# I Inverse laplace PDE

1. Sep 28, 2016

### fahraynk

I am trying to solve with Laplace Transforms in an attempt to prove duhamels principle but cant find the Laplace transform inverse at the end. The book I am reading just says "from tables"...

The problem :
$$U_t = U_{xx}\\\\ U(0,t)=0 \quad 0<t< \infty\\\\ U(1,t)=1\\\\ U(x,0)=0 \quad 0<x<1\\\\$$

The solution attempt :
$$SU(x,s) = U_{xx}(x,s)\\\\ U(1,s) = \frac{1}{S}\\\\ U = \frac{1}{S} \frac{e^{\sqrt{S}x}-e^{-\sqrt{S}x}}{e^{\sqrt{S}}-e^{-\sqrt{S}}} = \frac{1}{S} \frac{sinh(\sqrt(S)x)}{sinh(\sqrt{S}})\\\\$$
The inverse transform is the convolution $$1 \ast \mathcal{L}^{-1}(\frac{sinh(\sqrt(S)x)}{sinh(\sqrt{S}})$$

Does anyone know of a table where I can find this... The integral to actually compute it myself is... terrifying. Do I have to use the integral... if so... can someone show me how...

2. Oct 4, 2016