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Math Help - Inverse Laplace Transform

  1. #1
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    Inverse Laplace Transform

    Hey folks,

    I'm having trouble tackling the inverse laplace transform of the following term as I have only been able to obtain formulas for single term ratios of hyperbolic functions. I can't find anything for 2 terms like this in the numerator... Any help would be appreciated!

     \frac{T_o \cdot cosh(\sqrt{s} \cdot L)sinh\(\sqrt{s} \cdot x)}{s \cdot sinh(\sqrt{s} \cdot L)}

    where T_o and L are numerical constants.

    Any help please! Thanks!

    EDIT:

    Sorry I suppose I should have elaborated.

    This is part of a heat conduction problem, solving \frac{\partial T}{\partial t} =\alpha \frac{\partial^2 T}{\partial x^2}

    Solving by taking LT, solving the corresponding ODE in (x,s) and then inversing back to the solution in (x,t). So x is constant in terms of the inverse transform, but not constant for the solution.
    Last edited by Kasper; November 29th 2011 at 10:38 PM.
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  2. #2
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    Re: Inverse Laplace Transform

    T0 and L are constants.
    Obviously s is the variable.
    But what is x ?
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  3. #3
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    Re: Inverse Laplace Transform

    Quote Originally Posted by JJacquelin View Post
    T0 and L are constants.
    Obviously s is the variable.
    But what is x ?
    Sorry I suppose I should have elaborated.

    This is part of a heat conduction problem, solving \frac{\partial T}{\partial t} =\alpha \frac{\partial^2 T}{\partial x^2}

    Solving by taking LT, solving the corresponding ODE in (x,s) and then inversing back to the solution in (x,t). So x is constant in terms of the inverse transform, but not constant for the solution.
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  4. #4
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    Re: Inverse Laplace Transform

    Quote Originally Posted by Kasper View Post
    Hey folks,

    I'm having trouble tackling the inverse laplace transform of the following term as I have only been able to obtain formulas for single term ratios of hyperbolic functions. I can't find anything for 2 terms like this in the numerator... Any help would be appreciated!

     \frac{T_o \cdot cosh(\sqrt{s} \cdot L)sinh\(\sqrt{s} \cdot x)}{s \cdot sinh(\sqrt{s} \cdot L)}

    where T_o and L are numerical constants.

    Any help please! Thanks!

    EDIT:

    Sorry I suppose I should have elaborated.

    This is part of a heat conduction problem, solving \frac{\partial T}{\partial t} =\alpha \frac{\partial^2 T}{\partial x^2}

    Solving by taking LT, solving the corresponding ODE in (x,s) and then inversing back to the solution in (x,t). So x is constant in terms of the inverse transform, but not constant for the solution.
    The inverse Laplace transform would be very complicated, may be impossible to express in a finite number of standard functions.
    I am surprised that you obtained this kind of function of s from the Laplace transform of the PDE relatively to t.
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