Solving Partial differential equations using fourier series

Question

consider the following differential equation in the domain $\Omega := (0,2) \times(0,3)$ :

$$ -\Delta u = f \text{ in }\Omega$$ $$ \frac{\partial u}{\partial \overline n} = 0 \text{ on }\partial \Omega$$

$\overline n$ is a normal vector and $f(x,y) \in L^2(\Omega)$

Find a formal solution $u(x,y)$ using fourier series

My thoughts:- My first idea is should I assume the solution is of the form

$$u(x,y) = c_{k,l} \cos\left(\frac{k \pi x}{2}\right)\sin\left(\frac{l \pi y}{3}\right)$$

In the above assumption I assumed basis vectors.

because when differentiated $cos$ becomes $sin$ so the boundary conditions would be satisfied on the boundaries $\partial \Omega$ and at last find relation between the coefficient values of left hand and right hand side.

I some how feel my approach is not right ?

Can somebody help ?

Answer 1

You have found the eigenfunctions of the Laplace equation. Now you need to expand your right hand side using double Fourier series: $$f={\sum}{f_{k,l}cos(k\pi x/2)sin(l \pi y/3)}$$ Then, substituting your solution for $u$ in the equation $\Delta u =f$, we should equate $c_{k,l}=f_{k,l}$ which gives us $$c_{k,l}=\frac {f_{k,l}}{\lambda_{k,l}}$$ where $\lambda_{k,l}^2=(k \pi/2)^2 +(l \pi/3)^2$