$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
A Forward Euler scheme; integration by parts
Isolating the unknown \( u^{n+1} \) on the left-hand side:
$$
\int_{\Omega} u^{n+1}\baspsi_i\dx = \int_{\Omega}
\left\lbrack u^n - \Delta t \left( \dfc\nabla^2 u^n + f(\x, t_n)\right)
\right\rbrack v\dx
$$
Integration by parts of \( \int\dfc(\nabla^2 u^n) v\dx \):
$$ \int_{\Omega}\dfc(\nabla^2 u^n)v \dx =
-\int_{\Omega}\dfc\nabla u^n\cdot\nabla v\dx +
\underbrace{\int_{\partial\Omega}\dfc\frac{\partial u^n}{\partial n}v \dx}_{=0\quad\Leftarrow\quad\partial u^n/\partial n=0}
$$
Variational form:
$$
\begin{equation*}
\int_{\Omega} u^{n+1} v\dx =
\int_{\Omega} u^n v\dx -
\Delta t \int_{\Omega}\dfc\nabla u^n\cdot\nabla v\dx +
\Delta t\int_{\Omega}f^n v\dx,\quad\forall v\in V
\end{equation*}
$$
