• Support PF! Buy your school textbooks, materials and every day products Here!

Use the Fourier transform directly to solve the heat equation

  • Thread starter richyw
  • Start date
  • #1
180
0

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?
 

Answers and Replies

  • #2
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
14,539
1,150

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]
Don't you mean ##U(\omega,0) = \Phi(\omega,0)##?

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?
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}.##
 

Related Threads for: Use the Fourier transform directly to solve the heat equation

Replies
4
Views
3K
Replies
7
Views
3K
Replies
2
Views
8K
  • Last Post
Replies
4
Views
2K
Replies
10
Views
16K
Replies
3
Views
538
Top