1.6.3.2 Solution ppe-b

Question:

Check the separation of variables solution to the above problem,

$$u(x,y) = \sum_{n=1}^\infty f_n \sin(nx) \cosh(ny)$$

Here the “Fourier coefficients” $f_n$ are chosen so that they satisfy

$$f(x) = \sum_{n=1}^\infty f_n \sin(nx)$$

Can you immediately see that this separation of variables solution is probably no good?

Answer:

Plug the solution in the partial differential equation and all four boundary conditions. You will find that they are all satisfied.

If you are concerned about manipulating infinite sums, then you are perfectly right. For now simply assume that $f(x)$ is such that $f_n$ is zero above some largest value $n=n_{\rm {max}}$. Then the sums are finite and you can manipulate them in the usual ways.

You can see that this separation of variables solution is probably no good, but not from the simple checks above.