$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Method 1: Boundary function and exclusion of Dirichlet degrees of freedom
- \( \baspsi_i = \basphi_i \), \( i\in\If =\{0,\ldots,N=N_n-1\} \)
- \( B(x)=D\basphi_{N_n}(x) \), \( u= B + \sum_{j=0}^N c_j\basphi_j \)
$$
\begin{equation*}
\int_0^Lu'(x)\basphi_i'(x) dx =
\int_0^L f(x)\basphi_i(x) dx - C\basphi_i(0),\quad i\in\If
\end{equation*}
$$
$$
\begin{equation*}
\sum_{j=0}^{N}\left(
\int_0^L \basphi_i'\basphi_j' dx \right)c_j =
\int_0^L\left(f\basphi_i -D\basphi_N'\basphi_i\right) dx
- C\basphi_i(0)
\end{equation*}
$$
for \( i=0,\ldots,N=N_n-1 \).
