I want to solve the following PDE using Fourier series:
- $u(x,y): \Omega \to \mathbb{R}$,
- $\Omega=(0,\pi)\times (0,2\pi)$
- $u-3u_{xx}-u_{yy}= 3\sin(2x)-\sin(5x)$
$u_{xx}$ and $u_{yy}$ are second derivatives with respect to $x$ and $y$ respectively. The boundary conditions of the problem are:
- Dirichlet boundary condition in $x$ direction: $u(x=0,y)=u(x=\pi,y)=0$
- Neumann boundary condition in $y$ direction: $u_y(x,y=0)=u_y(x,y=2\pi)=0$
I chose my base functions for $u$ such that they satisfy boundary conditions: $\sin(nx)\cos(my/2)$. Therefore, the left-hand side of the equation would be: $$\sum a(n,m)\left[1+3n^2+\frac{m^2}{4}\right]\sin(nx)\cos(my/2)$$ My problem is that I should be able to write the right-hand side of the equation in form of $\sum b(n,m)\sin(nx)\cos(my/2)$, so that I can obtain $a(n,m)$. Any suggestions?