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Thread: Biharmonic Operator -Fourier Transform.

  1. #1
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    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.
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    Re: Biharmonic Operator -Fourier Transform.

    Quote Originally Posted by bluesboy91 View Post
    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 $\displaystyle u(x,y)$ would give the PDE

    $\displaystyle \Delta ( \Delta u(x,y))=0 \iff \frac{\partial^4 u}{\partial x^4}+2\frac{\partial^4 u}{\partial x^2 \partial y^2 }+\frac{\partial^4 u}{\partial y^4 } =0$

    If you take the fourier transform with respect to $\displaystyle y$ you get the ODE

    $\displaystyle \frac{\partial^4 \hat{u}}{\partial x^2}-2k^2\frac{\partial^4 \hat{u}}{\partial x^2 }+k^4\hat{ u} =0$

    This is a 4th order ODE in $\displaystyle x$ with charaterisitic equation

    $\displaystyle m^4-2k^2+k^4=0 \iff (m-k)^2(m+k)^2$

    So the solution will have the form

    $\displaystyle \hat{u}(x,k)=(c_1+c_2x)e^{kx}+(c_3+c_4x)e^{-kx}$

    I hope this helps, If this is not what you are looking for please clarify.
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  3. #3
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    Re: Biharmonic Operator -Fourier Transform.

    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!
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