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My professor personally asked me to solve the following problem :

Let $\Omega=(0,1)\times(0,1)\times\mathbb{R}$.

\begin{cases} -\triangle u=f & \text{in }\Omega,\\ u=0 & \text{on }\partial\Omega, \end{cases}

which is a Poisson equation on square column of infinite length. Any suggestion for me? If exists, please give a helpful recommendation textbook(or anything) for me.

kayak
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1 Answers1

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Hint: find a basis of eigenfunctions for the Laplacian with the given boundary conditions and express both $u$ and $f = 1$ as a combination of these eigenfunctions. You can then find the coefficients of $u$ (which are the unknowns) by matching the coefficients in the expansions of $-\Delta u$ with the coefficients of the expansion of $f$.

You can use the same method as in finding a solution of the nonhomogeneous heat equation (eigenfuction method).