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.
Does anyone know of any good sources, websites, books etc. That would be best for trying to become proficient in these topics in as short of a time period possible?
I have a good grasp on calculus concepts as is, but I'm undertaking a unit that involves these concepts in the application to...
for the right part of the notes, why the integral of (e^-su)f(u) from 0 to T will become integral of (e^-st)f(t) from 0 to T suddenly ? why not integral of (e^-s (t-nT) )f(t-nT) from 0 to T ? as we can see, u = t +nTd given/known dataHomework EquationsThe Attempt at a Solution
Homework Statement
Diagram for a vehicle suspension is given. Displacement of wheel is given by 'x' and and displacement of body is 'y'.
Spring constant, k = (7*10^4) Nm
Damping coefficient, c = (3*10^3) N/m/s
mass,m = 250kg
a) Make a Laplace Transform of system and utilize it to predict 'y'...
Homework Statement
Solve the following initial value problem using Laplace transforms: y' + 4y = 3t3 e−4t ; y(0) = 0 . Useful information: Recall that the Laplace transform of y 0 is pY − y(0), where Y is the Laplace Transform of y. The Laplace transform of tk e−at is k!/(p + a)k+1 . Confirm...
bear with me, i know that this question has been asked many time , but i would like a definite answer, now, starting off the external charge density on the outer surface of sphere WILL be uniform by unique solution of Laplace equation and letting the sphere be huge, so, electric field due to...
Hello everybody! I'm sorry if it's not the right section to post in. I'm trying to solve this exercise:
\frac{1}{2i\pi}*\int_{8-i\infty}^{8+i\infty}\frac{e^{s(t-5)}}{(s+4)^2}ds
The request is to find the result in function of t
I know i must use the Riemann inversion formula, and so the request...
(2.) Let the differential equation ¨x + 2 ˙x + 2x = 6 sin(t)U(t − 3π/2) , x(0) = 2, x˙(0) = 2
Solve for the position function x(t) using the Laplace transform:
I'm working on a few problems to find Laplace transforms and I got stuck on this one.
${U}_{3}(t){(t-3)}^{5/2}$
It looks different from the other I've been doing so I don't really know how to get started
Homework Statement
xy''+y'+xy=0, y'(0)=0, y(0)=0 Using the method of Laplace transforms, show that the solution is the Bessel function of order zero.
Homework Equations
-(d/ds)L{f(x)}
The Attempt at a Solution
The only thing I got out of this when trying to solve it was y=0. Obviously not...
Suppose that $u$ is the solution of the Laplace equation
$u_{xx}+u_{yy}=0$ in $\{(x,y)\in \mathbb{R}^2: x^2+y^2<1\}$
$u(x,y)=x$ for all $(x,y)\in \mathbb{R}^2$ such that $x^2+y^2=1.$
Find the value of $u$ in $(0,0).$ Use the property of median value.
Homework Statement
L-1{(2s2+3)/(s2+3s-4)2}
The Attempt at a Solution
I factored the denominator
f(t)=(2s2+3)/((s-1)(s+4))2
now I've tried partial fractions to get
(2s2+3)/((s-1)(s+4))2 = A/(s-1)2 + B(s+4)2
(2s2+3)=A(s+4)2 + B(s-1)2
by substitution, s=1 and s=-4
5=A(25)
A=1/5
35=B(25)...
Homework Statement
L-1{3/s1/2}
Homework EquationsThe Attempt at a Solution
3L-1{1/s1/2}
3L-1{(1/sqrt(π))(sqrt(π)/(sqrt(s))}
3/(sqrt(π))L-1{(sqrt(π))/(sqrt(s))}
3/(sqrt(π))(1/(sqrt(t))
This is what I got from the solution for this problem. What tipped them off to multiply by sqrt(π)? And...
Let $\Omega$ an open domain in $\mathbb{R}^2.$ Suppose that $u$ is an application $C^2$ that satisfy the Laplace equation $\Delta u=0$ in $\Omega.$ Let $\Omega_{(a,b)}=\{(x+a,y+b): (x,y)\in \Omega\}$ and define $v(X,Y)=u(X-a,Y-b)$ for all $(X,Y)\in \Omega_{(a,b)}.$ Show that $v$ is an...
Homework Statement
The differential equation given:
y''-y'-2y=4t2
Homework EquationsThe Attempt at a Solution
I used the laplace transform table to construct this equation,and then I did partial fraction for finding the inverse laplace transform.But I'm now stuck at finding the inverse laplace...
Homework Statement
Use the convolution property to obtain the inverse Laplace transform of
F(s)= e-3s * ((3s+15)/s2+25)
Homework EquationsThe Attempt at a Solution
= (3*(s/s2+52) + 15*(1/s2+52)) *e-3s
Using table of Laplace:
3*(s/s2+52) = 3*cos(5*t) = T7
15*(1/s2+52) = 15/5*sin(5*t) =T18
e-3s...
Homework Statement
[/B]
I know for t[u(t)-u(t-2)], we can simplify that to tu(t)-((t-2)+2)u(t-2) which then gives us tu(t)-(t-2)u(t-2)-u(t-2). Now, the laplace transform seems trivial but I am having problems with this equation:
sin(t)[u(t)-u(t-pi)]
Homework...
Hello everybody!
I know how to solve Laplace equation on a square or a rectangle.
Is there any easy way to find an analytical solution of Laplace equation on a trapezoid (see picture).
Thank you.
Homework Statement
Hello,
I have just started studying Laplace transformations and I am struggling to identify reverse Laplace transforms. I understand how to perform the transform, but going the other way is really confusing me.
i.e, given ##F(P)## find ##f(t)##.
If I have that ##F(P) =...
The ordinary differential equation, with initial values,shall be solved using Laplace transform. The ODE looks like this
\begin{equation}
y''(t')+2y''(t)-2y(t)=0
\end{equation}
And the initial conditions are
\begin{equation}
y(0)=y'(0)=0, y''(0)=0
\end{equation}
The problem is with the first...
Homework Statement
Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teacher is against using it..)
y'' + 2y' + 2y = 2 ; y(0)= y'(0) = 0
Homework Equations
Lf'' = ((s^2)*F) - s*f(0) - f'(0)
Lf' = sF - f(0)
Lf = F(s)
The...
Homework Statement
Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teach is against using it..)
y'' - 4y' + 3y = 0 ; y(0)=2 y'(0) = 8
Homework Equations
Lf'' = ((s^2)*F) - s*f(0) - f'(0)
Lf' = sF -...
Homework Statement
y'' + 3y' + 2y = r(t),
r(t) = u(t - 1) - u(t - 2),
y(0) = y'(0) = 0.
I need to solve this by convolution, which I know is commutative. The problem is that my calculation gives (f * g) =/= (g * f). Could someone please tell me where my mistake is?
Homework Equations...
Homework Statement
So I know 1/(s-a)=e^(a1), but why is say, 2/((s+4)^2) equal to 2xe^-4x? Do I just simply add an X if the numeration is a constant other than 1?
Homework Statement
Let
##f(t)=
\begin{cases}
\sin t , \; \; 0 \le t < \pi \\
0 , \; \; \; \; \; \text{else.}
\end{cases}##
Use Laplace transform to solve the initial value problem
##x'(t)+x(t)=f(t), \; \; \; x(0)=0.##
Homework Equations
Some useful Laplace transforms...
Homework Statement
I'm suppose to verify the given Laplace in (a) Cartesian (b) Sperical and (c) Cylindrical coordinates. (a) was easy enough but I need to know if I'm doing (b) and (c) correctly. I don't need a solution, I simply need to know if the my Spherical formula is correct, my...
Homework Statement
I uploaded the question as a picture and attached it.
Homework Equations
Unit step function -
u_c (t) =
\begin{cases}
1 & \text{if } t \geq c \\
0 & \text{if } t < c
\end{cases}
Impulse function -
δ(t) = \displaystyle\lim_{Δ\rightarrow 0} δ_Δ (t)
Multiplication Property...
Homework Statement
y'' + 4y = 8t^2 if 0 < t < 5, and 0 if t > 5; y(1) = 1 + cos(2), y'(1) = 4 - 2sin(2). Use the Laplace transform to find y.
Homework Equations
t-shift, s-shift, unit step function.
The Attempt at a Solution
I have been trying to solve it for hours, but keep getting the wrong...
Homework Statement
I uploaded the problem statements as a picture as well. I have completed these and was wondering if someone could check my work, and let me know if it is correct.
Problem 1.3:
Find the expression for the transfer function of this linear time-invariant causal system with...
Homework Statement
Here is an imgur link to my assignment: http://imgur.com/N0l2Buk
I also uploaded it as a picture and attached it to this post.
Homework Equations
u_c (t) =
\begin{cases}
1 & \text{if } t \geq c \\
0 & \text{if } t < c
\end{cases}
The Attempt at a Solution
Question 1.1 -...
Homework Statement
[/B]
Having a little trouble solving this fractional inverse Laplace were the den. is a irreducible repeated factor
2. The attempt at a solution
tryed at first with partial fractions but that didnt got me anywhere, i know i could use tables at the 2nd fraction i got as...
i have read many of the answers and explanations about the similarities and differences between laplace and Fourier transform.
Laplace can be used to analyze unstable systems.
Fourier is a subset of laplace.
Some signals have Fourier but laplace is not defined , for instance cosine or sine...
I am trying to numerically calculate the electric potential inside a truncated cone using the finite element method (FEM). The cone is embedded in cylindrical coordinates (r,phi,z). I am assuming phi-independence on the potential, therefore the problem is essentially 2D; I am working only with...
Can anybody direct me towards a proof of the second shifting theorem for Laplace transforms? I'm understanding how to use it but I can't figure out where it comes from. I've been learning from Boas, which doesn't offer much in way of proof for this theorem. Are there any good resources online...
Homework Statement
(didn't know how to make piecewise function so I took screenshot)
Homework EquationsThe Attempt at a Solution
My issue here with this problem is that I have absolutely no idea where to start... I have read through the textbook numerous times, and searched all over the...
Hello guys. I need an easy explanation regarding Laplace Transform and Fourier Transform. I know it is quite a mathematics question but I need an explanation in which it has something to do with engineering. I already search a bit about them but still cannot find and explanation that easy enough...
Hi, I'm dealing with an exam problem (which i have 8 days to solve, with no outside help limitations), which is to write a fortran95 program that solves linear systems.
the first part asks to find the determinant of a NxN matrix with the laplace expansion, implementing it as a recursive...
Hello,
When evaluating the step response of a circuit, the resulting Laplace representation is:
$\frac{I_{pd}}{s^2 C1 R1}$
If I look this up on a table of Laplace Transforms, this results in $\frac{I_{pd}*t}{C1 R1}$.
However, I'm struggling to solve this via partial fraction expansion--is...