$$
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\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
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\newcommand{\Ddt}[1]{\frac{D #1}{dt}}
\newcommand{\E}[1]{\hbox{E}\lbrack #1 \rbrack}
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\newcommand{\xpoint}{\boldsymbol{x}}
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\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\w}{\boldsymbol{w}}
\newcommand{\V}{\boldsymbol{V}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\F}{\boldsymbol{F}}
\newcommand{\stress}{\boldsymbol{\sigma}}
\newcommand{\strain}{\boldsymbol{\varepsilon}}
\newcommand{\stressc}{{\sigma}}
\newcommand{\strainc}{{\varepsilon}}
\newcommand{\I}{\boldsymbol{I}}
\newcommand{\T}{\boldsymbol{T}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\ii}{\boldsymbol{i}}
\newcommand{\jj}{\boldsymbol{j}}
\newcommand{\kk}{\boldsymbol{k}}
\newcommand{\ir}{\boldsymbol{i}_r}
\newcommand{\ith}{\boldsymbol{i}_{\theta}}
\newcommand{\iz}{\boldsymbol{i}_z}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
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\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
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$$
The collocation method
\( R=0 \) (i.e.,the differential equation) must be satisfied at \( N+1 \) points:
$$
\begin{equation}
-\sum_{j\in\If} c_j\baspsi_j''(\xno{i}) = f(\xno{i}),\quad i\in\If
\end{equation}
$$
This is a linear system \( \sum_j A_{i,j}=b_i \) with entries
$$
\begin{equation*} A_{i,j}=-\baspsi_j''(\xno{i})=
(j+1)^2\pi^2L^{-2}\sin\left((j+1)\pi \frac{x_i}{L}\right),
\quad b_i=2
\end{equation*}
$$
Choose: \( N=0 \), \( x_0=L/2 \)
$$ c_0=2L^2/\pi^2 $$