Pierre-Simon, marquis de Laplace (; French: [pjɛʁ simɔ̃ laplas]; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized and extended the work of his predecessors in his five-volume Mécanique Céleste (Celestial Mechanics) (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace.Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is also named after him. He restated and developed the nebular hypothesis of the origin of the Solar System and was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse.
Laplace is remembered as one of the greatest scientists of all time. Sometimes referred to as the French Newton or Newton of France, he has been described as possessing a phenomenal natural mathematical faculty superior to that of any of his contemporaries.
He was Napoleon's examiner when Napoleon attended the École Militaire in Paris in 1784.
Laplace became a count of the Empire in 1806 and was named a marquis in 1817, after the Bourbon Restoration.
For part (b),
I have tried finding the Laplace transform of via the convolution property of Laplace transform.
My working is,
##L[\cos^2 (2t)] = L[\cos 2t] * L[\cos 2t]##
##L[\cos^2 (2t)] = \frac{s}{s^2 + 4} * \frac{s}{s^2 + 4}##
##\int_0^t \frac{s^2}{(s^2 + 4)^2} dt = \frac{ts^2}{(s^2 +...
For this problem (b),
The solution is,
However, I don't understand how they got their partial fractions here (Going from step 1 to 2).
My attempt to convert into partial fractions is:
##\frac{2s + 1}{(s - 1)(s - 1)} = \frac{A(s - 1) + B(s - 1)}{(s - 1)(s - 1)}##
Thus,
##2s + 1 = A(s - 1) +...
Trying to model friction of a linear motor in the process of creating a state space model of my system. I've found it easy to model friction solely as viscous friction in the form b * x_dot, where b is the coefficient of viscous friction (N/m/s) and x_dot represents the motor linear velocity...
Could someone check whether my proof for this simple theorem is correct? I get to the result, but with the feeling of having done something very wrong :)
$$\mathcal{L} \{f(ct)\}=\int_{0}^{\infty}e^{-st}f(ct)dt \ \rightarrow ct=u, \ dt=\frac{1}{c}du, \
\mathcal{L}...
Is there a way to solve Laplace’s Equation on irregular domains if the domain’s shape is given by a function for example a 2D parabolic plate. I keep seeing numerical methods but I want to know is there an ANALYTICAL method to solve it on an irregular domain. If there isn't are there approximate...
Hi,
I was attempting example 27.1 question from the book: 'Information Theory, Inference, and Learning Algorithms'. It is about the Laplace approximation. I was confused about part (b) of the question and wanted to check my method if possible.
[EDIT]: The link to the book website (official) is...
Hello!
Im having some trouble with solving ODE's using Laplace transformation,specifically ODE's that require partial fraction decomposition.Now I know how to do partial fraction decomposition,and have done it many times on standard polynoms but here some things just are not clear to me.For...
Are there any practical applications of Laplace transform? I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if it has a pulse forcing function.
How can Laplace transforms be introduced so that students are...
Are there any practical applications of Laplace transform? I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if it has a pulse forcing function.
How can Laplace transforms be introduced so that students are...
Lets consider very simple equation ##x''(t)=0## for ##x(0)=0##, ##x'(0)=0##. By employing Laplace transform I will get
s^2X(s)=0 where ##X(s)## is Laplace transform of ##x(t)##. Why then this is equivalent to
X(s)=0
why we do not consider ##s=0##?
Hello!
Consider this transferfunction H(s);
$$ H(s) =\frac{s-1}{1-2(s^2-s)-As-\frac{A}{2}} $$
Now I need to determine A (note that A is coming from R) so that the impulse response h(t) (so in time domain) so that it contains components with $$te^{at} \sigma(t) $$.
Now I honestly really have...
Hi!
The problem clearly states that there is a surface charge density, which somehow gives rise to a potential.
The author has solved the Laplace equation in cylindrical coordinates and applied the equation to the problem.
So ##\nabla^2 V(r,\phi) = 0##, and ##V(a,\phi) = V_a(\phi)## (where...
\mathcal{L}^{-1}[\frac{e^{-5s}}{s^2-4}]=Res[e^{-5s}\frac{1}{s^2-4}e^{st},s=2]+Res[e^{-5s}\frac{1}{s^2-4}e^{st},s=-2]
From that I am getting
f(t)=\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)}. And this is not correct. Result should be
f(t)=\theta(t-5)(\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)})...
I have a f(t) that is, e^(-t) *sin(t), now I calculate the Laplace transformation, that is:
X(s) = 1 / ( 1 + ( 1 + s)^2 ) (excuse me but Latex seems not run ).
Now I imagine the plane with Re(s), Im(s) and the magnitude of X(s).
If i take Re(s) = -1 and Im(s) = 0, I believe I have X(s) = 1 ( s...
Hey! 😊
An ice cream parlour offers 12 different types of ice cream, including vanilla ice cream. There are 8 people passing by, each of whom chosses a ball of ice cream. Of course, the ice cream parlour has taken good precautions, so that there is enough ice cream from each variety.
Model the...
Hello ,
The Laplace operator equals
## \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} ##
so does it equal as well nable or Del operator squared ## \bigtriangledown^2## ?
where
## \bigtriangledown =\frac{\partial}{\partial...
I have tried to Fourier transform in ##x## and get the result in the transformed coordinates, please check my result:
$$
\tilde{u}(k, y) = \frac{1-e^{-ik}}{ik}e^{-ky}
$$
However, I'm having some problems with the inverse transform:
$$
\frac{1}{2\pi}\int_{-\infty}^\infty...
Here is the initial problem and my attempt at getting Laplace solution. I get lost near the end and after some research, ended up with the Bessel equation and function. I don't completely understand what this is or even if this i the direction I go in.
This is a supplemental thing that I want to...
The Laplace transform gives information about the exponential components in a function, as well as oscillatory components. To do so there is a need for the complex plane (complex exponentials).
I get why the MGF of a distribution is very useful (moment extraction and classification of the...
I was wondering how you work out what values of s a Laplace transform exists? And what it actually means? The example given in class is an easy one and asks to calculate the Laplace transform of 3, = 3 * Laplace transform of 1 = 3 * 1/s. Showing this via the definition, where does the range of s...
So, I know the direct definition of the Laplace Transform:
$$ \mathcal{L}\{f(t) \} = \int_0^\infty e^{-st}f(t)dt$$
So when I plug in:
$$\frac{\cos(t)}{t}$$
I get a divergent integral.
however:https://www.wolframalpha.com/input/?i=+Laplace+transform+cos%28t%29%2F%28t%29
is supposed to be the...
My attempt at finding this was via convolution theorem, where we take F(s) = 1/s^2 and G(s) = e^(-sx^2/2). Then to use convolution we need to find the inverses of those transforms. From a table of Laplace transforms we know that f(t) = t. But I am sort of struggling with e^(-sx^2/2). My 'guess'...
Hello everybody,
I am working on a Python project in which I have to make Bayesian inference to estimate 4 or more parameters using MCMC.
I also need to evaluate the evidence and I thought to do so through the Laplace approximation in n-dimensions:
$$ E = P(x_0)2\pi^{n/2}|C|^{1/2} $$
Where C...
I am interested in modeling a battery charging/discharging. I am starting off with a simple model using a voltage source in series with a parallel RC branch which is in series with a resistor. I will be measuring the open circuit voltage between the last series resistor and the bottom of the...
##\frac {\partial \vec F} {\partial x} ## + ##\frac{\partial \vec F} {\partial y} ## = vector which gives me a direction of the greatest increase of the greatest increase of the function, where ##\vec F ## = gradient of the function. If I multiple the first by ##\hat i## and the second by ##\hat...
hi guys
i am facing a little problem calculating this Laplace transform ## \mathscr{L}(\frac{e^{\alpha t}}{t})## , when calculate it using the method of the inverse Laplace transform its equal to
$$ ln{\frac{1}{s-\alpha}}$$
but then when i try to use the theorem
$$...
I've included the problem statement and a bit about the function but my main issue is with the equation after "then" and the one with the red asterisk. I don't understand why the Laplace transform for a u(t)*e^(-t/4) isn't (1/s)*(1/(s+1/4)). The book I am reading says it's(1/(s+1/4)).
Homework Statement:: Why is the heaviside function in the inverse laplace transform of 1?
Relevant Equations:: N/A
This is a small segment of a larger problem I've been working on, and in my book it gives the transform of 1 as 1/s and vice versa. But as I've looked online for help in figuring...
Suppose I'm solving
$$y''(t) = x''(t)$$ where $$x(t)$$ is the ramp function. Then, by taking the Laplace transform of both sides, I need to know $x'(0)$ which is discontinuous. What is the appropriate technique to use here?
Was just practicing some problems on the Fundamentals of Electric Circuits, and came across this question.
I understand I will have to transform to the s domain circuit, which looks something like this:
Then doing nodal analysis, I will get the following for the first segement
(10/s-V1)/1 =...
I struggle to find an appropriate inverse Laplace transform of the following
$$F(p)= 2^n a^n \frac{p^{n-1}}{(p+a)^{2n}}, \quad a>0.$$
WolframAlpha gives as an answer
$$f(t)= 2^n a^n t^n \frac{_1F_1 (2n;n+1;-at)}{\Gamma(n+1)}, \quad (_1F_1 - \text{confluent hypergeometric function})$$
which...
Hi again,
The previous problem was done using y′′(t)+2y′(t)+10y(t)=10 with with intial condition y(0⁻)=0.
In the following case, I'm using an initial condition and setting the right hand side equal to zero.
Find y(t) for the following differential equation with intial condition y(0⁻)=4...
Hi,
This thread is an extension of this discussion where @DrClaude helped me. I thought that it'd be better to separate this question.
I couldn't find any other way to post my work other than as images so if any of the embedded images are not clear, just click on them. It'd make them clearer...
Hi,
I 'm trying to find the Laplace transform of the following expression.
I used the following conversion formulas.
I think "1" is equivalent to unit step function who Laplace transform is 1/s.
I ended up with the following final Laplace transform.
Is my final result correct? Thank you...