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. engnrshyckh

    B Inverse Laplace transform

    I used partial fraction method first as: 1/s(s^2+w^2)=A/s+Bs+C/(s^2+w^2) I found A=1/w^2 B=-1 C=0 1/s(s^2+w^2)=1/sw^2- s/s^2 +w^2 Taking invers laplace i get 1/w2 - coswt But the ans is not correct kindly help.
  2. P

    MHB Solving Integral Equation w/ Laplace Transform - Abdullah

    We would need to recognise that the integral in the equation is a convolution integral, which has Laplace Transform: $\displaystyle \mathcal{L}\,\left\{ \int_0^t{ f\left( u \right) \,g\left( t - u \right) \,\mathrm{d}u } \right\} = F\left( s \right) \,G\left( s \right) $. In this case...
  3. P

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

    Taking the Laplace Transform of the equation gives $\displaystyle \begin{align*} s^2\,Y\left( s \right) - s\,y\left( 0 \right) - y'\left( 0 \right) + 7 \left[ s\,Y\left( s \right) - y\left( 0\right) \right] + 6\,Y\left( s \right) &= \frac{60\,\mathrm{e}^{-6\,s}}{s} \\ s^2\,Y\left( s \right) -...
  4. P

    MHB Seth's question via email about a Laplace Transform

    Since this is of the form $\displaystyle \frac{f\left( t \right)}{t} $ we should use $\displaystyle \mathcal{L}\,\left\{ \frac{f\left( t \right) }{t} \right\} = \int_s^{\infty}{F\left( u \right) \,\mathrm{d}u } $. Here $\displaystyle f\left( t \right) = \cosh{\left( 4\,t \right) } - 1 $ and so...
  5. P

    MHB Adam's question via email about Laplace Transforms

    Take the Laplace Transform of the equation: $\displaystyle \begin{align*} s\,Y\left( s \right) - y\left( 0 \right) + 11\,Y\left( s \right) &= \frac{3}{s^2} \\ s\,Y\left( s \right) - 5 + 11\,Y\left( s \right) &= \frac{3}{s^2} \\ \left( s + 11 \right) Y\left( s \right) &= \frac{3}{s^2} + 5 \\...
  6. P

    MHB Jun's question via email about Laplace Transform

    Upon taking the Laplace Transform of the equation we have $\displaystyle \begin{align*} s^2\,Y\left( s \right) - s\,y\left( 0 \right) - y'\left( 0 \right) + 4\,Y\left( s \right) &= -\frac{8\,\mathrm{e}^{-6\,s}}{s} \\ s^2 \,Y\left( s \right) - 2\,s - 0 + 4\,Y\left( s \right) &=...
  7. P

    MHB Massaad's question via email about Laplace Transforms

    Taking the Laplace Transform of the equation gives $\displaystyle \begin{align*} s^2\,Y\left( s \right) - s\,y\left( 0 \right) - y'\left( 0 \right) - 5\left[ s\,Y\left( s \right) - y\left( 0 \right) \right] - 6\,Y\left( s \right) &= -\frac{126\,\mathrm{e}^{-6\,s}}{s} \\ s^2\,Y\left( s \right) -...
  8. P

    MHB Mahesh's question via email about Laplace Transforms (2)

    This requires the convolution theorem: $\displaystyle \int_0^t{f\left( u \right) \,g\left( t- u \right) \,\mathrm{d}u } = F\left( s \right) \,G\left( s \right) $ In this case, $\displaystyle g\left( t - u \right) = \mathrm{e}^{-3\,\left( t - u \right) } \implies g\left( t \right) =...
  9. P

    MHB Mahesh's question via email about Laplace Transforms (1)

    Start by taking the Laplace Transform of both equations, which gives $\displaystyle \begin{cases} s\,X\left( s \right) - s\,x\left( 0 \right) + X\left( s \right) + 6\,Y\left( s \right) = \frac{6}{s} \\ s\,Y\left( s \right) - s\,y\left( 0 \right) + 9\,X\left( s \right) + Y\left( s \right) = 0...
  10. P

    MHB Alexander's question via email about Laplace Transforms

    The Heaviside function suggests a second shift, but to do that, the entire function needs to be a function of $\displaystyle t - 4$. Let $\displaystyle u = t - 4 \implies t = u + 4$, then $\displaystyle \begin{align*} \mathrm{e}^{5\,t} &= \mathrm{e}^{5\left( u + 4 \right) } \\ &=...
  11. S

    MHB Hello i would like some help with laplace transforms.

    hello if someone could please tell me if i am incorrect and where , and how to type it into a math program so it can understand it many thanks stephan2124 L -3e^{9t}+9 sin(9t) L-3e^{9t}+L 9 sin (9t) -3 Le^{9t}+9 L sin(9t) -3 (1/s-9) +9 (9/(s^2+9^2)) -3 (1/s-9) +9 (9/(s^2+81)) into a math...
  12. A

    MHB Laplace Convolution: f(t)=-5t^2+9

    f(t)=-5t^2+9\int_{0}^{t} \,f(t-u)sin(9u)du
  13. P

    MHB Dharshan's question via email about a Laplace Transform

    $\displaystyle \begin{align*} \mathcal{L} \left\{ 5\sin{ \left( 11\,t \right) } \sinh{ \left( 11\,t \right) } \right\} &= \mathcal{L} \left\{ 5\sin{ \left( 11\,t \right) } \cdot \frac{1}{2} \left( \mathrm{e}^{11\,t} - \mathrm{e}^{-11\,t} \right) \right\} \\ &= \frac{5}{2} \,\mathcal{L} \left\{...
  14. Jason-Li

    Comp Sci Laplace Transform of the input portion of this circuit

    So I have completed (a) as this (original on the left): I have then went onto (b) and I have equated T(s)=Z(s) as follows: and due to hence Does this look correct to you smarter people? Thanks in advance! All replies are welcome :)
  15. Kaushik

    B Understanding Laplace's Correction and the Adiabatic Process

    Laplace pointed out that the variation in pressure happens continuously and quickly. As it happens quickly, there is no time for heat exchange. This makes it adiabatic. But Newton believed it to be isothermal. Why isn't it isothermal but adiabatic? Why is there a change in temperature?
  16. George Keeling

    I What is a good formula for the Laplace operator?

    I have found various formulations for the Laplacian and I want to check that they are all really the same. Two are from Wikipedia and the third is from Sean Carroll. They are: A Wikipedia formula in ##n## dimensions: \begin{align} \nabla^2=\frac{1}{\sqrt{\left|g\right|}}\frac{\partial}{\partial...
  17. G

    MHB Laplace / inverse laplace transform

    Problem: Find a (limited?) solution to the diff eq. At the end of the solution, when you transform \frac{-1}{s+1} + \frac{2}{s-3} why doesn't it become -e^{-t} + 2e^{3t} , t>0 ?
  18. G

    MHB Solve integral with laplace transform

    So the task is to solve the following integral with laplace transform. Since t>0 we can multiply both sides with heaviside stepfunction (lets call it \theta(t)). What I am unsure about is what happens with the integral part and how do we inpret the resulting expression? What will it result...
  19. Boltzman Oscillation

    Engineering Help finding the damping ratio formula for this circuit

    The circuit to be analyzed is shown below: Since initial conditions are zero (from the instructions) I will use laplace transforms for the cirucit and I will use the MAME method to solve this circuit. The laplace transforms that are required will give me: $$E_g(s) = \frac{10}{s}$$ $$ L_3 =...
  20. M

    Calculating the torque of the Laplace force on this coil

    We first determine the Laplace force for each value ##\alpha##. $$F_{\alpha} = 5(0.3)(0.4)\sin(\alpha) = 0.6\sin(\alpha) \ \text{N}$$ We then calculate the torque at angle ##\alpha##. $$\tau_{\alpha} = \frac{a}{2} F_{\alpha} = 2.5 F_{\alpha} \ \text{N.m}$$ Then we just plug in ##\alpha## and...
  21. T

    MHB Solving PDE using laplace transforms

    [Solved] Solving PDE using laplace transforms Hey, I'm stuck on this problem and I don't seem to be making any headway. I took the Laplace transform with respect to t, and ended up with the following ODE: $\frac{\partial^2 W}{\partial x^2}-W(s^2+2s+1)=0$ and the boundry conditions for $x$...
  22. R

    Laplace inverse convolution

    e
  23. L

    A Laplace transform in spherical coordinates

    Summary: A 1963 paper by Michael Wertheim uses a Laplace transform in spherical coordinates. How is the resulting equation obtained? In 1963, Michael Wertheim published a paper (relevant page attached here), where he presented the following equation (Eq. 1): $$ y(\bar{r}) = 1 + n...
  24. PainterGuy

    I Solving a differential equation using Laplace transform

    Hi, I was trying to see if the following differential equation could be solved using Laplace transform; its solution is y=x^4/16. You can see below that I'm not able to proceed because I don't know the Laplace pair of xy^(1/2). Is it possible to solve the above equation using Laplace...
  25. E

    Can someone double check my solution to this Laplace Transform problem?

    My solution is in the file shown here
  26. E

    Help Proving a Complex Laplace Transform

    So I could just try using the definition by taking the limit as T goes to infinity of ∫ from 0 to T of that entire function but that would be a mess. I tried breaking it down into separate pieces and seeing if I could use anything from the table but I honestly have no clue I'm really stuck. I'd...
  27. F

    I Separation of Variables (PDE) for the Laplace Equation

  28. dRic2

    Simple electric potential and Laplace equation

    Imagine to be in 2 dimensions and you have to find the potential generated by 4 point-charges of equal charge located at the four corners of a square. To do that I think we simply add all the contributions of each single charge: $$V_i(x, y) = - \frac k {| \mathbf r - \mathbf r_i|}$$ $$ V(x, y)...
  29. SamRoss

    I Why is the Laplace transform unchanged when t is replaced with -t?

    In Mathematical Methods in the Physical Sciences by Mary Boas, the author defines the Laplace transform as... $${L(f)=}\int_0^\infty{f(t)}e^{-pt}{dt=F(p)}$$ The author then states that "...since we integrate from 0 to ##\infty##, ##{L(f)}## is the same no matter how ##{f(t)}## is defined for...
  30. cnh1995

    Physical Significance of the Laplace Transform

    I have used Laplace transform during my EE studies to solve differential equations and in control system analysis, but we were taught that as a tool kit to make the math easier. The physical meaning was never explained. I know basic time and frequency domain concepts (thanks to Fourier series)...
  31. majormuss

    Visualizing & Solving a 2D Laplace Eq problem (Polar Coordinates)

  32. C

    Engineering Advanced Circuits, Laplace Transform, Find Initial Conditions

    Vo(S) = [ N(s)Vi(s) + (- s2 + s - 2) ] / s3 + s2 + 1 ; can ignore (-s^2 + s - 2). From relevant equations: Vo(S) = [N(s)*Vi(s)]/(s^3 + s^2 + 1); -> (d3Vo(t)/dt3) + (d2Vo(t)/dt2) + Vo(t) = N(t)(dvi)/dt L[vi(t)] = t to s domain: [s3Vo(s) - s2Vo(0-) - SV'o(0-) - Vo''(0-)]Vo(s) + s2 - SVo -...
  33. Floro Ortiz

    A Inverse Laplace of an Overwhelming Function

    Hello, guys. I'm currently working on a physics problem that requires me to evaluate the inverse Laplace of the function in the attached file. When b = 0, "y" vanishes, and all one has to do is to look up the Laplace table for the inverse. However, non-zero b has been giving me a headache. I...
  34. Arman777

    The mean value of the cube, Force Field Laplace equation

    Homework Statement I have a value of $$ U=U_0+x (∂U/∂x)+y(∂U/∂y)+z (∂U/∂z)+1/2x^2(∂^2U/∂x^2)+1/2y^(2∂^2U/∂y^2)+...$$ We need to find the mean value of the U. So the answer is $$\overline{\rm U}\approx U_0+a^2/24(∇^2U)$$Homework Equations $$\overline{\rm U}=1/a^3 \int \int\int Udxdydz$$ The...
  35. L

    Laplace transform of sin(ωt)/[1+cos^2(ωt)]

    Homework Statement L{sin(ωt)/[1+cos^2(ωt)]} = Homework Equations d {arctan[cos(ωt)]} /dt = - ω•sin(ωt)/[1+cos^2(ωt)] The Attempt at a Solution ∫e^(-st)•[sin(ωt)/(1+cos²(ωt)] dt = -(1/ω)•∫e^(-st)•{arctan[cos(ωt)]}' dt = = (integrating by parts and taking Re(s) > 0) = = π/(4ω) -(s/ω)•∫...
  36. M

    A Laplace or Fourier Transform to solve a system of partial differential equations in thermoelasticity

    I've a system of partial diff. eqs. in thermo-elasticity, I can solve it using normal mode analysis method but I need to solve it using laplace or Fourier
  37. dRic2

    Simplifying Laplace Transform of Cosine with Angular Frequency and Phase Shift

    Homework Statement I have to find the L-transform of ##f(x) = cos(\omega t + \phi)## Homework Equations . The Attempt at a Solution The straightforward approach is to write ##cos(\omega t + \phi)## as ##cos(\omega t)cos(\phi) - sin(\omega t)sin(\phi)## and it becomes: $$Lf(s) = \frac {s...
  38. CptXray

    How can I solve a Laplace equation in a cube with mixed boundary conditions?

    Homework Statement There's a metal cunducting cube with edge length ##a##. Three of its walls: ##x=y=z=0## are grounded and the other three walls: ##x=y=z=a## are held at a constant potential ##\phi_{0}## . Find potential inside the cube. Homework Equations The potential must satisfy Laplace...
  39. M

    Mathematica Solving the Laplace Equation in weird domains

    Hi PF! I looked through the documentation on their website, but under the tab "Solve partial differential equations over arbitrarily shaped regions" I am redirected to a page that does not specify how to create a region. Any help is greatly appreciated. Also, if it helps, the domain is a...
  40. md nabil

    A Solving the Laplace equation over a trapezoidal domain

    can anyone help me on how I can map an isosceles trapezoid onto a rectangular/square domain.Actually I need to solve Laplace equation(delta u = 0) over this isosceles trapezoidal domain. Schwarz Christoffel mapping may help me. But can anyone give me any hint on this mapping procedure?
  41. J

    A Can I linearize this equation?

    By using the laplace transform: $f(t)=sin(Φ(t))$ I want it in the form: F(S)/Φ(S) The purpose is to linearize it in order to put it into a larger transfer function, so far my only solution is to simplify it using taylor expansion.
  42. A

    I Accuracy of the Normal Approximation to Binomial

    What is the preferred method of measuring how accurate the normal approximation to the binomial distribution is? I know that the rule of thumb is that the expected number of successes and failures should both be >5 for the approximation to be adequate. But what is a useful definition of...
  43. B

    Laplace Transform Time Shift Property

    Homework Statement I’m being asked to prove if and why (what instances in which) T<0 for the Laplace transform property of time shifting doesn’t hold. Homework Equations L{f(t-T)}=e^-aT* F(s) The Attempt at a Solution I know that for T<0 there are instances where the property cannot hold, but...
  44. evinda

    MHB Solve Laplace equation on unit disk

    Hello! (Wave) I want to solve the Laplace equation on the unit disk, with boundary data $u(\theta)=\cos{\theta}$ on the unit circle $\{ r=1, 0 \leq \theta<2 \pi\}$. I also want to prove that little oscillations of the above boundary data give little oscillations of the corresponding solution of...
  45. G

    A solution to the Laplace equation

    Hi, I am looking for the solution to the quadrant problem of the Laplace equation in 2 dimensions with Dirichlet boundary conditions \begin{equation} \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0 \end{equation} in the first quadrant ## x, y \geq 0 ## with boundary...
  46. D

    Problem finding the output voltage using Laplace transform

    Homework Statement The input signal of the circuit shown below is ##x(t)=2\sin (ω_ot + \pi/6)##. The switch in the circuit is controlled with a digital signal of the form ##s(t)=\sum_{k=-\infty}^{+\infty} (u(t+ε-kT_s) - u(t-ε-kT_s))##, ##\frac{2\pi}{T_s}=800\pi##, ##ε\to 0##, so that when the...
  47. MaxR2018

    Engineering RLC circuit solved with Laplace transformation

    Hi, i need some help here. Can you help me?:sorry: Here is the problem. Exercise statement: The switch have been closed for a long time y is opened at t=0. Using Laplace's transtormation calculate V0(t) for t ≥ 0 This is what i made to solve it: 1) I know while the switch is closed, the...
  48. evinda

    MHB Calculating the Inverse Laplace Transform for a Given Function

    Hello! (Wave) I want to find $f(t)$ if its Laplace transform is $F(s)=\frac{1}{s(s^2+1)}$. We use the following formula, right? $$f(t)=\frac{1}{2 \pi i} \lim_{T \to +\infty} \int_{a-iT}^{a+iT} e^{st} F(s) ds$$ But how can we calculate the integral $\int_{a-iT}^{a+iT} e^{st}...
  49. evinda

    MHB Boundary value problem for Laplace equation

    Hello! (Wave) Let $a,b>0$ and $D$ the rectangle $(0,a) \times (0,b)$. We consider the boundary value problem in $D$ for the Laplace equation, with Dirichlet boundary conditions, $\left\{\begin{matrix} u_{xx}+u_{yy}=0 & \text{ in } D,\\ u=h & \text{ in } \partial{D}, \end{matrix}\right.$...
  50. mjtsquared

    I Region of convergence of a Laplace transform

    If a Laplace transform has a region of convergence starting at Re(s)=0, does the Laplace transform evaluated at the imaginary axis exist? I.e. say that the Laplace transform of 1 is 1/s. Does this Laplace transform exist at say s=i?
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