Diagonal Dirichlet sine-mode heat model with constant forcing

Let $I=\{k\in \mathbb{N}:k\ge 1\}$ and $\mathcal{H}={\mathrm{\ell }}^2(I,\mathbb{R})$, with square-summability as in Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$. All modal coordinates below are indexed by $k\in I$. Define the diagonal operator $A$ by $D(A)=\{u\in \mathcal{H}:{\sum }_{k\ge 1}{k}^4\mathrm{\mid }{u}_k{\mathrm{\mid }}^2<\mathrm{\infty }\},(Au{)}_k=-k^2{u}_k\mathrm{.}$ Let $g\in \mathcal{H}$ be the fixed sequence $g_k=1\mathrm{/}k$ for odd $k$ and $g_k=0$ for even $k$; membership follows from The p-series for a real exponent p converges exactly when p is greater than one with exponent $2$. The split evolution is $u^{\mathrm{\prime }}=Au+g$. For $t\ge 0$, define the exact diagonal subflow $E_{t}u=(\exp (-k^{2}t)u_k{)}_{k\ge 1}$ and the exact translation subflow $B_{t}u=u+tg$. By The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$ and The exponential function is strictly increasing, $0<\exp (-k^{2}t)\le 1$, so each $E_t$ maps $\mathcal{H}$ to itself; each $B_t$ does so because $g\in \mathcal{H}$. The associated Strang modal step is $S_h=E_{h\mathrm{/}2}\circ B_h\circ E_{h\mathrm{/}2},h>0,$ where the rightmost map is applied first. Each coordinate of $E_t$ solves $u_k^{\mathrm{\prime }}=-k^2{u}_k$, and each coordinate of $B_t$ solves $u_k^{\mathrm{\prime }}=g_k$. This is a precisely defined sequence-space model; it records one Dirichlet sine-mode boundary mismatch and makes no general convergence assertion for unbounded PDE operators.