I have the steady state boundary value problem with periodic boundary conditions
$u-u''=f, u(-1)=u(1), u'(-1) =u'(1)$.
I need to show that the operator $L_p: C_p^2[ -1,1]\to C[-1,1], L_pu=u-u''$, where $C_p^2[ -1,1]=\{u\in C^[ -1,1]: u(-1)=u(1), u'(-1) =u'(1)\}$, is linear and symmetric. I am not sure how to define a vector product on $C_p^2[ -1,1]$. Can it be defined like this
\begin{align} (u,v)=\int_{-1}^1uv \end{align} ?
My next question is how to find non constant eigen vectors of $L_p$ that corresponds to given eigen value, let's say $\alpha_i=1+(i\pi)^2$. Should I solve differential equation? Also, could you give me a good reference on the spectral method for solving the problem \begin{align} L_pu=2+sin(4\pi x)+4cos(5\pi x). \end{align} I am not familiar with this type of problems, I just need to solve two of this kind for my project and I would appreciate if you could refer to similar solved examples so I can see and try to solve it. A hint would help as well.