Solving Poisson equation in a unit square

[gasper7]

Hey guys

I am trying to solve this Poisson equation, but i can't get a break through. Has anyone solved this before? I can't find ONE single example, where f instead of 0 (zero) is a pure function f(x,y). The main thing here is, that the problem has to be solved using Fourier series... This is the only good example I've found (http://math.la.asu.edu/~kuiper/502files/Laplace.pdf) but it's not explained how they got Umn,...

\(\displaystyle \Delta\, u=\,1,\, (x,y)\, \epsilon \,D,\,u(x,y)\,=\,0,\,(x,y)\,\epsilon\,\partial\,D\)

Basically, we're solving \(\displaystyle u_{xx}\,+\,u_{yy}\,=\,1\)

One possible solution for u is: \(\displaystyle u(x,y)\,=\,\frac{x^{2}\,+\,y^{2}}{4}\)

But the solution should be expanded into Fourier series, so I got stuck there. Hope someone will know how to solve this...

 

[HallsofIvy]

Hey guys

I am trying to solve this Poisson equation, but i can't get a break through. Has anyone solved this before? I can't find ONE single example, where f instead of 0 (zero) is a pure function f(x,y). The main thing here is, that the problem has to be solved using Fourier series... This is the only good example I've found (http://math.la.asu.edu/~kuiper/502files/Laplace.pdf) but it's not explained how they got Umn,...

\(\displaystyle \Delta\, u=\,1,\, (x,y)\, \epsilon \,D,\,u(x,y)\,=\,0,\,(x,y)\,\epsilon\,\partial\,D\)

Basically, we're solving \(\displaystyle u_{xx}\,+\,u_{yy}\,=\,1\)

One possible solution for u is: \(\displaystyle u(x,y)\,=\,\frac{x^{2}\,+\,y^{2}}{4}\)

I don't know what you mean by "possible solution". This clearly is NOT a solution to the equation.

But the solution should be expanded into Fourier series, so I got stuck there. Hope someone will know how to solve this...

Have you done anything at all? Surely the first thing you did was write
\(\displaystyle u(x, y)= \sum_{n=0}^\infty A_n(y) sin(n\pi x0)+ B_n cos(n\pi x)\)
put that into the differential equation so you get a system of equations for \(\displaystyle A_n(y)\) and \(\displaystyle B_n(y)\).