Use the Fourier transform directly to solve the heat equation

[richyw]

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),<br /> assuming that u is bounded and k > 0.<br /> <br /> <h2>Homework Equations</h2><br /> <br /> fourier transform<br /> inverse Fourier transform<br /> convolution thm<br /> <br /> <h2>The Attempt at a Solution</h2><br /> <br /> taking the FT of both sides i get <br /> $$U_t=-k w^2U-iw\mu U$$<br /> $$U(0,t)=\Phi(w,0)$$<br /> I solved the ode and got <br /> $$U(w)=e^{(\mu i w- w^2k)t}$$<br /> 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?[/tex]

 

[vela]

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),<br /> assuming that u is bounded and k > 0.<br /> <br /> <h2>Homework Equations</h2><br /> <br /> fourier transform<br /> inverse Fourier transform<br /> convolution thm<br /> <br /> <h2>The Attempt at a Solution</h2><br /> <br /> taking the FT of both sides i get <br /> $$U_t=-k w^2U-iw\mu U$$<br /> $$U(0,t)=\Phi(w,0)$$[/tex]

[tex] Don't you mean $U(\omega,0) = \Phi(\omega,0)$? <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I solved the ode and got <br /> $$U(w)=e^{(\mu i w- w^2k)t}$$<br /> 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? </div> </div> </blockquote>You left out the arbitrary constant when you solved for $U(\omega,t)$. You should have $U(\omega,t) = A(\omega) e^{(i\mu\omega-k\omega^2)t}.$[/tex]