Inverse Laplace Transfrom - Taylor Series/Asymptotic Series?!

chrissimpson

1. The problem statement, all variables and given/known data

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 ∞


2. Relevant equations

An asymptotic series..?!

3. 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!