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Inverse Laplace Transfrom  Taylor Series/Asymptotic Series?!by chrissimpson
Tags: inverse laplace 
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#1
Apr312, 01:49 PM

P: 11

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 


#2
Apr312, 05:19 PM

HW Helper
P: 1,583

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.



#3
Apr312, 06:01 PM

P: 11

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 nonhomogeneous PDE stuff!



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