We have the equation $$-\big(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\big)=1$$ with conditions $$u(1,y)=u(x,1)=0\\ \frac{\partial u}{\partial x}(0,y)=\frac{\partial u}{\partial y}(x,0)=0$$ where $$(x,y)\in\Omega=[0,1]\times[0,1].$$ I tried using separation of variables but got nowhere, if anyone could give me a hint on how to solve this i would be grateful.
2026-03-27 04:56:39.1774587399
Poisson equation on unit square?
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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 $f$.