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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Burgers Equation and Coupled Burgers Equation [closed]
What is the difference between the Burgers Equation and Coupled Burgers Equation 1D, 2D, and 3D forms?
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Calculate zeros (root of unit) by using partial fraction expansion and complex analysis, from David Kendall
I want to fill in the derivation of the following Harris's book
https://www.rand.org/content/dam/rand/pubs/reports/2009/R381.pdf
Specifically, I want to know how the author obtains $a_r$ after eq16.2 ...
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Annihilators & Laplace transform
Let the following differential equation :
$ \begin{align*}
\left[\frac{\text{d}}{\text{d}t}+\frac1{\tau}\right]V_{_\text{OUT}}&=\frac{A\sin\left(\omega \,t\right)}{\tau}\tag{1}\label{eq1}
\end{...
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Are bounded monotone functions piecewise continuous or at least their Laplace transform unique?
Suppose you have a function $h: \mathbb{R}_{+} \to \mathbb{R}_{+}$ that is monotone decreasing (thus bounded) and locally integrable. Can we conclude that it is piecewise continuous?
Obviously, due to ...
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How do i Solve this ? Find the output of the function if the input function is given [closed]
Find the output of the function if the input function is given as $$g(s)= \frac{s-5}{s(s+2)^2}.$$
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Difficult First Order ODE
The ODE is$$\frac{\mathrm dy}{\mathrm dx}+\frac{x}{y}=-x^2$$ I tried solving this ODE by creating a homogeneous form, exact differential form, Bernoulli's form(reducible to linear), but it did not ...
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Laplace transformation of double exponential involving a Poisson process
This is from a series of problems I am working on for self study:
Let us denote by $N(t)$, $t \geq 0$ a Poisson process with $N(0) = 0$ and parameter $\lambda$. Denote by $S_k$ its $k$-th jump time, ...
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What does Laplace transform say about the original function [closed]
I have a smooth function f(t) such that
$$F(s)=\int_0^\infty e^{-st}f(t)dt = 0$$
Laplace transform of this function is zero for all values of s. What does it imply about the function f(t)?
I managed ...
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Solution of a system of differential equations by the Laplace method
Original system of differential equations:
\begin{cases}
\dot{p_{2}}(t)=\mu p_{1}(t)-\lambda p_{2}(t)\\
\dot{p_{1}}(t)=\lambda p_{2}(t)-(\mu+\lambda)p_{1}(t)\\
\dot{p_{0}}(t)=\lambda p_{1}(t)\\
p_{2}(...
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Show that $\mathcal{L}^{-1}\left\{\frac{\Gamma(n)}{s^n}e^{-\frac{2a}{s}}{}_0F_1\left(n,\frac{a^2}{s^2}\right)\right\}(1)={}_0F_1\left(n,-a\right)^2$
I am trying to show that given $n, a > 0$
$$ G(s) = \frac{\Gamma(n)}{s^n} e^{- \frac{2a}{s}} {}_0F_1 \left( n, \frac{a^2}{s^2} \right) $$
the inverse Laplace transform of $G$ evaluated at $t=1$ is
...
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Is there a way to simplify this Laplace Transform and separate the p' from it?
I want to isolate dt/dp from the Laplace form to make it simpler. Is there a method to simplify and extract any variable from this Laplace form?
$$\mathcal{L}_t\left[\frac{p^{\prime}(t)}{(m p(t)+\...
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Uniqueness of 1st order linear ODE with step discontinuities in driving function
Consider the following first order ODE initial value problem with a discontinuous driving function composed of step functions:
$$y'(t) + y(t) = u_1(t) - u_2(t)$$
$$y(0) = 0$$
$$u_a(t) \equiv
\begin{...
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Mellin Inversion Theorem for integral transforms with different kernels.
I am studying the Mellin inversion theorem and its applications in various integral transforms. I understand that the Mellin transform uses the power function $ x^{s-1} $ as a kernel and that the ...
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Decaying functions on positive reals are Laplace transforms of signed measures
Given any function $f(t)$ that decays on $[0, \infty)$ to zero (but is not necessarily monotonic), is it possible to show that it can be represented as the Laplace transform of a signed Borel measure $...
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Determining the Laplace Transform of a triangle signal
I have been given the following signal:
I have written this using the Heaviside function. This gave me the following:
$$ x(t) = t[u(t)-u(t-1)] + t[u(t-1)-u(t-2)] = tu(t) - tu(t-2) $$
Applying the ...
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General solution of the wave equation using Laplace transform in the time domain
I have been exploring different methods of solving PDEs and in general various solutions of the wave equation under more general boundary conditions (like if the boundaries are ($-\infty$, $\infty$) ...
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finding the transfer equation of $\frac{H(s)}{Q_d(s)}$ in liquid control system with hydraulic integral controller.
On problem B-4-10
So my attempt is as follows:
from the liquid control system we have
$(q_d+q_i-q_o)dt=Cdh$ since $R=\frac{h}{q_0}$ and $R=0.5$ then $q_0=2h$.
By the equibilirium system on the lever ...
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Simplifying an inverse Laplace transform
I am reading a paper, in which the author claimed that the inverse Laplace transform
$$
L_1(\rho) = \frac{1}{2\pi i} \int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{s\rho}
\frac{\sin{2s\alpha_1} - s \...
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How would this partial differential equation (PDE) be classified? Can it be solved for any integer $a>0$?
I am trying to find the Laplace transformation of a multivariable function that satisfies the following equation:
$$
ax(t_1,t_2,...,t_n)=\sum_{i=1}^{n}{\lambda_it_i^ax(t_1,t_2,...,t_n)}+u(a)
\text{ ...
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Analytical solution of a coupled non-linear pde?
Hi, I want to find the solution to this non-linearly coupled PDE:
$\partial_t(L_x) = - \partial_r(L_x v^r) - \Omega L_y$;
$\partial_t(L_y) = - \partial_r(L_y v^r) + \Omega L_x$
Here,
$L_x(r,t)$ and $...
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Bromwich integral in evaluation of Green function of 1+1D Klein-Gordon equation
When I'm evaluating Green function of 1+1D Klein-Gordon equation using integral transform techniques, I have done in many ways of taking integral transform, I have got
\begin{equation}
G(x,x',t,t'...
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Proving Laplace Transform is Analytic
I was trying to prove that the Laplace Transform is analytic and to do so I tried to evaluate its derivative which is (this needs to be proven) $F'(s)=-\int_{0}^{\infty}tf(t)e^{-st}dt$ where f(t) is a ...
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Laplace transform region of convergence of two subsequent poles
Suppose we have a function:
$$
f(t)= e^{-4t} + sin(t)e^{-2t},
$$
where the Laplace transform is given by the standard lookup table of individual transforms:
$$
F(s)=\frac{1}{s+4}+\frac{1}{(s+2)^2+1},
$...
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Why do we call Laplace transform linear?
As per my current understanding, a linear function/operation is a mapping that satisfies:
$$ L\{f+g\} = L\{f\} + L\{g\} $$
$$ L\{kf\} = kL\{f\} $$
for all inputs it could accept. At first glance, it ...
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What is Laplace transform of $t^af(t)$ if Laplace transform of $f(x)$ is $F$.
What is the Laplace transform of $t^af(t),a>0$ if the Laplace transform of $f(x)$ is given to be $F(s)$. By definition, it should be like this:
$$\mathcal{L}\{t^a f(t)\}(s) = \int_{0}^{\infty} t^a ...
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Stuck on a linear system of DEs by Laplace transform
$x_1''-2x_2'+3x=0$,
$x_2''+2x_1'+3x_2=0$
where $x_1(0)=4,x_1'(0)=0,x_2(0)=0,x_2'(0)=0$
Applied Laplace transform to both equations
$4s=(s^2+3)X_1-2sX_2 ...(1)$
$8=(s^2+3)X_2+2sX_1$
can yall guide me ...
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Closed form of $\sum_{n=0}^{\infty}\frac{1}{(n!)^3}$
We know that $\displaystyle\sum_{n=0}^{\infty}\frac{1}{(n!)^1} = e$.
By considering the function $f(x,y) := \displaystyle\sum_{n=0}^{\infty} \frac{(xy)^n}{(n!)^2}$ and taking the Laplace transform in $...
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On two-sided Laplace transform in probability; how to show it is injective?
I am studying a theorem in probability that the Laplace transform of a (nonnegative) random variable determines the law of that random variable, which is equivalent to its injectivity. The book (...
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Laplace transform of $\sin(\omega t)$
I am learning about the Laplace transform and I know I got the answer to this example question wrong, but I'm trying to figure out if I just made a calculus or algebra type error, or if I'm ...
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Solving 1st order PDE including convolution
I'm studying Van Kampen's "Stochastic processes in physics and chemistry" and stuck to some exercise (p.78):
That is, solving
\begin{equation}
\frac{\partial P(y, t)}{\partial t}=\int_{-\...
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Using the Laplace transform, solve the system of linear differential equations with constant coefficients
Using the Laplace transform, solve the system of linear differential equations with constant coefficients
$ y' + z' = t, $
$ y'' - z = e^{-t}, $
$ y(0) = 3, $
$ y'(0) = -2, $
$ z(0) = 0, $
where $y(t)$...
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Calculate the Laplace transform of the function $\frac{\sin t}{t}$ and then use it to compute the integral
Calculate the Laplace transform of the function $\frac{\sin t}{t}$ and then use it to compute the integral
$$ \int_0^\infty \frac{e^{-at} - e^{-bt}}{t} \, dt, $$
where $a, b \in \mathbb{R}$.
Attempt: ...
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Let $F(z)$ be the Laplace transform of the function $f(t)$. Derive two formulas
Let $F(z)$ be the Laplace transform of the function $f(t)$. Derive the formulas:
$$
\mathcal{L} \left\{ \int_0^t f(u) \, du \right\} = \frac{F(z)}{z}
$$
and
$$
\mathcal{L} \left\{ \frac{f(t)}{t} \...
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Laplace transform of exponential functions with derivatives.
I have been trying to calculate the Laplace transform of these troublesome exponential functions:
Having $\alpha \in \mathbb{R^+}$
1.$\mathcal{L}\left\{e^{n \alpha t}\frac{f(t)}{t^2} \right\},n \in \...
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Laplace transform of probability density functions
I have a curious observation motivated by some applications of queueing theory. Let $f:[0,+\infty)\to\mathbb R^+$ be a PDF so that $\int_0^{+\infty}f(t)dt=1$. Let's say that $f$ is bounded and ...
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General solution for linear Volterra-like integral equation?
A linear Volterra integral equation looks like this (see the wiki)
\begin{align}
x(t) = f(t) + \int_0^t K(t, s)x(s)~\mathrm{d}s.
\end{align}
If the Kernel function $K$ is of the form $K(t, s) = K(...
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Convert the Laplace transform of the Bessel function to a Fourier transform
I want to calculate the Fourier transform of the function $f(t)$, defined as $f(t)=0$ if $t<0$ and $f(t)=J_{n}(t)$ if $t\ge0$, in which $J_{n}(t)$ is the Bessel function of the first kind. That is, ...
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How to derive the 1/s rule for Laplace transform of an integral?
How do we derive the rule $\mathcal{L}\{ \int_0^t f(\tau) d\tau\}=F(s)/s$ where $F(s)=\mathcal{L}\{f(t)\}$ and the symbol $\mathcal{L}$ represents the Laplace transform operator ?
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Inverse Laplace transform of product of three exponentials
Consider the Laplace Transform of the form
$$
\operatorname{F}\left(s\right) =
\frac{1}{\left(s - s_{1}\right)^{\large a_{1}}
\left(s - s_{2}\right)^{\large a_{2}}
\...
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How to show the laplace trasnform of a translated piecewise function
I'm learning laplace transforms, and righ now I'm on their properties.
I am given the following problem:
I've done the obvious of ∫ f(t-c) exp(-st)ds but I get stuck at this point on how to integrate....
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Laplace transform of special function
The Confluent hypergeometric function of first kind (aka Kummer's function) is defined as
$${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a%
\right)}\int_{0}^{1}e^{zt}t^{a-...
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Need help with a Laplace transform / probability theory question
Exercise 7.1 here:
For the first part, it's easy to see for n = 1 and I've tried a few different things to get the integrals to match for n = 2 (and suspect the solution is some type of induction), ...
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Prove that the convolution of the signals and its time reversal is an odd signal.
Suppose signal $g(t)$ is obtained by time reversal of signal $f(t)$ for all times $t$. Prove that the convolution of the signals $f$ and $g$ is an odd signal.
My attempt at proof
Given: $g(t)=f(-t)\...
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Compute the integral: $\int_{0}^{\infty}e^{-x}\text{Ei}(-x)dx$
My idea was to use the Laplace transform using this identity:
$$\mathcal{L}\left\{\int_{0}^{t}f(x)dx\right\}=\frac{\mathcal{L}\{f(t)\}}{s}=\frac{F(s)}{s}$$
My first approach was utilizing the proof of ...
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Reconciliation of different results regarding Laplace transform of $\ln(1+t)$
I want to calculate the Laplace transform of $\ln(1+t)u(t)$. We know from
Laplace transform of natural logarithm
that $\mathcal{L}\{\ln t \;u(t)\} = -\frac{\gamma+\ln s}{s}$ where $\gamma$ is the
...
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I Have a Question About the Laplace Inverse Transformation Process.
The equation that needs to be inverse transformed[1]
$$
\frac{s + (s + 1)(s^2 + 1)}{(s^2 + 1)(s^2 + 2s + 2)}
$$
The modified equation 1 for the Laplace inverse transformation[2]
$$
\frac{s^3 + s^2 + ...
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Why do we care about convergence of the Laplace transform?
When I took elementary differential equations, with the textbook of Boyce & DiPrima, I learned about using the Laplace transform to solve some initial value problems. I also took a course in ...
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Laplace Transform solve ODE second order
Find ODE second order using Laplace Transform
$y" + 3y' + 4y = 3t $
$; y(0) = 2$
$; y'(0) = 2$
( This is imaginary roots)
I have found the answer using undetermined coefficient but i got stuck when ...
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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(...
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Inverse Laplace Transform of $e^{-as}/s^2$ for $a>0$
I am trying to compute the inverse Laplace transform of $$F(s) = \frac{e^{-as}}{s^2}$$ for $a > 0$.
I computed it as follows:
$$\mathcal{L}^{-1}_{s\to t} \left\{\frac{e^{-as}}{s^2}\right\} = \text{...
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