$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Dirichlet boundary conditions
Dirichlet condition at \( x=0 \) and Neumann condition at \( x=L \):
$$
\begin{align*}
u(\x,t) &= u_0(\x,t),\quad & \x\in\partial\Omega_D\\
-\dfc\frac{\partial}{\partial n} u(\x,t) &= g(\x,t),\quad
& \x\in\partial{\Omega}_N
\end{align*}
$$
Forward Euler in time, Galerkin's method, and integration by parts:
$$
\begin{equation*}
\int_\Omega u^{n+1}v\dx =
\int_\Omega (u^n - \Delta t\dfc\nabla u^n\cdot\nabla v)\dx -
\Delta t\int_{\partial\Omega_N} gv\ds,\quad \forall v\in V
\end{equation*}
$$
Requirement: \( v=0 \) on \( \partial\Omega_D \)
