Questions tagged [laplace-transform]
The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform.
2,584 questions
-3
votes
0answers
30 views
Solve initial value problem $y'' + 9y = f(t)$ using Laplace transform [on hold]
$y'' + 9y = f(t),$
$y(0) = 0$ and $ y'(0) = 1$
$f(t) = 1$ if $0 \leq t < 3\pi $
$f(t) = 0$ if $3\pi \leq t < \infty $
0
votes
0answers
19 views
Initial value problem , system of equations involved , solve using Laplace transform
Can anyone help me solve this problem, I could not find a similar solution on the web.
Solve the following problem :
Initial value problem,
System of equations involved ,
Using Laplace transform
$...
0
votes
0answers
17 views
PDF for the ratio of two random variable
Let $X_1$ and $X_2$ two random variable with pdfs
$$
f_{X_i}(t)=\oint_{c}^{}g_i(t)(a_it)^{-s}ds
$$
for $a_i>0$. I would like to dirive the pdf of
$$
Z=\frac{X_1}{X_2}
$$
I found the following ...
0
votes
0answers
15 views
Calculate Laplace transform of $I_0(\sqrt{x}) \ln(I_0(\sqrt{x}))$
Calculate the Laplace transform of the function
$$I_0(\sqrt{x}) \ln(I_0(\sqrt{x})) $$
where $I_0(u)$ is the modified Bessel functions of zero-th order.
I have been trying to solve this but It ...
0
votes
1answer
19 views
Possible ways of solving second order ODE IVP.
The problem is the following:
y''(t) + 2y'(t) + 5y(t) = H(t-2) - H(t-1) (Where H(t) is the Heaviside function)
y(0)=0, y'(0)=0 (Initial values)
I started working with the Laplace transform and ...
0
votes
0answers
16 views
Finding the Laplace transform of the Laguerre polynomial
I have tried to find the Laplace transform of $L_n(t)$, which is the $n$th order Laguerre polynomial defined by
$$L_n(t) :=e^t \frac{\mathrm{d}^n}{\mathrm{d}t^n}(t^n e^{-t}).$$
I have managed to get
...
3
votes
0answers
38 views
+50
Clarification of Solution of PDE Laplace Transform Problem
I have the following problem:
The transverse displacement $u(x, t)$ of a semi-infinite elastic string satisfies
$$\dfrac{\partial^2{u}}{\partial{t}^2} = c^2 \dfrac{\partial^2{u}}{\partial{x}^...
-3
votes
0answers
16 views
I want MATLAB codes that can plot the graph of θ =2.6575*exp(-0.8875*y)-3*erf(0.942*y)-4.3332*erf(0.942*y) with boundary conditions. [on hold]
I obtained this MATLAB codes [θ=2.6575*exp(-0.8875*y)-3*erf(0.942*y)-4.3332*erf(0.942*y)] from a temperature equation ∂θ/∂t=1/Pr((∂^2 θ)/(∂y^2 ))+MEc u^2 which I solved in fluid mechanics using ...
0
votes
2answers
38 views
Solving integral equation $y(t) = e^t ( 1 + \int_0^t e^{-\tau} y(\tau)d \tau )$ with Laplace transform
Solve the integral equation $$y(t) = e^t \bigg(1+ \int_0^t e^{-\tau}\ y(\tau) d \tau \bigg)$$
The apporach that stands out to me is using the convolution theorem and laplace transforms. Im able to ...
0
votes
0answers
24 views
Inverse Laplace Transform Mixing Sine and Exponential
I have a doubt about Laplace transforms. Most concretely, about the existence of an inverse.
I know already these facts
$$\mathcal{L}^{-1}(\exp(-\sqrt{s}))=\frac{\exp({-\frac{1}{4 t}})}{2 \sqrt{\pi ...
-1
votes
0answers
21 views
Inverse Laplace transform of $\frac{1-s^2e^{2s}}{(s-2)^2(s^2+is+2)}$ [on hold]
Inverse Laplace transform of $\frac{1-s^2e^{2s}}{(s-2)^2(s^2+is+2)}$
can't confirm my result with Wolfram
-1
votes
1answer
35 views
Using Convolution Theorem to Find the Inverse Laplace Transform of $\frac{s}{(s - 2)^{3/2}(s^2 + 1)}$
I am trying to use the convolution theorem to find the inverse Laplace transform of
$$\dfrac{s}{(s - 2)^{3/2}(s^2 + 1)}$$
The convolution theorem states that
$$f * g = \int_0^t f(\tau)g(t - \tau) ...
0
votes
1answer
50 views
Inverse Laplace transform of $\left(\frac{1+\alpha s}{1+\alpha(s-s_0)}\right)^p$
I am hoping to identify the function $f(t)$ that has the following Laplace transform,
$$
\tilde f(s)=\int_0^\infty f(t)e^{-st}dt=\left(\frac{1+\alpha s}{1+\alpha(s-s_0)}\right)^p
$$
where $\alpha,...
0
votes
0answers
26 views
Differential Equations: Switching from a sum of unit step functions into a piecewise function
I've completed a question involving unit-step functions and would like verification on my working out and answers, particularly for the last part, as we are asked to express the solution explicitly ...
-4
votes
0answers
20 views
Please can someone help me find the inverse Laplace transformof u(y,s)=e^(-y√(Ps+F )) [on hold]
I solved the partial differential equation ∂u/∂t=1/P ((∂^2 u)/(∂y^2 )-Fu) using Laplace transform and obtained the general solution u(y,s)=e^(-y√(Ps+F)) . I now want the inverse Laplace transform of ...
3
votes
1answer
40 views
The Laplace transform of $t^{-3/2}e^{-1/t}$ is denoted $F(s)$. Show that $\frac{dF}{ds} = -\frac{F}{s^{1/2}}$.
The Laplace transform of $t^{-3/2}e^{-1/t}$ is denoted $F(s)$. Show that
$$\frac{dF}{ds} = -\frac{F}{s^{1/2}}$$
not sure how to get this.
I tried differentiating $F(s) = \int_{0}^{\infty}t^{-3/2}...
0
votes
1answer
43 views
Laplace Transform of $t^{t^2}$ [closed]
I have been trying to model a dynamic system and I came up with a differential equation which involved me finding the laplace transform of $t^{t^2}$. I have tried all the theorems I am familiar with, ...
1
vote
1answer
34 views
Finding Inverse Laplace Transform of a product series
I need to compute the inverse Laplace transformation of the following equation.
\begin{align*}
f(s)&=\frac{A}{\prod_{i=1}^{L}(s+a_i)^m} \\
&=\frac{A}{(s+a_1)^m\,(s+a_2)^m\cdots (s+a_L)^m}.
\...
-1
votes
0answers
23 views
Compute the inverse Laplace transformation [closed]
Need help computing the inverse Laplace transformation of the following equation.
\begin{align*}
f(s)&=\frac{A}{\prod_{i=1}^{L}(s+a_i)^m} \\
&=\frac{A}{(s+a_1)^m\,(s+a_2)^m\cdots (s+a_L)^m}.
...
0
votes
0answers
47 views
Improper Integral using Laplace
So I want to calculate the following integral using the laplace transform:
$$\int_0^\infty \frac{dt}{(t^2+1)^2} $$
I used the theorem:
$$\int_0^\infty f(t)G(t) dt=\int_0^\infty \mathcal{L}[f(t)](x)...
0
votes
0answers
17 views
How to understand the Laplace transform of function?
Assume, we have the function as below
$$I_{m} = \sum_{j=1}^{3} \sum_{d_{i}\in \Phi}P_{D}h_{i}G{j}P(||d_{I}||),$$
where $d$ is the Euclidean distance. Using the equation above, the Laplace transform ...
0
votes
0answers
18 views
Inverse Laplace Transform in Thermal Problem with Newton Cooling Condition
I need to compute the inverse Laplace transform of the following function: $$\hat u(x,s)=\frac{\exp(-\sqrt{s}x)}{s(1+\Lambda^{-1}\sqrt{s})},$$ where $x\geq0$ and $\Lambda>0$.
This function comes ...
1
vote
2answers
40 views
Inverse Laplace transform of $\frac{40.5}{s(s^2-9)}$ using convolution theorem
Find the inverse Laplace transform of $$\frac{40.5}{s(s^2-9)}$$ using the convolution theorem.
I see how you can solve this using partial fractions, but apparently it's supposed to be easier if you ...
0
votes
1answer
68 views
Using Laplace for PDE
The transverse displacement $u(x, t)$ of a semi-infinite elastic string satisfies:
$$\frac{\partial^2u}{\partial t^2} = c^2 \frac{\partial^2u}{\partial x^2}$$
$ x > 0, t > 0$
with initial ...
0
votes
0answers
50 views
Laplace transform and Fourier transform
I used the Fourier and Laplace transforms to solve a series of equations. Now I have to use the inverse of these conversions to get the wave function. My question is
$$L^{-1}(\frac{d}{dk}\psi(k,s))=?\...
1
vote
0answers
39 views
How to evaluate the inverse Laplace transform of $F(p) = \lambda\tanh (\lambda)/p$ where $\lambda=\left(\frac{p(1+p)}{2(1+2p)}\right)^{1/2}$
I am trying to solve a system of partial differential equations arising from a mathematical physical problem using the Laplace transform approach.
In order to retrieve the solution in the time domain, ...
4
votes
0answers
39 views
On the solution of a linear system of differential equations for the unknown series coefficients
In a mathematical physical problem, I came across the following linear system of differential equations (obtained upon using Fourier series expansion):
\begin{align}
\frac{\mathrm{d} \rho_n}{\mathrm{...
0
votes
1answer
22 views
What is the Laplace transform of this convolution-like integral?
We have the following integral in the time domain :
$$\int_0^\infty e^u\cos{(2(u-t))}y(u)du$$
This is very similar to the convolution integral. I'm not sure if the $2$ scaling in the cosine function ...
-6
votes
1answer
29 views
Laplace transform of zero …how it will be solved [closed]
What is Laplace transform of zero?
How it will solved?
Explain with simple example
0
votes
0answers
14 views
Prove that a function could be a Laplace transform of another function
I'm preparing for some exams and this is a question that is usually asked.
Given a function $F(s)$ prove that it can be a Laplace transform of a function $f(t)$
While we've been given the conditions ...
1
vote
0answers
21 views
Laplace transform signal
Is this correct ?
$$x(t)= \begin{cases} e^{-5t}, & 0≤t<5 \\ 0, & \text{otherwise} \end{cases}$$
\begin{align}
L[x(t)] &=\int_{-\infty}^∞x(t)e^{-st}dt \\
&=\int_{0}^5e^{-5t}e^{-st} ...
2
votes
0answers
74 views
Laplace transfom of $\int_t^{\infty} \frac{f(u)}{u}du$
I want to know the Laplace transform of the function: $\mathcal{L} \int_t^{\infty} \frac{f(u)}{u}du$.
I tried the following method
$\int_0^{\infty} e^{-ts}\int_t^{\infty} \frac{f(u)}{u}du dt=\int_0^{...
1
vote
2answers
38 views
Can the Inverse Finding the Laplace transform of $\frac{2s + 1}{s(s + 1)(s + 2)}$ without using partial fractions?
I'm wondering if we can perhaps using the convolution theorem to find the inverse Laplace transform of $\dfrac{2s + 1}{s(s + 1)(s + 2)}$? I can find it using partial fraction decomposition, but it is ...
2
votes
2answers
23 views
Inverse Laplace Transform of $s^{-2}(s^2 + 1)^{-1}$ Using Convolution Theorem?
I am trying to find the inverse Laplace transform of $s^{-2}(s^2 + 1)^{-1}$. I could multiply these together and use partial fraction decomposition, but, unless I am mistaken, I think there is another ...
0
votes
1answer
29 views
Proof of Laplace Transform of *Real Number* Powers
As I understand it, the theorem and proof for the Laplace transform of positive integer powers is as follows:
Theorem
Let $t^n : \mathbb{R} \to \mathbb{R}$ be $t$ to the $n$th power for some $...
0
votes
2answers
22 views
Inverse laplace transform of $\frac{1}{(s+a)(s+b)}$ [closed]
Been trying to find the inverse laplace transform of $$\frac{1}{(s+a)(s+b)}$$ where $a$ and $b$ are constants.
Most obvious thing to do is try a partial fraction decomposition, but it just becomes a ...
3
votes
1answer
59 views
“Damped” wave equation with Fourier method
The problem I was trying to solve is the following PDE problem
$$\begin{cases} \partial_{tt}^2 u = \partial_{xx}^2 u -\gamma\partial_t u \\[5 pt]
u(0,t)= u(\pi, t) = 0 \\[5 pt]
u(x,0) = (\sin2x)^4 -...
0
votes
1answer
44 views
Generalized Fresnel Integral using Laplace
So I wanted to solve the following integral:
$$\int_0^\infty \sin{(x^2) dx}$$
I did it by using the Laplace transform of the function:
$$I(t) = \int_0^\infty \sin{(tx^2) dx}$$
$$\mathcal{L} [I(t)]...
-1
votes
0answers
18 views
Inverse Laplace Transform of irrational function $\frac{1}{{a{s^\alpha } + bs + c}}$
I'm looking for a way to inverse this transformant
$$F(s) = \frac{1}{{a{s^\alpha } + bs + c}}$$
where $a,b,c,\alpha $ are real numbers and $\alpha > 0$.
Is there any technique to do it?
...
1
vote
1answer
21 views
Confused About Change of Integration Limits in Convolution Proof
My textbook has proved the convolution theorem as follows:
Theorem
If $f(t)$ and $g(t)$ are two functions of exponential order (so that their Laplace transforms exist), and writing $\mathcal{...
0
votes
2answers
21 views
Questions About Textbook Proof of Convolution Theorem
I am reading a textbook on Laplace transforms.
In the proof of the convolution theorem, the author starts by writing the following:
$$\mathcal\{ f(t) * g(t) \} = \int_0^\infty e^{-st}\int_0^t f(\...
3
votes
2answers
66 views
Using contour integration for the inverse Laplace transform to find the inverse transform of $\dfrac{s}{s^2 + a^2}$
I am trying to use the contour integration formula for the inverse Laplace transform, find the inverse transform of $\dfrac{s}{s^2 + a^2}$.
My textbook says that the solution is $\cos(at)$, but it ...
0
votes
2answers
31 views
Inverse Laplace transform of $\dfrac{s}{(s + 1)^2 - 4}$
I am trying to find the inverse Laplace transform $\mathcal{L}^{-1} \left\{ \dfrac{s}{(s + 1)^2 - 4} \right\}$. My textbook says that the solution is $e^{-t} \cosh(2t) - \dfrac{1}{2}e^{-t}\sinh(t)$.
...
0
votes
1answer
17 views
Laplace transform and delay
Hi guys if i have to transform $e^{t-1}u(t-1)$ , why the result is $\frac{e^{-s}}{(s-1)}$ and not $\frac{e^{-(s-1)}}{(s-1)}$ if the $e^{t-1}$ multiply all the "function" ?
0
votes
1answer
17 views
$\mathcal{L} ( (t+5)U(t-1) ) $
$\mathcal{L} ( (t+5)U(t-1) ) = U(t-a)f(t-a) = e^{-as} F(s) $
a= 1
I am having trouble dealing with $f(t-a)$ and finding $F(s)$
$f(t-1) = e^{t-3} = e^{(t-1)-3+1} = e^{(t-1)-2} $
$f(t) = e^{t-2} $ ...
2
votes
0answers
28 views
Help Solving Textbook Heat Conduction Laplace Transforms PDE Problem 2
This problem is related to this question. If you can answer this, then you might be able to also answer the other question, so please have a look at it.
I am trying to solve the following problem:
$$...
0
votes
2answers
67 views
Solve DE by Laplace transformation: $y'' + y = 8e^{-2t} \sin t$?
The differential equation is as follows
$$\begin{cases}
y'' + y = 8e^{-2t} \sin t \\
y(0)=0 \\y'(0)=0
\end{cases}$$
How do I solve it by Laplace transformation?
In my solution I've taken the ...
1
vote
1answer
58 views
What are practical applications of Laplace Transforms in the real world [closed]
In lay man's terms what really in the real world are practical examples of how Laplace transforms are used to solve basic mathematical problems in mechanical engineering I would really appreciate a ...
6
votes
2answers
96 views
Help Solving Textbook Heat Conduction Laplace Transforms PDE Problem
I am trying to solve the following problem:
$$\dfrac{\partial{\phi}}{\partial{t}} = \dfrac{\partial^2{\phi}}{\partial{x}^2} - \cos(x), \ x > 0, t > 0$$
$$\phi(x, 0) = 0, \ x > 0$$
$$\phi(0,...
1
vote
1answer
35 views
What Is the Laplace Transform of $\frac{\partial{\phi}}{\partial{t}} = \frac{\partial^2{\phi}}{\partial{x}^2} - \cos(x)$, where $\phi = \phi(x, t)$?
I was wondering what the Laplace transform of $\dfrac{\partial{\phi}}{\partial{t}} = \dfrac{\partial^2{\phi}}{\partial{x}^2} - \cos(x)$, where $\phi = \phi(x, t)$, is?
I know the Laplace transform of ...
