# Inverse Laplace Transform

• Nov 29th 2011, 07:04 PM
Kasper
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!

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

where $\displaystyle T_o$ and $\displaystyle L$ are numerical constants.

EDIT:

Sorry I suppose I should have elaborated.

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

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.
• Nov 29th 2011, 10:21 PM
JJacquelin
Re: Inverse Laplace Transform
T0 and L are constants.
Obviously s is the variable.
But what is x ?
• Nov 29th 2011, 10:31 PM
Kasper
Re: Inverse Laplace Transform
Quote:

Originally Posted by JJacquelin
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 $\displaystyle \frac{\partial T}{\partial t} =\alpha \frac{\partial^2 T}{\partial x^2}$

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.
• Nov 30th 2011, 10:52 PM
JJacquelin
Re: Inverse Laplace Transform
Quote:

Originally Posted by Kasper
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!

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

where $\displaystyle T_o$ and $\displaystyle L$ are numerical constants.

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