|Apr3-12, 01:49 PM||#1|
Inverse Laplace Transfrom - Taylor Series/Asymptotic Series?!
1. The problem statement, all variables and given/known data
Find the inverse laplace transform of W(x,s), i.e. find w(x,t).
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!
|Apr3-12, 05:19 PM||#2|
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.
|Apr3-12, 06:01 PM||#3|
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!
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