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"...(adsbygoogle = window.adsbygoogle || []).push({});

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...

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# I Inverse laplace PDE

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