Inverse Laplace Transform

**Kasper** · November 29th 2011, 07:04 PM

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.

**JJacquelin** · November 29th 2011, 10:21 PM

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

**Kasper** · November 29th 2011, 10:31 PM

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.

**JJacquelin** · November 30th 2011, 10:52 PM

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.

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