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.

View More On Wikipedia.org
Recent contents Top users
  1. M

    Laplace Transform with Integral Convolution

    Homework Statement Determine the Laplace Transform of ∫(from 0 to t) (t-τ)cos(2(t-τ))e-4τ dτ using Laplace Transform tables. Homework Equations I know the basic convolution theorem is (f*g)(t) = ∫f(τ)g(t-τ)dτ The Attempt at a Solution I'm not sure if this is double convolution...
  2. P

    How Do Laplace Transforms Help Understand Frequency and Amplitude in Signals?

    Hi. We are learning about Laplace transforms at uni and I must say that this is a real pain. I have one questions concerning the concept of Laplace transforms, and also a question concerning a specific transform. The task is to make a Laplace transform of: t*sin(2t). I could do an integration by...
  3. B

    MHB Second order differential help (Laplace)

    Hey guys , having a bit of bother getting a solution for this question. Any help would be greatly appreciated! There is a Picture attached showing how far I have got ..
  4. D

    Calculate the range of S using Laplace Transform

    Homework Statement find the range of values for s: f(t)=e^(-t/2)u(t)Homework Equations The Attempt at a Solution what I did initially was to take the integral from 0 to infinity of e^(-st)e^(-t/2)u(t) dt this gave me 2/(2s+1) which when I sub in -.5 to find a divide by 0 or discontinuity then...
  5. D

    MHB How Does Electric Potential Behave Across a Circular Boundary in Electrostatics?

    The inside of a circle of radius \(R\) is held at a constant potential \(V_0\) in the xy plane. The outside is held at a constant potential \(0\). Show that that the electric static potential for \(z > 0\) is \[ T(r, z) = V_0R\int_0^{\infty}\mathcal{J}_0(kr)\mathcal{J}_1(kR)e^{-kz}dk. \] Since...
  6. N

    MHB How Do You Compute the Inverse Laplace Transform of a Power Series?

    Please refer to the attachment. For part a) so far I have: $e^x = 1 + \frac{x}{1!} + ...+ \frac{x^n}{n!}$ So $S^\frac{-1}{2}e^\frac{-1}{S} = S^\frac{-1}{2}(1 -\frac{1}{S} + \frac{1}{2!S^2} - \frac{1}{3!S^3} + \frac{1}{4!S^4} + ... - ...)$ I don't think my $S^\frac{-1}{2}$ on the outside is...
  7. A

    Laplace transforms for the transient analysis of networks

    Homework Statement Use Laplace transforms to derive an expression for the current flowing in the circuit shown in the figure, given that i = 0 when t=0 Homework Equations Expression for the current in an LR series circuit I(t) = V / R (1-e-Rt/L) V - Volts (6V) R - Ohms (10Ω) L...
  8. A

    Laplace transforms for transient analysis

    Homework Statement A capacitor of 0.1 F and a resistor of 5 Ω are connected in series; the combination is applied to a step voltage of 20V. Determine the expression for the: (a) current that flows in the circuit and (b) the voltage across the capacitor in time domain. Homework Equations...
  9. N

    MHB Find Laplace Transform of $ t^2+1)^2 $ without Expansion

    How do you find the laplace transform of this without expanding it? $ L(t^2+1)^2 $
  10. D

    MHB Why are there 2 cosines used in the general solution for this domain?

    The domain is \(0\leq x < \infty\) and \(0\leq y\leq b\). \begin{align*} \phi(x, 0) &= f(x)\\ \phi(x, b) &= 0 \end{align*} I want to show that \[ \phi(x, y) = \frac{1}{\pi}\int_0^{\infty} \int_{-\infty}^{\infty}f(\xi) \frac{\sinh[u(b - y)]}{\sinh(ub)} \cos[u(\xi - x)]d\xi du \] Why isn't the...
  11. T

    Laplace Transform - Learn & Calculate Transforms Easily

    Please ignore this thread.
  12. T

    Solve Laplace Transform of e^-te^tcost

    Homework Statement $$L\{ { e }^{ -t }*{ e }^{ t }cost\}$$ Homework Equations The Attempt at a Solution $$L\{ { e }^{ -t }*{ e }^{ t }cost\} \\ =L\{ \int _{ 0 }^{ t }{ { e }^{ -\tau }{ e }^{ t-\tau }cos(t-\tau )d\tau } \} \\ =\frac { L\{ { e }^{ t }cost\} }{ s } \\ =\frac {...
  13. T

    Trying to find inverse laplace transforms for some problems

    Homework Statement $${ { L } }^{ -1 }\{ \frac { s }{ { ({ s }^{ 2 }+1) }^{ 2 } } \} +{ { L } }^{ -1 }\{ \frac { 1 }{ { ({ s }^{ 2 }+1) }^{ 2 } } \}$$ Homework Equations The Attempt at a Solution I used ##{ { L } }\{ { t }^{ n }f(t)\} ={ (-1) }^{ n }\frac { { d }^{ n } }{ d{ s...
  14. U

    Inverse Laplace Transformation of s/(s^2+2s+5)

    Homework Statement I have some problem finding the inverse laplace transformation of the function: \frac{s}{s^2+2s+5} Homework Equations http://math.fullerton.edu/mathews/c2003/laplacetransform/LaplaceTransformMod/Images/Table.12.2.jpg The Attempt at a Solution I tried to...
  15. Y

    Laplace equation in polar coordinates.

    \nabla^2 u=\frac {\partial ^2 u}{\partial x^2}+\frac {\partial ^2 u}{\partial y^2}=\frac {\partial ^2 u}{\partial r^2}+\frac{1}{r}\frac {\partial u}{\partial r}+\frac{1}{r^2}\frac {\partial ^2 u}{\partial \theta^2} I want to verify ##u=u(r,\theta)##, not ##u(x,y)## Because for ##u(x,y)##, it...
  16. A

    Inverse Laplace transform (Initial Value Problem)

    Homework Statement I'm stuck trying to find out the inverse Laplace of F(s) to get y(t) (the solution for the differential equation): Y(s) = 1 / [ (s-1)^2 + 1 ]^2 The Attempt at a Solution I tried using a translation theorem and then apply the sine formula, but the denominator...
  17. C

    Laplace transforms to solve initial value DE / partial fractions

    Hey guys, i have read many posts on physics forums but this would be my first post. I am quite stuck so any help would be much appreciated. Homework Statement Use Laplace transforms to solve the initial value problem: f''(y) + 4f'(y) +8y = u(t-1) where y(0) = 1 and y'(0) = -1 Solve...
  18. Y

    Can the Laplace Transform Be Determined for Specific Functions?

    Homework Statement Can people help me on these two questions, please. Q1) Does f(t) have a Laplace transform F(s) for sufficiently large real value of s, where f(t) = et/(t4-1). Q2) Either find a function f(t) for which F(s) = L{f(t);t→s} = es, or explain why no such function f(t)...
  19. D

    MHB Laplace on a plate with corners excluded

    We have a square plate with length \(100\). Three side are \(0\) degrees and one side is kept at \(100^{\circ}\). I left \(T(100, y) = 100\) and the other than were zero. Small areas near the two corners must be consider excluded. So I took this as the corners at \((100,100)\) and...
  20. S

    Laplace Transform Question

    In my differential equations book (Edward and Penny) there are many examples of Laplace transforms being applied to linear differential equations with constant coefficients and no examples of them being applied to linear differential equations with variable coefficients. My question is, can this...
  21. B

    Laplace Transformation Convolution Integral

    Homework Statement I need to find the laplace transformation of the following function (and it's ok to leave it expressed as an integral). After doing the initial steps and algebra I got Y(s)= g(t)/(s+2)^2 + 7(1/(s+2)^2)+ 2(1/(s+2)^2) the answer is y(t)=2e^-2t +te^-2t...
  22. N

    Use Laplace to solve this differential equation

    Homework Statement Obtain the solution of the differential equation x'' + w2nx = t My use of L refers to the Laplace Homework Equations The Attempt at a Solution L{x'' + w2nx = t} I decided to do the Laplace of each part individually starting with x'' L{x''} =...
  23. N

    Derive Laplace Transform of the Third Derivative

    Homework Statement Derive he Laplace Transform of the third derivative of f(t). Homework Equations The Attempt at a Solution So, I'm not at all sure how to do this. I think I can start with: L{f'''(t)} = But I'm honestly not sure how this works. Any guidance would be...
  24. N

    Finding the Laplace Transformation of a Piecewise Function

    Homework Statement Obtain the Laplace transformation of the function defined by f(t) = 0 t<0 = t2e-at t>=0Homework Equations The Attempt at a Solution I'm a little unsure of what I'm doing here, so bear with me. L {t2e-at} = ∫inf0 t2e-at dt = ∫0inf t2e-(a+s)tdt How do I integrate...
  25. B

    How do I use integration by parts to find the Laplace transformation of tsin(t)?

    Homework Statement Find the Laplace transformation of the following function by using iterations of integration by parts: f(t) = tsin(t) Homework Equations The Attempt at a Solution I know how to do integration by parts (as learned in calculus) but have never seen a funtion...
  26. M

    Inverse Laplace Transform of a product of exponential functions

    I am reviewing some material on Laplace Transforms, specifically in the context of solving PDEs, and have a question. Suppose I have an Inverse Laplace Transform of the form u(s,t)=e^((as^2+bs)t) where a,b<0. How can I invert this with respect to s, giving a function u(x,t)? Would the inverse...
  27. Y

    Does Laplace and Poisson's equation have to be for static condition?

    I read from the PDE book about Laplace equation in static condition ie ##\frac {\partial U}{\partial t}=0##. But is it true that even if U is time varying ie ##U=U(x,y,z,t)##, you can still have Laplace and Poisson's equation at t=k where k is some fixed value. ie...
  28. I

    Laplace transform of unit step function

    Homework Statement The Attempt at a Solution I know that u(t) is a unit step function and holds a value of either 0 or 1. In laplace transform, when we integrate f(t) from 0 to infinity, we take u(t) to be 1. In this case, since u(t) is u(-t), does this mean it holds a value of 0? Does not...
  29. S

    Find the inverse Laplace transform?

    Homework Statement Find the inverse Laplace transform of e^(-3pi*s)/(s^2+2s+3). Homework Equations I know that you're supposed to factor out the e^(-3pi*s) and the other part becomes 1/(s+1)^2+2 but how do you get the answer? I'm confused. The Attempt at a Solution The answer is...
  30. S

    Inverse Laplace Transform of this expression?

    Homework Statement Find the Inverse Laplace Transform of \frac{1}{(s^{2} + 1)^{2}} Homework Equations The Attempt at a Solution I tried using partial fractions but it didn't work. It looks like a cosine transform, but I don't know what else to do. Help please :(
  31. M

    Understanding Laplace Transform's Spectrum of Damped Sinusoids

    Hi All, in a previous post on the physical meaning of Laplace's Transform I found the following statement " The fundamental Laplace transform pair is H(t), the Heaviside step function, and 1/s, its spectrum of damped sinusoids. Note that the spectrum is weighted towards low frequencies...
  32. T

    Laplace transform,partial fraction problem

    This is more of a pre calc question but it dose however come from diff eqs, just in case I have made fundamental mistakes, i have posted it here. I have been studying this topic for few days by myself, never had any problems with algebra until here. really appreciate all of your help...
  33. J

    Using laplace transforms to solve integrals

    Homework Statement ##\int_0^\infty \frac{a}{a^2+x^2} dx## Homework Equations All the basic integration techniques. The Attempt at a Solution So, I saw this problem and wanted to try it using a different method then substitution, which can obviously solve it pretty easy. Since it is a very...
  34. S

    Find the inverse Laplace transform?

    Homework Statement Find the inverse Laplace transform of F(s)=(8s^2-4s+12)/(s(s^2+4)). Homework Equations A/s+(Bs+C)/(s^2+4) 8s^2-4s+12=A(s^2+4)+(Bs+C)(s)=As^2+4A+Bs^2+Cs=s^2(A+B)+Cs+4A 8=A+B C=-4 A=3 B=5 L^-1 (3/s)+L^-1 ((5s-4)/(s^2+4)) =3+ (Now I'm stucked.) The Attempt...
  35. S

    Find the inverse Laplace transform?

    Homework Statement Find the inverse Laplace transform of F(s)=(2s-3)/(s^2-4). Homework Equations I don't want to find the answer by looking at the Table. F(s)=2s/(s^2-4)-3/(s^2-4) The Attempt at a Solution The answer is f(t)=2 cosh 2t - (3/2) sinh 2t.
  36. Z

    Engineering Finding the Transfer Function of an RC Circuit in the Laplace domain

    Hi, Basically I have circuit with an input Vi (s) across a capacitor C which is in parallel with a resistor R1. And these 2 components are in series with another resistor R2 (please see attached drawing). The question states: Show that the transfer function of the circuit is: Vo/Vi...
  37. A

    Laplace & Fourier: When to Use?

    greetings, Can anyone tell me when we should use Laplace transform and Fourier transform? It seems both of them are equal except σ . thanks in advanced.
  38. E

    Optimal Controller Gain for Desired Response in 2nd Order Laplace Transforms

    Homework Statement Solve the DE for y(t) with the IC's y(0)=20.8m/s and y'(0)=0 if the input is a step function scaled by the desired velocity Vo. vd(t)=Vou(t). Assume the desired velocity Vo=27.8m/s Homework Equations y''(t) + (D/M)y'(t) + (K/M)y(t) = (K/M)vd(t) M = 1,000kg D = 100kg/s K...
  39. J

    Solving Laplace Transform for t-tU(t-1)

    I attached the problem as a word document. I'm stuck trying to determine the laplace transform for t-tU(t-1). I know I'm supposed to work with 1/s^2(s+2) and solve for A, B,C. I got B=1/2, A=-1/4, and C=1/4 when 1=(As+B)(s+2)+Cs^2. The answer to the problem is y= 1/4 + 1/2t +1/4 e^-2t -[1/4...
  40. J

    Creating intuition about Laplace & Fourier transforms

    Hey everyone, I've been reading up a bit on control systems theory, and needed to brush up a bit on my Laplace transforms. I know how to transform and invert the transform for pretty much every reasonable function, I don't have any technical issue with that. My only problem is that some...
  41. J

    How to Simplify and Solve the Inverse Laplace Transform of 1/(s^2+s-20)?

    Having difficulty with L-1 {1/(s^2+s-20)}: Should I make it L-1 {1/(s+5)(s-4)}? I'm stuck.
  42. I

    Inverse laplace transform of this simple function?

    what is the inverse laplace transform of (2s)(1/(s-2))? could i use the identity ∫f(T)g(t-T)dT=F(s)G(s)? i was hesitant so i figured i'd just ask before i continue..
  43. Chris L T521

    MHB ?'s question at Yahoo Answers regarding Laplace Transforms

    Here is the question. Here is a link to the question: Can you show the steps of this differential equations problem? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  44. alyafey22

    MHB The Laplace of unfamiliar function

    I found the following integral \int_{0}^{\infty} \, e^{-at} \sin(bt) \frac{\ln t}{t}\, dt This thread will be dedicated to solve the integral , any ideas will always be welcomed (Cool). Serious steps will be made in the next threads . It seems to involve Laplace identities the value returned...
  45. A

    Vout(t) using laplace and differentiation

    Homework Statement Homework Equations V(L)=L*di/dt laplace(u(t))=1/s The Attempt at a Solution was just wondering if i did this right. converted to the s domain, then wrote voltage equation around loop, in terms of current I(s): V(s)=R*I(s)+L(\frac{dI(s)}{dt}-iL(0-))...
  46. M

    What is the significance of 'sufficiently large' values in Laplace Transform?

    In the attachment that I added I highlighted the portion I am questioning. I will define L[f(t)](s) to be the laplace transform of the function f(t). f(t) = e^t L[f(t)](s) = 1/(s-1). The laplace transform is defined for all values s≠1. L[f(t)](2) = 1. Question: "What do they mean by...
  47. N

    Solve Laplace Transform: te^t*H(t-1)

    Homework Statement Laplace transform: te^t*H(t-1) Homework Equations L{te^t} = 1/(s-1)^2 L{H(t-1)} = e-s/s The Attempt at a Solution Using e^(at)f(t) -> F(s-a) with..e^(1t)H(t-1) We can get, e^(-1s-1)/(s-1) = e^(-s-1)/(s-1) [I think?] Thus applying tf(t) -> -dF/ds Derive[e^(-s-1)/(s-1)]...
  48. C

    Express Laplace Transform of y(t) in given form.

    Homework Statement y(t) solves the following IVP y''(t) + 2y'(t) + 10y(t) = r(t) y(0) = 2 y'(0) = 3 r(t) = 0 if t < 0 t if 0 ≤ t ≤ 1 0 if t > 1 Demonstrate that the laplace transform of y(t) is Y(s) = \frac{2s+7}{s^{2}+2s+7} + \frac{e^{-s}}{s(s^{2}+2s+7)} +...
  49. D

    Laplace equation in rectengualar channel (Fluid mechanics)

    Homework Statement Estimate the speed a potential flow in gravity field would have in direction y in rectangle channel with depth h [/iteh] and length l . The fluid is incompressible and on the surface z = 0 we have boundary condition \dfrac{\partial^2 \phi}{t^2} + g\dfrac{\partial...
  50. T

    My engineering faculty is having us rote-memorize Laplace transforms

    Hi, Laplace transforms are a fundamental tool in electrical engineering and control systems engineering. Unfortunately my University is having us read books that use Laplace Transforms, but I've never learned the theory rigorously! For example the book I'm currently reading state briefly that...
Back
Top