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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Non-negative functions on $[0,+\infty[$ whose Laplace transform is well-defined on $]0,1]$
I am trying to determine the set $E$ of non-negative Lebesgue-measurable functions $f$ on $[0,+\infty[$ whose Laplace transform is defined on $]0,1]$, that is, such that the Lebesgue integral $\int_0^{...
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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^{...
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What are the rules for solving differential inequalities using Laplace Transforms?
I am looking to solve a non-autonomous differential inequality of the form
$$\frac{dV}{d\tau}-\epsilon V-\frac{E^2}{2\epsilon}\leq\frac{A^2}{2\epsilon}\tau^2+\frac{2AE\tau}{\epsilon},$$where $V:\...
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Modelling of electrical RLC circuit, using Laplace transform
enter image description here
Can someone explain the process to obtain the equation 5 and 6, from figure 5?
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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 ...
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Where is the Laplace transform an isomorphism?
I have understand that the Fourier transform is an isomorphism between the Schwartz space.
Is the Laplace transform some isomorphism between some function space?
For example, I understand that if $f(z)...
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How would I simplify this Laplace transform?
I was reading an article about using a Laplace Transform on a fractional differential equation and attempted to derieve a formula that was just skipped over to.
The Laplace transform is as follows:
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Laplace transform using the transform of the derivative.
I wanted to compute the Laplace transform of $$f(t)=\begin{cases}t & 0\le t\le 1\\
0& \text{otherwise} \end{cases}$$ using the derivative. So I computed
$f'(t)=\begin{cases}1 & 0\le t\le ...
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Is DiracDelta function $\delta(t)$ equals zero in the neighborhood of $0$?
I know that DiracDelta function equals zero everywhere except at $0$, but what about $\delta(0^-)$ and $\delta(0^+)$?
Should they be evaluated as $\delta(0)$ or to be equal $0$ ?
I ask this question ...
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A non-local equation on Laplace transforms.
I wish to find all probability distributions $\mu$ on $\mathbb R^+$ whose Laplace transform:
$\varphi(z) = \int_{0}^{\infty} e^{-zx} \mu(dx), z \geqslant 0,$
solves the equation:
$\varphi(z) = \beta \...
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Solve Laplace Transformation using Convolution
Find $\displaystyle \mathcal{L}\left (\int\limits_0^t e^{-2\tau}\sin(3\tau)\;d\tau \right )$. I know we can solve this problem by solving the integral and then finding the Laplace Transformation. Is ...
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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?...
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Solve $f(t)=e^t+e^t\int_0^te^{-\tau}f(\tau)\mathrm{d}\tau$
The question:
$$\begin{equation*} f(t)=e^t+e^t\int_0^te^{-\tau}f(\tau)\mathrm{d}\tau \end{equation*}.$$
I believe we need to take the Laplace transform of all terms. I am getting stuck with this part:
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Laplace transform of $f(t^{n})$ for $n>3$
In this previous question, it was shown that the Laplace transform of $f(t^2)$ can be expressed in terms of the Laplace transform $F$ of $f$, namely:
$$ H_2(s) = \int_0^\infty f(t^2) e^{-st} dt = \...
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Piecewise to Heaviside
I really need help with this exercise, I try by days.
Use Heaviside in this exercise:
$$f(t) = \begin{cases} 2t^2 + 4t +1 ,& 0 \leq t < 6 \\4t^2 -16t+50 ,& t \geq 6\end{cases}$$
for show L{...
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Complex exponential, getting constants from partial fraction decomposition in Laplace domain (used s = j*omega) prove / explain please
I came across this in a control engineering textbook: consider the transfer function $G(s)$ as the following ratio of functions, where the denominator is a polynomial in s:
$$G(s) = \frac{p(s)}{q(s)} =...
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Commutative law in convolution of laplace transfer
This question needs to use t-shifting and convolution in inverse laplace transfer.
I tried to consider t-shifting first, then did convolution as shown in below.
Since there is u(t-a) function, I ...
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Solving nonhomogeneous ordinary differential equation by use of Laplace transform
Given the differential equation
$$ 𝑦′′ − 5𝑦′ + 6𝑦 = e^{−𝑡}, \quad 𝑦(0) = 𝑐_1, \, 𝑦’(0) = 𝑐_2 $$
how can I use the Laplace transform to solve this nonhomogeneous differential equation ?
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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 ...
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Laplace transform of a PDE
I am learning how to use Laplace transforms to solve PDEs. In case of an expression like
\begin{equation}
\int_{r}^{\infty} f(x,t) g(x) dx
\end{equation}
the resulting Laplace transform (with respect ...
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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 $$ \frac{1}{2\sqrt{3}}\Big(\sin(\sqrt{3}t)-\sqrt{3}t\cos(\sqrt{3}t)\...
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An unzipping problem
Imagine a continuous one-dimensional line, which is duplicated exactly once. Duplication starts at random spacetime points. Once a point is duplicated, it starts a double duplication wave moving in ...
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If $f \neq 0$, do we have $\int_{ x_i \geq 0} f(x) e^{-\sum_{i=1}^n c_i x_i^2} d^nx \neq 0$ for some $c \in (0,\infty)^n$?
Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a bounded measurable mapping. Next, define
\begin{equation}
\mathbb{R}^n_+ := \{ x \in \mathbb{R}^n \mid x_i \geq 0 \text{ for all } i=1,\cdots,n\}
\end{...
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Extending Laplace transform to multivariate cases?
Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a measurable $L^\infty$ function. Assume further that $f$ is not zero a.e. and $f(-x)=f(x)$ for a.e. $x \in \mathbb{R}^n$.
Then, I wonder if we can conclude ...
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Write a piecewise function with unit steps
enter image description here
As the image shows I have to write the piecewise function as a unit step function thing. I have tried to do this:
$$1\cdot\Bigg(u(t-0)-u(t-1)\Bigg) + c\cdot \Bigg(u(t-1)\...
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Laplace transform of normal distributions and their convolution
Hi (apologies for my english), i have been interested in the CLT for the past weeks, and already have proved that the convolution of 2 normal distributions is another normal distribution using the ...
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Solve the following differential equation by the Laplace transform method: $y''+y'= 8x^2$
Solve the following differential equation by the Laplace transform method:
$y''+y'= 8x^2, \\ y(0)=0 , \\ y'(0)=1$
By applying the Laplace transform to both sides:
$L\{y''+y'\}=L\{8x^2\}$
$L\{y''+y'\}=...
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What are some good reference books for signals?
The title might be a bit misleading, but basically I've been self studying differential equations so I could apply them in electronic circuit design. I was wondering what books could be recommended on ...
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What is the Expected Time of Crystallization?
Consider the process of periodic 1D crystallization, where multiple sites initiate crystallizing waves at random with speed $v$. The fraction of the crystallized substance, at position $x$ and time $t$...
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How to derive $\mathcal{L}\{ \log^2(1+t)\}(s) = \log(2/s^3 + 2/s^2 + 1/s)$?
I aim to determine the Laplace transform of
$\log^2(1+t)$.
Apparently, Mathematica gives me a closed formula
$\log(2/s^3 + 2/s^2 + 1/s)$. Any idea how to obtain this formula?
I though run into a ...
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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 ...
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What does it mean for a distribution to have support on the real axis?
What does it mean for a distribution to have support on the
real axis? I'm ask this because I just saw the following:
If $d$ is a distribution with support on the real axis, the Laplace-Borel ...
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Coupled functions $f,g$ satisfy $Lf(\lambda) = \frac{\beta}{\beta + \lambda} Lg(\lambda)$, solve for $f$
Let $\beta, \delta \geq 0$, $\beta + \delta > 0$. Let $f(t), g(t) : \mathbb{R}_{\geq 0} \rightarrow [0,1]$ satisfying the following equations:
$$f(t) = \int_{0}^t \beta e^{-\beta (t-s)} g(s) ds$$
$$...
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Formula for Laplace transform of the jump probability in a continuous time random walk
I am trying to understand the basics of continuous time random walks, and this formula has no explanation as far as I can find:
$$\hat{p}_{n}(s) = \hat{\rho}(s)^n\cdot\frac{1-\hat{\rho}(s)}{s}$$
Where ...
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Laplace transform with scaled time variable
We know that by definition,
$L{f(t)}=\int_{0}^{\infty}e^{-st}f(t)dt$. But how is $L{f(at)}=\int_{0}^{\infty}e^{-st}f(at)dt$? Shouldn't $e^{-st}$ be replaced by $e^{-s(at)}$ in the 2nd case since $at$ ...
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Infinite integral of $e^{-\alpha t^\beta} \cos(\omega t)$
I am attempting to calculate the integral
\begin{equation}
I=\int_0^{\infty} e^{-\alpha t^\beta} \cos(\omega t) dt,
\end{equation}
where $\alpha,\beta >0$, and $\omega$ are just parameters.
My ...
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Continuous power series coefficient function
Suppose $f(x)$ is analytic and infinitely differentiable at $x=1$.
$F(x)$ is another function such that, for all nonnegative $\alpha<1$, this infinite sum $F(\alpha)x^{\alpha}+F(α+1)x^{\alpha+1}+F(...
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Difficulty With An Inverse Laplace Transform
I am a physics student modeling a quantum mechanical system for my undergraduate research. I am currently solving for the time-dependent coefficients $c_1(t), c_2(t), c_3(t)$ on a super position of ...
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How do I solve the following inverse Laplace transform?
I was solving a physics problem and eventually came upon this transform and I could not find a way to resolve it. $c$ is just a constant. Could anyone help me on this? I appreciate the help.
$$
\...
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Sum of two sinusoids
Let $A_1 , A_2 , w_1, w_2 $ are positives numbers If for all real t , $s_1(t)=A_1\cos(w_1t+\phi_1)$ , $s_2(t)=A_2\cos(w_2t+\phi_2)$ and $s=s_1 +s_2$
We know if $w_1 = w_2$, we can write $s$ as $s(...
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Solutions and graphs of partial integro differential equations
Consider
$$\partial_tu+ \partial_xu-D\partial_{yy}u=0,$$
subject to the initial condition $u(x,y,0) = v(x,y)$ and the boundary conditions at the infinities, $ u(0,0,t) = u(\pm\infty,\pm\infty,t) = 0,$ ...
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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 ...
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First-passage time distribution in Laplace space?
I'm struggling to understand the reasoning between moving between two steps in a reaction scheme for a paper I am reading. For this (from the description), the probabilities over different paths are ...
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Sum the inverse of the successor of the square of the natural numbers
I was thinking if I could solve the sum $\sum_{n=1}^\infty \frac{1}{n^2+1}$, and I reached using the laplace transform of the sine at the integral $\int_0^\infty \frac{\sin(t)}{e^t-1} dt$. I entered ...
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Using Laplace- and Fourier-Transformation for solving dynamic beam equation
let's assume we have a cantilever beam of length L, which is clamped at left end (x=0) and is loaded with a time-varying force $F(t)$ at the free end (x=L). Further, the bending stiffness is $EI=const$...
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Incorrect Transfer Function - Control Systems
I want to find the transfer function for a system described by this non-linear state-space model:
\begin{equation}
X(t) = \begin{bmatrix}
y(t)\\
v(t)\\
...
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Solving IVP using Laplace Transforms
I would like to solve the IVP
$$y'' + 2y' + 3y = 3t$$
with $y(0)=0, y'(0)=1$ using transform methods. Could someone check my work is correct, as my solution differs from that in the text?
Solution.
...
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Derivative of the Laplace's transform of a function.
I'm writing a dissertation about the probabilistic interpretation of Laplace's transform and I need to prove a formula about the first derivative.
Let $f \, : \, \left[0,+\infty\right)\, \to \mathbb{R}...
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How is the inverse Laplace-Borel transform defined?
How is the inverse Laplace-Borel transform defined?
Si $\mu\in (\mathcal{O}(\mathbb{C}))^{\ast}$ (The dual space of the entire functions or analytic functionals) then the Laplace-Borel $\mathcal{L}\mu(...
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