$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Example with two Dirichlet values; boundary function
$$
\begin{equation*}
B(x) = \sum_{j\in\Ifb} U_j\basphi_j(x)
\end{equation*}
$$
Here \( \Ifb = \{0,N_n\} \), \( U_0=C \), \( U_{N_n}=D \); \( \baspsi_i \) are
the internal \( \basphi_i \) functions:
$$ \baspsi_i = \basphi_{\nu(i)}, \quad \nu(i)=i+1,\quad i\in\If =
\{0,\ldots,N=N_n-2\} $$
$$
\begin{align*}
u(x) &= \underbrace{C\cdot\basphi_0 + D\basphi_{N_n}}_{B(x)}
+ \sum_{j\in\If} c_j\basphi_{j+1}\\
&= C\cdot\basphi_0 + D\basphi_{N_n} + c_0\basphi_1 + c_1\basphi_2 +\cdots
+ c_N\basphi_{N_n-1}
\end{align*}
$$
