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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 ...
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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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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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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.
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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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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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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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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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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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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 ...
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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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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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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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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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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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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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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 $\Bbb{R}$ .
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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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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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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 ...
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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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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 ...
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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{...
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“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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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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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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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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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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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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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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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 ...
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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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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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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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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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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^...
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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 ...
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$\lim_{n \to \infty} E[e^{-X*\frac{t}{n}}]^n \approx \lim_{n \to \infty} E(1-\frac{E[X] t}{n})^n = e^{-E[X]*t}$

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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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 $\...
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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) ...
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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{...
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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 ...
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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(...
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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 ...
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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\...
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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. ...
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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 ...
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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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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)...