= Solution
{c}
Use $V=\{v\in H^1(0,1):v(0)=0\}$ and its continuous piecewise <linear subspace> $V_h=\operatorname{span}\{\phi_1,\ldots,\phi_N\}$. The condition at $x=1$ is natural, so the last <hat function> remains free. Set $u_h^0$ to the $L^2$ <orthogonal projection> of $u_0$ onto $V_h$: the stated $L^2$ initial data need not have nodal values.
For each $m$, find $u_h^{m+1}\in V_h$ such that
$$
\boxed{\left(\frac{u_h^{m+1}-u_h^m}{\Delta t},v_h\right)
+((u_h^{m+1})',v_h')+(u_h^{m+1},v_h)=0\quad(v_h\in V_h).}
$$
This applies <Backward Euler method> to diffusion and reaction together. The step <bilinear form> is coercive, so each step has a unique solution.
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