Solving a 4th order pde of the form $\Delta^2 u=0$

Asked Jun 14 '17 at 04:47

Active Jun 15 '17 at 07:53

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I am attempting to solve a 4th order pde in two variables on a square

$$u_{xxxx}+2u_{xxyy}+u_{yyyy}=0$$

with some Neumann boundary condition on the bottom $u_y=\sin x$ and $u_x=0$ on the top we may assume $u_{yy}=u_{xx}$.

What I have tried was to use separation of variables, $u=XY$

The problem is that when I do that I get the equation

$$X^{(4)}Y+2X^{(2)}Y^{(2)}+XY^{(4)}=0$$

I decided to divide by $XY$ but that leaves the middle term to be problematic. Can someone give me some advice for possible approaches here?

asked Jun 14 '17 at 04:47

Kori

wait does this mean u is a multivariable function ?? – Saketh Malyala Jun 14 '17 at 08:48
yes it is multivariable function. – Kori Jun 15 '17 at 07:22
As far as I know, the biharmonic equation is only separable in planar polar coordinates. – Dmoreno Jun 16 '17 at 21:09
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You may find this question useful as well https://math.stackexchange.com/questions/687089/biharmonic-equation-on-a-square-fourier-series-solution-needed – Dmoreno Jun 16 '17 at 21:17

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