What is Laplace: Definition and 1000 Discussions

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.

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  1. I

    Inverse Laplace transform of 1/s

    Homework Statement Homework Equations N/A The Attempt at a Solution The left hand side (red box) is the data sheet provided to us in the exam. The right hand side (blue box) is Wolfram Alpha. The data sheet says that the inverse Laplace transform of 1/s is equal to u(t) (i.e. the unit step)...
  2. S

    Is Solving Laplace's Equation for a Dielectric Cylinder Straightforward?

    Consider an infinitely long hollow dielectric cylinder of radius a with the electricpotential V = V0 cos φ on the surface of the cylinder where φ is an angle measured around the axis of the cylinder. Solve Laplace’s equation to find the electric potential everywhere in space.Do you just plug V...
  3. Z

    Inverse Laplace Help: F(s)=e^-4s(s^2/(s^2+9))

    Homework Statement F(s) = \frac{s^2e^{-4s}}{s^2+9} Homework Equations \mathcal{L} ({f(t-a)\mathcal{H}(t-a)}) = e^{-as}F(s) The Attempt at a Solution F(s) = e^{-4s}(\frac{s^2}{s^2+9}) using \mathcal{L} {f(t-a)\mathcal{H}(t-a)} = e^{-as}F(s) f(t) = cos(3t-12)\mathcal{H}(t-4) I do...
  4. E

    Step function Laplace transform - can someone check my work?

    Homework Statement Homework EquationsThe Attempt at a Solution 1 ##3u(t)-3u(t-2)## ##\frac{3}{s}-\frac{e^{-2s}3}{s}## 2 ##\frac{5t}{2}u(t)-\frac{5t}{2}u(t-2)## ##\frac{5}{2s^2}-\frac{5e^{-2s}}{2s^2}## 3 I have no idea how to do this one.
  5. Dopplershift

    Partial Differentiation Laplace Equation Question

    Homework Statement Consider the Laplace Equation of a semi-infinite strip such that 0<x< π and y>0, with the following boundary conditions: \begin{equation} \frac{\partial u}{\partial x} (0, y) = \frac{\partial u}{\partial x} (0,\pi) = 0 \end{equation} \begin{equation} u(x,0) = cos(x)...
  6. E

    Determine the Laplace transform for the following functions

    Homework Statement Can someone check my work? Homework EquationsThe Attempt at a Solution 1. ##\frac{1}{s+2}+\frac{1}{s^2+1}## 2. ##\frac{2}{s}+\frac{3}{s+4}## 3. ##\frac{s*sin(-2)+cos(-2)}{s^2+1}## 4. ##\frac{1}{(s+1)^2}## 5. Don't really know how to do this one...
  7. E

    Engineering Solving a circuit with Laplace

    Homework Statement Assume zero initial conditions Step 1. Write the nodal equations to find i(t) in the time domain. Step 2.Solve the differential equation obtained in step 1 using laplace to obtain i(t). Homework EquationsThe Attempt at a Solution Convert to sdomain ##7e^{-6t}## becomes...
  8. D

    Inverse Laplace transform for 1/(350+s) * X(s)

    Hi, everyone, the question is as below: Find the inverse Laplace transform to 1/(350+s) * X(s). 's' is the Laplace variable and 'X(s)' is also a variable. I inverted 1/(350+s) and X(s) separately and multiplied them together directly. But this seems not giving me the correct answer. Could...
  9. W

    Understanding the LaPlace Transformation: A Guide for Solving Complex Integrals

    Dear PF members. I am requesting again your help as I keep struggling with the LaPlace transformation. I have this exercise to do(please see below) We know that L[f(t)]= integral from 0 to infinity of f(t)*e^(-st) dt thus in our case, L[f(t)]= integral from 0 to infinity of sin(t)*e^(-st) dt...
  10. whatphysics

    Engineering Math: Laplace Transform

    Not homework question, just need clarification and explanation. How did the person get from the left equation to the right side. I know he's just simplifying. But he didn't include steps and I've been trying to work out how to no avail. Any help on how this person simplified the LHS to RHS? Thanks!
  11. J

    Laplace and ROC of function(- [e^(-at)]u(-t))

    Homework Statement If laplace of [e^-(at)] u(t) is 1/(s+a) and ROC is s > -a Find laplace and ROC of -e^(-at) u(-t) Homework Equations Laplace is integral over minus infinity to plus infinity of f(t) e^(-st) dt The Attempt at a Solution Well i integrated f(t) over the limits with e(-st)u(-t)...
  12. C

    I Laplace transformation of nested function

    Hello! I want a formula (if there exists) to find the Laplace transformation of a nested function; a function within a function For example what is the LT of θ(f(t)), where θ is the step function? Is there already a formula for such things or should I follow the definition integrating etc..? I...
  13. gmm

    Engineering Circuit Analysis with Laplace Transforms

    Homework Statement Homework Equations V=IR All of them actually The Attempt at a Solution So I Started off by transforming the voltage source into the 's' domain vs(s) = (4/s) -(4/s)*e-.5t I know the initial conditions are zero, in other words at t=0, the voltage and currents at the...
  14. D

    I How Do You Derive the Kernel of the Laplace Equation with Boundary Conditions?

    Hi! Everyone. I encounter some trouble in deriving the kernel of Laplace equation with prescribed boundary conditions. Given the following preposition: $$T(x, y) = \int_{-\infty}^{\infty}dx'\frac{y/\pi}{(x-x')^{2}+y^2}F(x')...[1]$$ satisfies the Laplace equation for ##x\in(-\infty, \infty)##...
  15. D

    I Inverse Laplace to Fourier series

    I have the following laplace function F(s) = (A/(s + C)) * (1/s - exp(-sα)/s)/(1 - exp(-sT)) I think that the inverse laplace will be- f(t) = ((A/C)*u(t) - (A/C)*exp(-Ct)*u(t)) - ((A/C)*u(t-α) - (A/C)*exp(-C(t-α))*u(t-α)) and f(t+T)=f(t) Now I want to find the Fourier series expansion of f(t)...
  16. kostoglotov

    Which inverse Laplace form can I use?

    Homework Statement I have the second order diff eq: Solving by Laplace transform gets me to: I could use the inverse laplace transform that takes me back to e^{at}cos(bt) with b=0, but that only solves for the homogeneous (complementary) part of the equation, it won't reproduce the dirac...
  17. RJLiberator

    ODE: Solving using Laplace Transform

    Homework Statement Solve: y''+λ^2y = cos(λt), y(0) = 1, y'(π/λ) = 1 where t > 0 Homework EquationsThe Attempt at a Solution I start off by taking the Laplace transform of both sides. I get: L(y) = \frac{s}{(s^2+λ^2)^2}+\frac{sy(0)}{s^2+λ^2}+\frac{y'(0)} {s^2+λ^2} Now take the inverse...
  18. Pouyan

    Finding inverse of a Laplace transform by convolution

    Homework Statement find the inverse Laplace transform of the given function by using the convolution theorem Homework Equations F(s) = s/((s+1)(s2)+4) The theorem : Lap{(f*g)(t)} = F(s)*G(s) The Attempt at a Solution I know how to find it the answer is : we have 1/(s+1) * s/(s+4) and the...
  19. T

    Problem with solving laplace equation with a charged ring

    Homework Statement A ring charge of total charge Q and radius a is concentric with a grounded conducting sphere of radius b, b < a. Determine the potential everywhere. The ring is located in the equatorial plane, so both the sphere and the ring have their center at the same spot. Homework...
  20. Hanyu Ye

    I The difference between capillary number and Laplace number?

    It seems that both the capillary number and the Laplace number represents the relative effect of viscous forces and surface tension. Can anybody explain the difference between them? Thanks a lot. I have attach the Wikipedia- links to these two concepts...
  21. Pouyan

    Laplace Transform of this function

    Homework Statement We want to find the Laplace transform for f(t): 0 for t≤2 and (t-2)2 for t≥2 Homework Equations I know that Lap{uc f(t-c)} = e-csLap{f(t)}=e-csF(s) I rewrite f(t)=0+g(t) where g(t) = 0 for 0≤t<2 and (t-2)2 for t≥2 so we can write f(t)=g(t)= u2(t)*(t-2)2...
  22. F

    I Solving PDE with Laplace Transforms & Inverse Lookup

    I am trying to solve with Laplace Transforms in an attempt to prove duhamels principle but can't find the Laplace transform inverse at the end. The book I am reading just says "from tables"... The problem : $$ U_t = U_{xx}\\\\ U(0,t)=0 \quad 0<t< \infty\\\\ U(1,t)=1\\\\ U(x,0)=0 \quad...
  23. E

    A Triple Product in Laplace Transform

    Hello - I'm not sure this is where this should go, but I'm working with Laplace Transforms and differential equations, so this seems as good a place as any. Also, I doubt this is graduate level math strictly speaking, but I went about as high as you can go in calculus and linear algebra during...
  24. tomwilliam2

    Laplace transform -- By parts?

    Homework Statement ##A\dot{x} + By = 0## ##C\dot{y} + Dx = 0##Homework Equations ##\int u'v = uv - \int uv'## The Attempt at a Solution This is a system of linear DE: ##A\dot{x} + By = 0## ##C\dot{y} + Dx = 0## Where the constants A-D are non-zero and x and y are functions of time. This is...
  25. C

    Establishing equations for a worm screw mechanism

    Hey guys :) So I'm looking to form equations that I can apply a Laplace transformation to. The mechanism specifically is a trailer jack - it converts rotational motion to linear motion. And its high torque provides a mechanical advantage to lifting heaving loads. Can anyone help me form...
  26. P

    MHB Collin's question via email about a Laplace Transform

    As the Heaviside function is a function of t - 4, that means all other terms must also be functions of t - 4. The sine function is, but the exponential isn't. However with a little manipulation, we get $\displaystyle \begin{align*} f\left( t\right) &= \mathrm{H}\,\left( t - 4 \right) \,\sin{...
  27. P

    MHB Collin's question via email about solving a DE using Laplace Transforms

    Taking the Laplace Transform of both sides we have $\displaystyle \begin{align*} \mathcal{L}\,\left\{ y'' + 4\,y \right\} &= \mathcal{L}\,\left\{ \mathrm{H}\,\left( t - 7 \right) \right\} \\ s^2\,Y\left( s \right) - s\,y\left( 0 \right) - y'\left( 0 \right) + 4\,Y\left( s \right) &=...
  28. P

    MHB Collin's questions via email about Inverse Laplace Transforms

    As the denominator is a function of s + 3, it suggests a shift had to have been utilised. As such, we also need the numerator to be a function of s + 3... Let $\displaystyle \begin{align*} u = s + 3 \end{align*}$, then $\displaystyle \begin{align*} s = u-3 \end{align*}$ and thus...
  29. P

    MHB Henry's question via email about an Inverse Laplace Transform

    It's not entirely obvious what to do with this question, as the denominator does not easily factorise. However, if we realize that $\displaystyle \begin{align*} s^4 + 40\,000 = \left( s^2 \right) ^2 + 200^2 \end{align*}$ it's possible to do a sneaky completion of the square... $\displaystyle...
  30. C

    Question on Laplace Transform of a constant voltage source

    Hello. I am reviewing the use of the Laplace Transform to do circuit analysis and I am slightly confused about the transform of a constant voltage source. For example, let's say we have a constant voltage source V1(t) applied to a circuit for a long time - let's say it reaches steady state. We...
  31. Delta2

    Do Moving Charges Create Magnetic Fields, or is it Something Else?

    Ok so when we have a current carrying conductor inside a magnetic field there would be Laplace force ##L\times Bi## which is the macroscopic form of the microscopic Lorentz force ##v\times Bq## in a large number of electrons ( or it is not ?) But also there will be hall voltage which will...
  32. M

    Laplace transform for a piezoelectric acceletometer

    Homework Statement FIGURE 4(a) represents a system to measure acceleration (i.e. an accelerometer). It shows a piezoelectric crystal that is connected to an amplifier and display via a length of coaxial cable2.A piezoelectric current is produced when the crystal is distorted by an applied...
  33. B

    I Laplace transform using differential equations

    Hi members, Laplace transform using differential equations.(see attached PDF file) My question d/ds(s^2 y- s Y(0)-Y'(0).)... Y(t)=sin(sqrt(t)) Y(o)=0 Now Y'= cos(sqrt(t)/2sqrt(t) Y'(0)=infinity d/ds (Y'(0)=?? can it be treated as a constant or can we change limit and differentiation??I...
  34. S

    Engineering Eigenfunctions of the vector Helmholtz equation

    Hi everyone, I'm looking for a reference book that treats the theory behind the eigenfunctions solution of the so called vector Helmholtz equation and its Neumann and Dirichlet problems. I've already found a theory inside the last chapter of Morse & Feshbach's Methods of theoretical physics...
  35. F

    Laplace transform of (e^-s) / [ (s)(s-3) ]

    Homework Statement find the laplace transform of (e^-s) / [ (s)(s-3) ] since there's (e^-s) which can be found in L { f(t-a) H(t-a) } = (e^-(as)) F(s) , so , i found a = 1 , then i found F(s) = 1/ [ (s)(s-3) ] , formula : i have attached the working below , is it correct ? btw , the...
  36. V

    Laplace transform to solve a nonhomogeneous equation

    Mod note: Moved from a Homework section can i use the Laplace transform to solve a nonhomogeneous equation if i have these Initial condition s(x) and s(-x)
  37. J

    Low Pass Filters and Laplace Transform

    Homework Statement Given that r(t) = L^-1 (Inverse laplace) *H(S) and by making the link between the time-domain and frequency-domain responses of a network, explain in detail why the ideal “brick-wall” lowpass filter is not realisable in practice. [/B]Homework EquationsThe Attempt at...
  38. R

    Inverse Laplace Transform

    Homework Statement Given the Laplace transform $$F_L(s) = \frac{1}{(s+2)(s^2+4)},$$ by using the complex inversion formula compute the inverse Laplace transform, ##f(t),## for the following regions of convergence: (i) ##Re(s)<-2;## (ii) ##-2<Re(s)<0;## (iii) ##Re(s)>0.## Homework Equations...
  39. MAGNIBORO

    I Laplace Transform of tanh(x)?

    hi, sorry for the bad english. exist a Laplace Transform of tanh(x)? i know math of high school, so sorry if it is a question a little silly thanks
  40. S

    MHB Laplace Transform Help: Solving for Unknown Variables in Electrical Circuits

    Evening All I have had a go at a laplace transform and got stuck. $$\frac{d^2v}{dt^2}+\frac R L \d v t+\frac 1{LC}v=\frac 1{LC}V_0$$ $$R=12 \Omega, L=0.16H, C=10^{-4}F, V_0=6V, v(0)=0, v'(0)=0$$ so subbing these in i get $$\mathscr L \left[ \frac {d^2v}{dt^2}+75\d v t+62500 v...
  41. Dor

    Solve Laplace equation on rectangle domain

    Homework Statement I'm having issues with a Laplace problem. actually, I have two different boundary problems which I don't know how to solve analytically. I couldn't find anything on this situations and if anybody could point me in the right direction it would be fantastic. It's just Laplace's...
  42. astrodeva

    Solving Laplace Equations using this boundary conditions?

    The equation is Uxx + Uyy = 0 And domain of solution is 0 < x < a, 0 < y < b Boundary conditions: Ux(0,y) = Ux(a,y) = 0 U(x,0) = 1 U(x,b) = 2 What I've done is that I did separation of variables: U(x,y)=X(x)Y(y) Plugging into the equation gives: X''Y + XY'' = 0 Rearranging: X''/X = -Y''/Y = k...
  43. T

    First order Laplace transform

    Homework Statement A process can be represented by the first order equation (4δy(t)/δt) + y(t) = 3u(t) Assume the initial state is steady (y = 0 at t = –0). (a) Determine the transfer function of this process in the s domain. (b) If the input is a ramp change in u(t) = 4t, determine the...
  44. P

    MHB Emad's question via email about Inverse Laplace Transform

    The closest Inverse Laplace Transform from my table is $\displaystyle \begin{align*} \mathcal{L}^{-1}\,\left\{ \frac{2\,a\,s\,\omega}{\left( s^2 + \omega ^2 - a^2 \right) ^2 + 4\,a^2\,\omega ^2 } \right\} = \sin{ \left( \omega \, t \right) } \sinh{ \left( a \, t \right) } \end{align*}$ so we...
  45. F

    Laplace transform of (2s^2 +10s) / ((s^2 -2s +5)(s+1))

    Homework Statement (2s^2) +10s / (s^2 -2s +5 )(s+1) , I have checked the partial fraction , it's correct , but according to the ans it's (e^t)[(3cos2t + 2.5sin2t)] - (e^-t), but my ans is (e^t)[(3cos2t + 4sin2t)] - (e^-t)Homework EquationsThe Attempt at a Solution
  46. RJLiberator

    PDE: Laplace (?) Problem? Sturm Liouville?

    Homework Statement Solve ∇^2u=0 in D subject to the boundary conditions u(x,0) = u(0,y) = u(l,y) = 0, u(x,l) = x(l-x) where D = {(x,y): 0≤x≤l, 0≤y≤l} Homework EquationsThe Attempt at a Solution So, I've looked at the notes and the book and have a gameplan to attack this problem. However...
  47. ltkach2015

    Dartmouth Extended Laplace Tables -- Not general enough? item26.a

    Homework Statement [/B] http://www.dartmouth.edu/~sullivan/22files/New%20Laplace%20Transform%20Table.pdf (see item 26a) homogenous solution to underdamped in amplitude phase form: (see attached image) 2. Relevant info - non zero initial conditions: x(t=0) = xo AND dx/dt(t=0) = vo - unforced...
  48. J

    I Heaviside's Operational Calculus and Laplace Transform

    Hello everyone, I was studing Heaviside's operators for solving ODE, which I strongly recommend to have a look because it helps a lot when the differential equations have "exotic" inhomogeneous terms, but it is a method that works and you do not know exactly why. Some biographies tell that...
  49. S

    I What are the Laplace transforms of powers of y?

    Let's say you have a function y(t). You know how derivatives of y have their own Laplace transforms? Well I was wondering if powers of y such as y^2 or y^3 have their own unique Laplace transforms as well. If so , how do you calculate them (because plugging them into the usual integral doesn't...
  50. chrispypatt

    Final Value Theorem Rule Clarification

    My homework problem is as follows: Consider the Laplace transform shown below. (4s3+15s2+s+30)/(s2+5s+6) a. What is the value of f(t=0) and f(t=∞)? Use the initial and final value theorems. b. Find the inverse transform f(t). Use this expression to find f(t=0) and f(t=∞) and compare with the...
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