I'm trying to solve the biharmonic equation $\nabla^4 \phi = 0$ for $\phi(x,y)$ over a square domain $0 \le x \le L_x$, $0 \le y \le L_y$, for (currently) general boundary conditions.

However, I end up obtaining an answer that doesn't make sense, even before I apply any boundary conditions.

To the best I know, any arbitrary continuous function $\phi(x,y)$ concerned over the square domain can be expressed using a 2D extension of the half range Fourier series. That is,

$$\phi(x,y) = \sum_{j=0}^\infty \sum_{k=0}^\infty \hat{\phi}_{jk} \, \mathrm{c}_j^x(x) \, \mathrm{c}_k^y(y)$$


$$\hat{\phi}_{jk} = \frac{4}{L_x L_y} \int_0^{L_x} \int_0^{L_y} \phi(x,y) \cos \left(\frac{\pi j}{L_x}x\right) \cos \left(\frac{\pi k}{L_y}y\right) \, \mathrm{d}x \, \mathrm{d}y, $$

$$c_j^x(x) = \cases{ \frac{1}{2} & \text{for $j=0$} \\ \cos \left(\frac{\pi j}{L_x}x\right) & \text{for $j \ne 0$} }$$


$$c_k^y(y) = \cases{ \frac{1}{2} & \text{for $k=0$} \\ \cos \left(\frac{\pi k}{L_y}y\right) & \text{for $k \ne 0$} }$$

Substituting into the biharmonic equation

$$\frac{\partial^4 \phi}{\partial x^4} + 2\frac{\partial^4 \phi}{\partial x^2 y^2} + \frac{\partial^4 \phi}{\partial y^4} = 0$$

and performing appropriate partial differentiation yields

$$\sum_{j=0}^\infty \sum_{k=0}^\infty \Bigg( \left(\frac{\pi j}{L_x}\right)^2 + \left(\frac{\pi k}{L_y}\right)^2 \Bigg)^2 \hat{\phi}_{jk} \, \mathrm{c}_j^x(x) \, \mathrm{c}_k^y(y) = 0$$

Then, by noting that

$$\frac{4}{L_x L_y} \int_0^{L_x} \int_0^{L_y} \mathrm{c}_j^x(x) \, \mathrm{c}_k^y(y) \cos \left(\frac{\pi m}{L_x}x\right) \cos \left(\frac{\pi n}{L_y}y\right) \, \mathrm{d}x \, \mathrm{d}y = \delta_{jm} \delta_{kn}$$

where $\delta_{pq}$ is the Kronecker delta,

it can be seen that

$$\sum_{j=0}^\infty \sum_{k=0}^\infty \Bigg( \left(\frac{\pi j}{L_x}\right)^2 + \left(\frac{\pi k}{L_y}\right)^2 \Bigg)^2 \hat{\phi}_{jk} \, \delta_{jm} \delta_{kn} = 0$$


$$\Bigg( \left(\frac{\pi m}{L_x}\right)^2 + \left(\frac{\pi n}{L_y}\right)^2 \Bigg)^2 \hat{\phi}_{mn} = 0$$

Therefore, this implies that

$$\left(\frac{m}{L_x}\right)^2 + \left(\frac{n}{L_y}\right)^2 = 0 \quad \text{or} \quad \hat{\phi}_{mn} = 0$$

and the former condition can only hold if $m=n=0$

In order words,

$$\hat{\phi}_{mn} = 0 \quad \text{for all $j \ne 0$ and $k \ne 0$}$$,

which implies the only solution is a constant function. I haven't even enforced any boundary conditions, so clearly this conclusion must be nonsense.

I'm not sure at what part I've slipped up. Is the expression for the half range Fourier series incorrect and not general enough?


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