- #1

joshmccraney

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## Homework Statement

$$u_{xx}=u_t+u_x$$ subject to ##u(x,0)=f(x)## and ##u## and ##u_x## tend to 0 as ##x\to\pm\infty##.

## Homework Equations

Fourier Transform

## The Attempt at a Solution

Taking the Fourier transform of the PDE yields

$$

(\omega^2-i\omega) F\{u\}= \partial_t F\{u\}\implies\\

F\{u\} = \exp(t/4)F\{f(\omega)\}\exp((\omega-i/2)^2t)\implies\\

u = \exp(t/4)\int_{-\infty}^\infty f(\omega)\exp((\omega-i/2)^2t)\exp(i\omega x) \, d\omega

$$ From here I don't know how to apply convolution. Can anyone help?