# Thread: Inverse Laplace Transform

1. ## 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.

2. ## Re: Inverse Laplace Transform

T0 and L are constants.
Obviously s is the variable.
But what is x ?

3. ## Re: Inverse Laplace Transform

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 $\frac{\partial T}{\partial t} =\alpha \frac{\partial^2 T}{\partial x^2}$

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.

4. ## Re: Inverse Laplace Transform

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!

$\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.