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## Homework Statement

Use the Fourier transform directly to solve the heat equation with a convection term

[tex]u_t =ku_{xx} +\mu u_x,\quad −infty<x<\infty,\: u(x,0)=\phi(x),

assuming that u is bounded and k > 0.

## Homework Equations

fourier transform

inverse fourier transform

convolution thm

## The Attempt at a Solution

taking the FT of both sides i get

[tex]U_t=-k w^2U-iw\mu U[/tex]

[tex]U(0,t)=\Phi(w,0)[/tex]

I solved the ode and got

[tex]U(w)=e^{(\mu i w- w^2k)t}[/tex]

but now I am a bit confused on the next step, is this where I want to get my initial condition involved, or do I want to try and get it back as u(x,t) using inverse FT. I can see that my solution is a gaussian multiplied by another function of F, so I think I might be able to use convolution thm?