Question

Heat equation with convection.

\[u_t=ku_{xx}+\mu u_x\]\[u(x,0)=\phi(x)\]on \(-\infty,\infty)\) with k>0.

I am supposed to solve this by using the fourier transform directly.

Answer 1

so I will use U(w,t) to denote the FT of u(x,t).

Answer 2

I have \[U_t-k(iw)^2U-\mu i w U = 0\]\[U_t+kw^2U-\mu i wU=0\]

Answer 3

which apparently is already wrong? my textbook ends up with \[U_t+kw^2U+iw\mu U=0\]I was just wondering where I am going wrong here?

Answer 4

Then after that, I am having trouble getting the solution by directly using the fourier inverse. I don't think I can use the convolution thm, because it asks for foruier transforms directly.

Answer 5

humm im used to heat equation for 2 dimention u would use
del^2 = 0 right ?
in 2d does it work ?

Answer 6

Oh hang on, I realized it was all taken to the left side. I'll read that again. lol

Answer 7

@Mashy :3

Answer 8

It seems like the book is wrong here / has a typo. I don't see where the sign would disappear at the end there and become positive on \( -\mu i w U \).

Answer 9

ok so my work looks good so far?

Answer 10

then solving the ODE I have \[U(w,t)=U(w,0)e^{wt(-kw+i\mu)}\]

Answer 11

and then I am stuck trying to get back to u(x,t) without using the convolution thm.

Answer 12

I am not sure that we are restricting convolution theorem from the process by having to use the Fourier transform directly. At least, I have been incapable of finding a Fourier transform explanation that was without convolution theorem to obtain the inverse (last viewed http://www.physics.ucc.ie/fpetersweb/FrankWeb/courses/PY4118/heat.pdf ), nor could I think of any way to do this myself without it... o.o

Are you sure it absolutely does not want convolution theorem?

Answer 13

Anyways, I will be off for a few hours. If you haven't resolved the issue, please bump the question or tag another helper to get back to you. Good luck with the maths! :)