Thread: Biharmonic Operator -Fourier Transform.

Hi guys,

just wondering if someone out there could give me a hand with the following problem.

u(x,y) multiplied by a biharmonic operator =0.

The question is asking us to prove that this can be satisfied by S(x,k) = (S"(x,k)-k^2*S(x,k)) = 0 (which is straight forward enough)

but then to solve the function with respect to the boundary conditions:

u_x(0,y)=0 and u_y(0,y)=f(y) using Fourier Transforms.

I then have to solve the system using the convolution theorem.

if anyone is a bit of a whiz with this sort of thing, could you give me a hand?

id really appreciate it.

Thanks guys.

Originally Posted by bluesboy91

Hi guys,

just wondering if someone out there could give me a hand with the following problem.

u(x,y) multiplied by a biharmonic operator =0.

The question is asking us to prove that this can be satisfied by S(x,k) = (S"(x,k)-k^2*S(x,k)) = 0 (which is straight forward enough)

but then to solve the function with respect to the boundary conditions:

u_x(0,y)=0 and u_y(0,y)=f(y) using Fourier Transforms.

I then have to solve the system using the convolution theorem.

if anyone is a bit of a whiz with this sort of thing, could you give me a hand?

id really appreciate it.

Thanks guys.

Your question is really hard to understand as written. When you say multiplied to do mean acted on?

Biharmonic operator acting on would give the PDE



If you take the fourier transform with respect to you get the ODE



This is a 4th order ODE in with charaterisitic equation



So the solution will have the form



I hope this helps, If this is not what you are looking for please clarify.

Hi mate,

Thanks for the help. I'm still unsure of a few things. (that solution did clarify
An initial problem I was facing).

First is,

Using the BC/IC: u_x(0,y)=0 and u_y(0,y)=f(y) how do I transform these conditions (more specifically the second one as it depends on y which is to be transformed) and then use that to find the constants c_i(k)?

Second, after getting that solution, solving the system via the convolution theorem. ( I obviously haven't attempted this one as I can't do the first part).

Thanks heaps!