F(s)\}=\frac{1}{2\...
- 2,145
1
vote
0
answers
67
views
Using Laplace transform to solve $y'(t)+y(t-1)=t^2$
I am quite stuck with the following differential equation:
$$y'(t)+y(t-1)=t^2 \quad \text{with} \: y(t)=0 \; \text{for} \; t\leq 0.$$
I would like to use Laplace transform to solve this one. Using ...
- 73
1
vote
0
answers
37
views
Branch cuts for inverse hyperbolic function
this is a post connected to my original question; see here for full explaination:
Solution to diffusion/Smoluchowski equation using an inverse Laplace transform
Basically, I am working on a new way to ...
- 39
2
votes
0
answers
24
views
Find an inverse Laplace Transform using partial fractions
Problem:
Find the $\mathscr{L}^{-1}\{H(s)\}$ of the following function:
$$ H(s) = \dfrac{1}{s^2+5s+6} $$
Answer:
First we use partial fractions to rewrite the fraction.
\begin{align*}
\dfrac{1}{s^2+5s+...
- 3,541
1
vote
1
answer
51
views
Inverse Laplace transformation of $\sinh(ks)$ [closed]
What is the analytic solution to the following inverse Laplace transformation:
$$ f(t)=L^{-1}{\Big\{ {\sinh(ks)} \Big\}} $$
where $k$ is a constant.
- 461
1
vote
0
answers
29
views
What is the inverse Laplace transformation of ${w\over {w^2+s^2}}\cdot {e^{bs}-e^{-bs}\over{e^{as}-e^{-as}}}$
What is the analytic expression of $f(t)$ which is defined as the following inverse Laplace transformation:
$$ f(t)=L^{-1}{\Bigg\{ {w\over {w^2+s^2}}\cdot {e^{bs}-e^{-bs}\over{e^{as}-e^{-as}}} \Bigg\}}...
- 461
0
votes
1
answer
45
views
Inverse Laplace Transform of $\frac{\omega n^2}{s((s+z\omega n)^2+\omega n^2(1-z^2))}$
I've to solve the inverse Laplace transform of the following function, and am unable to find a good starting point. Any hints in the right direction would be appreciated. Thanks.
$$\mathcal{L}^{-1}\...
- 6,406
0
votes
0
answers
40
views
How do we evaluate the inverse laplace transform?
Here's two answered and upvoted questions that do not mention anything about a region of convergence in their premises.
Usage of inverse Laplace transform
Inverse Laplace Transform properities
It is ...
2
votes
1
answer
64
views
Inverse Laplace transform of $\frac{3}{(s^2+9)^2}$ by contour integration
I've been trying to prove the result $$\mathcal{L}^{-1} \{\frac{3}{(s^2+9)^2}\}(t)=\frac{1}{18}
(\sin(3t)-3t\cos(3t))$$
by a contour integral. So $$\begin{align}\mathcal{L}^{-1} \{\frac{3}{(s^2+9)^2}\}...
- 451
0
votes
1
answer
87
views
When is a function the Laplace transform of some random variable?
The more general question will be what is the criteria for a function $f(t)$ to be the Laplace transform of some random variable $\xi$, i.e. $\mathbb{E}e^{-t\xi}=f(t)$?
A more specific question is why ...
- 1,011
11
votes
0
answers
362
views
Solution to diffusion/Smoluchowski equation using an inverse Laplace transform
I am studying a new formula that extracts solution to the diffusion-Smoluchowski equation and is rooted on the theory of complex calculus. Namely, the formula looks like
\begin{equation}
P(t) = \...
- 39
0
votes
1
answer
41
views
What is the Laplace transform of this function?
Let us say I have a function in the Laplace domain F(s) whose inverse Laplace transform is f(t). Is there some theorem/identity/derivation such that I can find the inverse Fourier transform of F(s)/(s-...
- 11
2
votes
1
answer
149
views
Inverse Laplace Transform of $\frac{e^{-\sigma\sqrt{x}}}{\sqrt{x}}$
Is there any way for me to solve the inverse Laplace Transform of $\frac{e^{-\sigma\sqrt{x}}}{\sqrt{x}}$? Here is my attempt to solve this:
$$
\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty} \frac{e^{-\...
- 79
2
votes
0
answers
86
views
Problem with Residue Theorem for Laplace Inversion
Arising out of my analytical solution to the 2-dimensional transient diffusion equation, I have to invert the following Laplace transform to get a surface flux in physical space (I have greatly ...
- 1,116
0
votes
0
answers
50
views
A question about finding the Laplace inverse transform
Hellow, I want to find the Laplace inverse transform of the following equation,
$$\frac{1}{2\pi i}\int^{i\infty+\epsilon}_{-i\infty+\epsilon}ds \frac{e^{st}}{s+ia+b(1-e^{-2cs})+\frac{b}{2}(3ie^{-cs}-...
- 3
4
votes
1
answer
79
views
Series Formula for Inverse Laplace Transform related to The Telegrapher's (Heaviside's) Equations
A problem related to The Telegrapher's (Heaviside's) Equations
$$V_x = -LI_t - RI$$
$$I_x = -CV_t - GV$$
gives rise to a Laplace Transform
$$Z(s)=\sqrt{\frac{R+s L}{G +s C}}$$
This problem has been ...
3
votes
1
answer
73
views
Model-free ultra-local function approximation
I've been reading a lot about model-free control and I came across the concept of the ultra-local model. There is a really intricate approach outlined here but I'm having an issue with one part ...
- 33
1
vote
1
answer
104
views
Laplace transform of the derivative of a function with jump discontinuity at t=0
While trying to find the inverse laplace transform of $F(s)=s\log\frac{s-1}{s+1}$, I first calculated at the inverse laplace transform of $$G(s)=\log\frac{s-1}{s+1}$$ which came out to be $$g(t)=\frac{...
- 35
0
votes
1
answer
48
views
How can I solve Laplace Tranformation of $1/s^{5/2}$?
I have just started Laplace Transformation
And I came across a problem which contains $1/s^{5/2}$
How to solve it? I know $\mathcal L\{t^n\}= n!/s^{n+1}$
Please say how to solve it.
0
votes
1
answer
64
views
Inverse Laplace transform of a function involving Gaussian
$F(s) = \cfrac{F_0(s+a)}{1-a F_0(s+a)} $
where $F_0(s)$ is Laplacian transform, given by:
$F_0(s) = \mathcal{L}[\exp(-t^2 \beta)] $, and
$\beta$ and $a$ are real numbers
I am interested in inverse ...
- 1
0
votes
0
answers
31
views
Inverse Lapace transform of $F(s)=\frac{I_0(r\sqrt{n^2+s})}{s I_0(\sqrt{n^2+s})}$
While trying to solve this diffusion-reaction problem, I encountered the following Laplace inversion problem:
\begin{align}
\mathcal{L}^{-1}_s\left\{\frac{I_0(r\sqrt{n^2+s})}{s I_0(\sqrt{n^2+s})} \...
- 517
0
votes
1
answer
89
views
Calculate Inverse Laplace Transform $H(s)=\frac{1}{s^4-s^2}$
Calculate Inverse Laplace Transform $$H(s)=\frac{1}{s^4-s^2}$$
I am trying to solve this by convolution but I don't know how to move forward. I have come this far.
$$H(s) = \frac{1}{s^4-s^2}$$
$$h(t) =...
- 45
0
votes
1
answer
32
views
How can I solve this constant value 5 using inverse Laplace?
I'm having a hard time understanding this problem. Please help me to solve this problem
Evaluate $\mathcal L^{-1}\left\{5 + \frac s{s^2+9}\right\}$
https://i.ibb.co/TK3jbqK/Screenshot-2022-06-28-...
- 13
0
votes
2
answers
43
views
Inverse Laplace transform of $\frac{1}{e^{-s}+1}$ [closed]
I would like to know how to perform the following inverse Laplace transform:
\begin{equation}
\mathcal{L}^{-1}\left\{\frac{1}{ e^{-s} + 1}\right\}\,.
\end{equation}
This expression somehow relates the ...
- 19
0
votes
1
answer
64
views
A question on laplace transform
I try to solve the following question on Laplace transform
$$L(\{ \int_0^{t}e^{-x^2}\})$$
I solved as following:
$$L(\{ \int_0^{t}e^{-x^2}\})=\frac{1}{s}L(\{e^{-x^2}\})=\frac{1}{s}L(\{\sum_0^{\infty}\...
- 649
0
votes
0
answers
29
views
Simplification Error Inverse Laplace Transform
I am trying to build a model of a electronic circuit and to solve the differential equation using the laplace transformation. With the help of numeric simulation I know what one factor has to be and ...
- 11
0
votes
1
answer
89
views
Use Residues to find the inverse Laplace transform $F(s)=\frac{2s^3}{(s^2-4)}$
Use Residues to find the inverse Laplace transform $F(s)=\frac{2s^3}{(s^2-4)}$.
The answer from the text book is $f(t)=\cosh^2(t)+\cos^2(t)$.
But my result is $2\cos^2(t)\cdot \cosh^2(t)$. Which is ...
1
vote
1
answer
53
views
Help solving this heat differential equation -u'' = 𝛿(x-1.5) ; delta
1.4 Numerical Methods
Solve the boundary value problem (BVP) by modeling the temperature in a well
conducting metal rod of length 3 in the presence of a point heat source in the
middle of the rod in ...
- 67
3
votes
1
answer
132
views
Inverse Laplace transform of $\dfrac{e^s}{s(e^s+1)}$ [duplicate]
The original problem is to solve $$\mathcal{L}^{-1}\left\lbrace\frac{e^s}{s(e^s+1)}\right\rbrace.$$ Doing partial fractions $$\frac{e^s}{s(e^s+1)}=\frac{1}{s}-\frac{1}{s(e^s+1)}$$ the problem reduces ...
- 449
3
votes
1
answer
124
views
Inverse Laplace transform of $\frac{1}{2s}\coth\left(\frac{s}{2a}\right)$
I would like to see how to compute the inverse Laplace transform
$${\mathscr L}^{-1}\left(\frac{\coth (s/(2a))}{2s}\right).$$
I am interested in the proof, not just the answer. I am reading the book &...
- 111
1
vote
0
answers
57
views
Find a function $g$ such that $(1*g)(t)=t^{-1/2}$.
Is there some function $g: (0,+\infty) \to \mathbb{R}$ satisfying
$$(1*g)(t)=t^{-1/2}$$
for all $t>0$? Here the sign $*$ represents the convolution. I tried apply the Laplace transform both site of ...
0
votes
1
answer
49
views
A strange trigonometric inequality
I am trying to check that:
$$f(t) = t^2(t-3) + 2 e^{-t/2} + e^{t/2}\left(\sqrt{3} \sin \frac{\sqrt{3}t}{2} + \cos \frac{\sqrt{3} t}{2} \right) - e^{-t/2} \left(\sqrt{3} \sin \frac{\sqrt{3}t}{2} + 3\...
- 50.4k
1
vote
1
answer
70
views
Finding Inverse Laplace Transform of exponential and quadratic term together?
I'm having trouble finding the inverse Laplace transform of:
$$G(s) = \frac{1}{2s^2}e^{-Cs}$$
where C is a constant.
I know that the inverse Laplace transform of $s^{-2}$ on its own is $t$, and that ...
- 300
1
vote
2
answers
88
views
Finding the righthand system from the Laplace transform $G(s) = \frac{1}{1- e^{-sT}}$
I want to find the impulse response $g(t) \in \mathbb{C}$, that it's two-sided Laplace transform is:
$$\mathcal{L}\{ g(t)\} = G(s)=\frac{1}{1-e^{-sT}}$$
I tried to find $g(t)$, by finding the inverse ...
0
votes
1
answer
22
views
Integral Equation and inverse laplace
When we get the equation in (5.81), we get complex expressions while taking the inverse laplace transform after moving to the next step. Is there any way to get rid of this complex expression?
user1006476
1
vote
2
answers
105
views
Showing $\int_{-\infty}^\infty e^{-tu}\frac{\sinh(\frac{u}2(\pi-2\gamma))}{\sinh(\frac{u}2\pi)}du=\frac{\sin2\gamma}{\sin(\gamma+t)\sin(\gamma-t)}$
For $-\gamma<t<\gamma$ and $0<\gamma<\frac{\pi}{2}$, I want to verify the identity
$$\int_{-\infty}^{\infty} e^{-t \lambda} \frac{\sinh\left(\frac{\lambda}{2}(\pi-2\gamma)\right)}{\sinh\...
- 13
3
votes
1
answer
74
views
Inverse Laplace Transform of $e^{-x\sqrt{s}}/\left[\left(s-a^2\right)\left(\sqrt{s}+a\right)\right]$
I'm interested in the following Laplace inversion problem
$$f(t,x)=\mathcal{L}^{-1}\left\{\frac{e^{-x\sqrt{s}}}{\left(s-a^2\right)\left(\sqrt{s}+a\right)}\right\}$$
with $a$ real and positive, as ...
- 517