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Using convolution for Laplace transform

  1. Apr 3, 2007 #1
    1. The problem statement, all variables and given/known data
    Using convolution theorem for Laplace theorem,, show that


    2. Relevant equations
    inverse Laplace transform (1/(S^3/2*(s-1)) = (2*e^t)/Pi^1/2 intregral (from 0 to t) e^-x*x^1/2dx.


    3. The attempt at a solution

    The inverse Laplace above is a product of 1/s^3/2 and 1/(s-1)

    and both terms are the Laplace transform of 2/Pi^1/2*t^1/2 and e^t respectively.

    changing variable t = u gives:2/Pi^1/2*u^1/2 and e^u

    using convolution: integral (from 0 to infinity) e^-st integral (from 0 to t)
    (2/Pi^1/2*u^1/2 )(e^u)du dt.

    I dont know how to go about with the integral from here onwards..Help!
     
  2. jcsd
  3. Apr 4, 2007 #2

    HallsofIvy

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    The "convolution theorem" you are told use says that the Laplace transform of a convolution of functions is the product of the Laplace transforms of the functions. Isn't that exactly what you need here?
     
  4. Apr 5, 2007 #3
    I am sorry but I dont really get you. Laplace transform of a convolution of functions is the product of the Laplace transforms of the functions.

    Isnt that what I am trying to do. finding the product of the inverse Laplace (which as given) and converting the products to its functions.

    Is there a rule or formula for convolution theorem of Laplace theorem?
     
  5. Apr 5, 2007 #4

    HallsofIvy

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    Good. You are saying that [itex]\frac{1}{s^{3/2}}\frac{1}{s-1}[/itex] is the product of the Laplace transforms of [itex]\frac{2}{\sqrt{\pi t}}[/itex] and [itex]e^t[/itex]. You also know that the Laplace transform of a convolution is the product of the individual functions. So [itex]\frac{1}{s^{3/2}}\frac{1}{s-1}[/itex] is the Laplace transform of the convolution of [itex]\frac{2}{\sqrt{\pi t}}[/itex] and [itex]e^t[/itex]. If you can show that the convolution of those function is equal to [itex]\frac{2e^t}{\sqrt{\pi}}\int_0^t e^{-x\sqrt{x}}dx[/itex], you are done!

    By the way, what "Precalculus" class deals with the Laplace transform?
     
  6. Apr 5, 2007 #5
    My problem now is to prove it because now the function has integral form 0 to t where does that come from?

    Is there a convolution rule that i need to know to be able to prove it?

    And im so sorry if I post it in the wrong section. Under which section should post this Laplace problems? thx.
     
  7. Apr 5, 2007 #6

    HallsofIvy

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    First what is the DEFINITION of the convolution of two functions? I think that is the formula you need.

    Second, this should be in "Calculus and Beyond". I will go ahead and move it.
     
  8. Apr 5, 2007 #7
    Oh I just found an appropriate section for this problem. Sorry for posting the problem at the wrong area. I'll make sure Ill post any problem/s in the appropraite sectionnext time.
     
  9. Apr 5, 2007 #8
    There was some typing error, in the original equation.

    inverse Laplace transform (1/(S^3/2*(s-1)) = (2*e^t)/Pi^1/2 intregral (from 0 to t) (e^-x)*(x^1/2)dx.

    The inverse Laplace above is a product of 1/s^3/2 and 1/(s-1)

    and both terms are the Laplace transform of (2/Pi^1/2)*t^1/2 and e^t respectively.

    Using convolution theorem,
    integral (from 0 to infinity) e^-st (integral (from 0 to t) f (t - x)*g(x)dx)dt

    putting the Laplace transform it becomes
    integral (from 0 to infinity) e^-st (integral (from 0 to t) (e^ (t - x))*(2/Pi^1/2)*x^1/2dx)dt

    But can I do this,
    (integral (from 0 to infinity) e^-st dt)*(integral (from 0 to t) (e^ (t - x))*(2/Pi^1/2)*x^1/2dx)dt

    If i can do that then it is already proven
    (2*e^t)/Pi^1/2 intregral (from 0 to t) (e^-x)*(x^1/2)dx.
     
  10. Apr 5, 2007 #9
    Well, actually i don't think I can do the following:
    (integral (from 0 to infinity) e^-st dt)*(integral (from 0 to t) (e^ (t - x))*(2/Pi^1/2)*x^1/2dx)dt

    I think the integration
    integral(from 0 to infinity) of e^-st*(integral (from 0 to t) (e^ (t - x))*(2/Pi^1/2)*x^1/2dx)dt using integration by parts

    integration of e^-st (from 0 to infinity), will give 1.

    hence and leaving (integral (from 0 to t) (e^ (t - x))*(2/Pi^1/2)*x^1/2dx)

    am I right? I hope I am making sense.
     
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