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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.

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Finding the function $g(t)$ given its Laplace transform $F(1/s)$

I am trying to find a function g(t) given that its Laplace transform is F(1/s), where F(s) is the Laplace transform of another function f(t). I know that if f(t) has the Laplace transform F(s), then f(...
Dr Potato's user avatar
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Inverse Laplace transform of $\frac{s}{2\sqrt{s^2-1}}e^{-\sqrt{s^2-1}|k|}-\frac{e^{-s|k|}}2$

To compute the Fourier transform described in this question, I need to perform the inverse Laplace transform of $$\tilde F_1(s)=\frac{s}{2\sqrt{s^2-1}}e^{-\sqrt{s^2-1}|k|}-\frac{e^{-s|k|}}2$$ with ...
Adam's user avatar
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Laplace Transform of a Second Order Systems Response to Triangular Pulse

I've been trying to derive the time domain response of a second order system to triangular pulse input using Laplace Transformation but even if I seem to be able to derive it Simulink simulations ...
Ahmet Burak's user avatar
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Continuity of Inverse Laplace transform on suitable subspaces

I came across of the following relation. Taking a sum of exponentials (SOE) $E(x)=\sum_{j=1}^N b_j e^{\gamma_j x}$, with $b_j\in \mathbb C$ and $\gamma_j \in \mathbb C_-$ (left part of the complex ...
Salfalur's user avatar
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Find the Inverse Laplace Transform of Θ(s)

I am asked to find the inverse Laplace transform of the following: $$Θ(s)=\frac{2^{n+1}n!}{s^{n+1}}$$ It is my understanding that the following are true: $$L^{-1}(F(ks))(t)=\frac{1}{k}f(t/k)$$ $$L^{-1}...
goingtonu's user avatar
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31 views

Inverse Laplace transform and residues.

$u(t)=\mathcal{L}^{-1}u(s)=\int_{\sigma-i\infty}^{\sigma+i\infty} \mathrm{e}^{st} \mathcal{L}u(s)\,ds,\quad t\geq 0$ $u(t)=\mathcal{L}^{-1}u(s)=\sum_{s \text{ in poles } } \text{Residues}(\mathrm{e}^{...
eraldcoil's user avatar
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4 votes
4 answers
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Transition probability density function for a non-trivial diffusion process.

Let $\mu$ and $\sigma > 0$ and $\beta_1 \ge 0 $ and $\beta_2 \ge 0$ be real numbers. Consider a stochastic process $X_t$ that satisfies the following stochastic differential equation: \begin{...
Przemo's user avatar
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29 views

Relation between the inverse Laplace and inverse Mellin transforms

If we have the answer to the inverse Melin transformation of an expression, can we arrive at the inverse Laplace transform of that expression? $$M^{-1}\left (\frac{1}{\Gamma (c+s)\Gamma (d-s)} \right ...
3pi.sahagh's user avatar
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Laplace transforms coinciding on a vertical line

It is a well known result that if two Laplace transforms $L(f_1)$ and $L(f_2)$ coincide for a periodic set of values along an horizontal line, then it follows that $f_1=f_2$ a.e. I was wondering if I ...
proofromthebook's user avatar
4 votes
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Find the inverse Laplace transform of F(s) = 1/(s+exp(-sτ)), where τ is a positive real parameter.

I'm looking for the inverse Laplace transform of $$F(s) = \frac{1}{s + e^{-s\tau}}$$ where τ is a positive real parameter. I am trying to use general inverse formula of Laplace transformation to solve ...
Kevin's user avatar
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3 votes
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The distribution of the first hitting time for the Constant Elasticity of Variance process.

The Constant Elasticity of Variance (CEV) process is a one dimensional diffusion process given by the following stochastic differential equation. \begin{equation} d X_t = \mu X_t \cdot dt + \sigma X_t^...
Przemo's user avatar
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2 votes
1 answer
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Solving the PDE through Laplace Transform Method

I have a particular PDE as shown below: $$ \frac{\partial u}{\partial t} = \nu \frac{\partial^2 u}{\partial x^2} + xe^{-\gamma x} $$ with Boundary conditions as shown, $$ \nu, \gamma >0 ~~~~1)~u(0, ...
KNVCSG's user avatar
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Solving a ordinary differential equation with initial conditions using the Laplace Transform

Problem: Use the Laplace form to solve the following initial-value problem. $$ y'' + 6y' + 8y = 16 $$ with $$ y(0) = 0 $$ $$ y'(0) = 10 $$ and the independent variable is $t$. Answer: \begin{align*} s^...
Bob's user avatar
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2 answers
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Interpretation of the Following Summation

I wanted to invert the following Laplacian expression which was not possible with simple manipulation. $$ \frac{s}{s^2 + \omega^2}\exp\bigg(-\sqrt{\frac{s}{\alpha}}x\bigg) $$ Thus, I used the Taylor ...
KNVCSG's user avatar
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1 answer
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Numerical Inverse Z-Transform - Abate and Whitt

I'm trying to implement an inverse Z-Transform using the Fourier series based technique by Abate and Whitt: The Fourier-series method for inverting transforms of probability distributions. Numerical ...
Tazorraxx's user avatar
2 votes
2 answers
71 views

Finding the analytical solution of a first order system with pure time delay

I have a simple system and I am searching for the solution for f(t): $$\frac{\partial f(t)}{\partial t} = c_1 \left( f(t) + g(t) + c_2 \right)$$. It turns out that, in this system $g$ can be related ...
Damkohler's user avatar
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1 answer
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Multiplication of a time-domain sinusoid to a s-domain (Laplace) signal?

I am confused between the transformations between the time-domain and the frequency domain. I have a signal y(t) which is a sum of multiple sinusoids. I band-pass filter this signal to extract one ...
Ayush Sharma's user avatar
1 vote
1 answer
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Is this derivation of the inverse Laplace transform solid?

I tried to recreate the proof of the inverse Laplace transform formula based on the blurry memory from an electrical engineering book I've read and I would like a real mathematician to look at it and ...
Tigozawr's user avatar
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1 answer
55 views

Solution of an IVP through Laplace transform

Let $𝑦(𝑡)$ be the solution of the initial value problem $$y''+4y=\begin{cases} t, & 0\leq t\leq 2\\ 2, & 2<t<\infty \end{cases}.$$ Also, it is given that $$y(0)=0, y'(0)=0.$$ Given ...
PAMG's user avatar
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Can you find the Inverse Laplace for this Voltage- Angular Displacement function (seen in the picture down Below)? [closed]

Can you find the Inverse Laplace for this function (seen in the picture down Below) Additionally show how to do this on Matlab. I used Matlab and got the answer below, which I think is wrong, but I ...
Jack Jones 's user avatar
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How pathological can a Laplace transform be?

Almost every treatment of the Laplace transform that I come across talks about "the poles" of the Laplace transform function $F(s)$, thereby seeming to implicitly assume that $F(s)$ is ...
tparker's user avatar
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Has anybody ever heard about this theorem? $L\{f'(t)\}(0)+f(0)=k$

Does anybody know the following theorem? $y(0)+f(0)=k$. Let $f:D\rightarrow C$ be a Laplace function with additional constant $k$, and $y(s)=L\{f'(t)\}$. If $y(0)$ is defined then: $y(0)+f(0)=k$.
Alessandro Pini's user avatar
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Inverse Laplace transform of $ \bar{U}(r,s) = \frac{1}{s} \frac{K_0 \left(\sqrt{s/\alpha} \, r \right)}{K_0 \left(\sqrt{s/\alpha} \, b \right)}$ [duplicate]

What is the inverse Laplace transform of $$ \bar{U}(r,s) = \frac{1}{s} \frac{K_0 \left(\sqrt{s/\alpha} \, r \right)}{K_0 \left(\sqrt{s/\alpha} \, b \right)} \tag{12}. $$ ? Related (but not providing ...
sancho.s ReinstateMonicaCellio's user avatar
1 vote
1 answer
34 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 ...
Nero's user avatar
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1 answer
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inverse laplace transform of $\frac{1}{s+b}e^{-x\sqrt{\frac{s}{k}}}$

I am attempting to find the inverse laplace transform of $\frac{1}{s+b}e^{-x\sqrt{\frac{s}{k}}}$ The solution should be $$\frac{e^{-bt}}{2} ( {e^{x\sqrt{\frac{-b}{k}}}\ erfc\left(\frac{x+2kt\sqrt{\...
Gabsmacked's user avatar
1 vote
0 answers
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Recreate a function from its Laplace transform

Suppose I have a function $f(t)$, whose domain is $(0,\infty)$. I take Laplace transform of the function: $F(s)=\int_0^{\infty}e^{-st}f(t)dt$. It exists only for $Re(s)>0$. If $s$ is real and $s\...
eMathHelp's user avatar
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1 vote
0 answers
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Inverse Laplace transform of modified bessel function $K_0$

From mathematica, I get $\mathscr{L^{-1}}(\frac{K_0(as)}{s}) = Log[\frac{t}{a} + \sqrt{-1+\frac{t^2}{a^2}}]$ when $a \leq t$. Then we know $ \mathscr{L^{-1}} (s F(s)) = \frac{d}{dt} f(t) + \mathscr{L^{...
Ziqin He's user avatar
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3 votes
0 answers
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Where did I go wrong in using the residue theorem to find the inverse Laplace transform of this function?

I used the residue theorem to solve the inverse Laplace transform of: $$f(t)=\mathcal L^{-1}\Bigg( {s e^{zs} \over (k-s)^2(k+s)^2}\Bigg) \tag 1$$ where $k$ and $z$ are non-negative. I have two poles ...
FriendlyNeighborhoodEngineer's user avatar
1 vote
0 answers
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to find inverse laplace transform of $f(s)=s\ln\left|\frac s{\sqrt{s^2+1}}\right|$

I tried the differential property of the Laplace transform to the logarithm component, then I tried the “multiplying by s” property, but the answer would be infinity. What do you think is the solution?...
Mohammed Mohammed's user avatar
1 vote
1 answer
61 views

How to find the inverse Laplace transform of this expression?

While solving an ODE using the Laplace transform, I ran into the following problem: $$f(t) = \mathcal L^{-1}\Bigg( {se^{cs} \over (e^s+e^{-s})(k-s)^2(k+s)^2}\Bigg) \tag 1$$ where $c$ and $k$ are ...
FriendlyNeighborhoodEngineer's user avatar
1 vote
1 answer
53 views

Complex exponential, getting constants from partial fraction decomposition in Laplace domain (used s = j*omega) prove / explain please

I came across this in a control engineering textbook: consider the transfer function $G(s)$ as the following ratio of functions, where the denominator is a polynomial in s: $$G(s) = \frac{p(s)}{q(s)} =...
Mr Phase Locked Loop's user avatar
1 vote
0 answers
59 views

Combined Use of S-Shifting and T-Shifting in Laplace Inverse Transform

Please see here to see question & my solution Here are question & solutions I tried. How should one approach problems where both s-shifting and t-shifting are required? Is there a specific ...
Handon's user avatar
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0 answers
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How do you calculate inverse Laplace transform of a fraction

I need to compute the inverse Laplace transform $$\mathcal{L}^{-1}\Big[\frac{3}{(s^2+3)^2}\Big].$$ I already know that the answer is $$ \frac{1}{2\sqrt{3}}\Big(\sin(\sqrt{3}t)-\sqrt{3}t\cos(\sqrt{3}t)\...
Lancet S.'s user avatar
  • 515
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0 answers
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Difficulty With An Inverse Laplace Transform

I am a physics student modeling a quantum mechanical system for my undergraduate research. I am currently solving for the time-dependent coefficients $c_1(t), c_2(t), c_3(t)$ on a super position of ...
AD203's user avatar
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2 votes
0 answers
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Verifying the inverse Laplace transform for a production-inventory problem: total expected backlogs when demand is Poisson

I am entirely self-taught when it comes to Laplace transforms, and I am seeking an independent opinion on my attempt to work out how to arrive at the below expression (note: I am interested ...
DrEti's user avatar
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0 answers
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Inverse Laplace transform of $\frac{\sinh(\sqrt z a)}{\sqrt z}$

I was trying solve this inverse Laplace transform, given by $\frac{\sinh(\sqrt z a)}{\sqrt z}$, with $a \in \mathbb{R}$. But i dont have any good idea to solve this. Please someone have a idea?
Impetus's user avatar
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1 vote
1 answer
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Stuck on Inverse Laplace transform, trying to convert convolution to ODE

I want to express a convolution of a known signal $b(t)=c(t)\ast h(t)$ as an ODE $b'(t)$ instead. The kernel $h(t)$ takes a double exponential form $Au(t)(1-e^{-T_{on} t})e^{-T_{off} t}$. After ...
kjohnsen's user avatar
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1 vote
0 answers
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Inverse Laplace transform of hyperbolic functions

can anyone tell me how can I seek the inverse Laplace transform of equation $F(s)=\frac{p}{s}-\frac{p}{s}\frac{sinh(a\sqrt{s})}{sinh(b\sqrt{s})}$, where $p$, $a$, and $b$ are positive constants ...
Maozhu Peng's user avatar
6 votes
2 answers
386 views

An inverse Laplace transform and its asymptotics

I am wondering if there is an analytic expression for the inverse Laplace transform $f(t):=\mathcal L^{-1}[F](t)$ for $F(s):=\frac1{s(\cosh{\sqrt{2s}}-1)}$. If there is no such analytic expression, ...
Hans's user avatar
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2 votes
1 answer
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Defining a fractional derivative operator over Laplace transforms

$$\Large{ \text{Introduction} }$$ I'm interested in solving methods for fractional-differential-equations (FDEs) without specifying the nature of the fractional derivatives. In doing so, I keep coming ...
Kevin Dietrich's user avatar
3 votes
1 answer
125 views

Inverse Laplace transform of $F(s) = \exp(-s)/s$ via its infinite series expansion

To invert $\frac{s}{{(s - 2)(s + 3)}}$ one might split it as: $$ \frac{s}{{(s - 2)(s + 3)}}\; = \;\frac{A}{{(s - 2)}}\; + \;\frac{B}{{(s + 3)}} $$ solve for $A$ and $B$ and invert the fractions in the ...
Carlos Gouveia's user avatar
4 votes
1 answer
235 views

Closed form of $\frac1s\int_0^\infty e^{-s(\tanh(t)+at)}dt=\frac1s\int_0^1e^{-st}(1-t)^{\frac{as}2-1}(1+t)^{-\frac{as}2-1}dt$ to invert $\tanh(x)+ax$?

$\def\L{\operatorname L} \def\M{\operatorname M} $ Our goal is inverting $f(x)=\tanh(x)+ax$ via Laplace inversion. The Laplace transform of $h(x)=f^{-1}(x)$ is: $$\L_s(h(t))=\int_0^\infty e^{-st}h(t)...
Тyma Gaidash ٠'s user avatar
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0 answers
15 views

Inverse bilateral Laplace transform: behaviour at time t=0

For the purpose of my research on stochastic processes on graphs I need to compute the behaviour of a correlation function at time $t=0$, $C(t=0)$. I was able to compute its bilateral Laplace ...
Mattia Tarabolo's user avatar
0 votes
1 answer
45 views

Find the Inverse Laplace Transform:

Please help me in finding Inverse Laplace transform of : $$ \frac{a(s^{2}-a^{2})}{s^4+4a^{4}} $$ I have tried id by following ways but need help in further approach $$ splitting\ \ \ s^4+4a^4=(s^2+2as+...
Dev Keshwani's user avatar
1 vote
0 answers
49 views

inverse Laplace transform of an integral

find $$u(x,t)$$ Given $$ U(x,s)=\frac{s+2}{(s+1)}{\int_{-\infty}^{\infty}f(x)cosh((s+1)(x-y))dx}$$ where U is the Laplace transform of the function u. I tried substituting $$cosh((s+1)(x-y))=\frac{e^{(...
ochem1's user avatar
  • 59
2 votes
2 answers
50 views

Initial value problem using Laplace Transform

I have to solve the IVP: $$\begin{cases} y''+by'-cy=1\\ y(0)=y_0\\ y'(0)=y'_0 \end{cases}$$ Suppose $$Y(s)=\frac{s^2+2s+1}{s^3+3s^2+2s}$$ We have to find $b,c,y_0$ and $y'_0$ I have found that: $$(s^2+...
amspsingh04's user avatar
1 vote
1 answer
442 views

Solve the boundary-value problem with Laplace transform

Solve $$\frac{d^2U}{dx^2} - sU = A, \qquad (0<x<1)$$ subject to the boundary conditions $$\frac{dU}{dx}(0) = 0, \qquad U(1) = 0.$$ The part I'm stuck on is that after taking Laplace transforms, ...
ochem1's user avatar
  • 59
0 votes
2 answers
82 views

moving the branch cut in an inverse Laplace transform

I would like to calculate the inverse Laplace transform of the function $e^{-\sqrt s}$. I understand that this is given by $$ {\cal L}^{-1} [e^{-\sqrt s}] = \frac{1}{2\pi i} \int_{-i\infty}^{i\infty} ...
proteus7's user avatar
2 votes
0 answers
105 views

What is the inverse Laplace transform of $\text{erfi}(\sqrt{s}\cdot x)$?

The inverse of function $\text{erf}(\sqrt{s}\cdot x)$ is relatively easy to obtain: $$\mathcal{L}_s^{-1}\left[\text{erf}\left(\sqrt{s}\cdot x\right)\right](t)=\delta (t)-\frac{|x| \cdot \theta \left(t-...
gpmath's user avatar
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1 vote
0 answers
58 views

What is the inverse Laplace transform of $I_{0} (\sqrt {as+b} )$? [closed]

What is the inverse Laplace transform of $I_{0} (\sqrt {as+b} )$? Note: $I_{0}$ is the modified Bessel function of first kind with zero-order. $a$ and $b$ are constants. $s$ the Laplace parameter
Saajid's user avatar
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