Inverse Laplace Transfrom - Taylor Series/Asymptotic Series?

[chrissimpson]

Homework Statement

W(x,s)=(1/s)*(sinh(x*s^0.5))/(sinh(s^0.5))

Find the inverse laplace transform of W(x,s), i.e. find w(x,t).

Answer:

w(x,t)= x + Ʃ ((-1)^n)/n) * e^(-t*(n*pi)^2) * sin(n*pi*x)

summing between from n=1 to ∞

Homework Equations

An asymptotic series..?!

The Attempt at a Solution

Taking the Taylor series for sinh 'simplifies' the expression to:

W(x,s)=(1/s)*(x + x^3 + x^5 +x^7 ...)

You can write this as:

W(x,s)=(x/s)+(1/s)*(x^3 + x^5 + x^7...)

W(x,s)= (x/s) + Ʃ(x^2n+1)/s

summing between n=1 and ∞

This is where I think I begin to run into trouble (unless I already have!) as I think I would now form:

w(x,t)= x + Ʃ(x^2n+1)


There is some mention in a similar question that an asymptotic series was used, but I can't work out how to make this fit in with the question!


Any help would be really appreciated!

Chris

 

[hunt_mat]

I got involved with things like this when I was looking at plasma physics, you want to use a branch cut in you inverse transform and usae the Bromwich inversion formula.

 

[chrissimpson]

Cheers! I'll have a look into that. Any idea how painfully difficult this will be - I'm coming towards the (current) limit of my mathematical capabilities with this non-homogeneous PDE stuff!