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For $U^{\prime \prime }=(\alpha I-K_h)U$ with symmetric positive definite $K_h$, all-data displacement is bounded for all time on a fixed grid exactly when $\alpha <{\lambda }_{\min }(K_h)$. Diagonalization of a matrix reduces this assertion to oscillators. At equality, an initial velocity in a zero-frequency mode produces linear growth; above it a growing hyperbolic mode is available. A lower bound on ${\lambda }_{\min }(K_h)-\alpha$ uniform in the mesh gives a uniform displacement estimate. This all-time statement is stronger than displacement stability of a symmetric semidiscrete wave equation on fixed finite time intervals.

Set $h_x=1/(N+1)$ and $J=2N+1$. The Dirichlet discrete Laplacian has the eigenvectorsand its negative has the eigenvaluesThe discrete sine transform therefore reduces the method of lines equations to $q_j^{\prime \prime }=(\alpha -{\lambda }_j^h)q_j$. Each coefficient is an oscillator, a linear function, or a hyperbolic function according to the sign of ${\lambda }_j^h-\alpha$. A finite collection of oscillators is bounded in every fixed-grid norm. A zero-frequency mode can grow linearly, and a negative-frequency-square mode can grow exponentially. Consequently,is necessary and sufficient for all-time boundedness of a semidiscrete reaction wave equation on this grid. Equality allows bounded displacement only when the initial velocity has zero projection onto the first eigenvector; above it, the growing components must be absent in every unstable mode. A strictly positive spectral gap also gives a time-independent displacement bound uniform over grids whose gaps are bounded below.

The discrete threshold is strictly smaller than ${\pi }^2/4$, because $\sin s<s$ for $s>0$, and tends to ${\pi }^2/4$ as $N\to \infty$. A coarse grid can therefore introduce long-time growth into a continuum problem with $\alpha <{\pi }^2/4$. This is distinct from displacement stability of a symmetric semidiscrete wave equation on fixed finite time intervals. As in part (a), if existence of the printed limit is required for every initial datum, no $\alpha$ qualifies: any one mode admits oscillatory, secular, or growing initial data, depending on its coefficient.

The printed formula is not a consistent spatial discretization of the stated PDE. The original PDF genuinely has $1/\Delta x$, rather than $1/(\Delta x{)}^2$. Writing $h=\Delta x$, the Taylor expansion givesso it approximates a diffusion coefficient tending to zero. Moreover, $m=1,\dots ,M$ and $h=1/(M+1)$ cover $(0,1)$, whereas the given Dirichlet boundary conditions belong to $(-1,1)$. No value at the interior location $x=0$ was prescribed. These are actual source defects, not repairs justified by the TeX conversion.

A conditional analysis of the literal ordinary differential equation system is still possible. If it is closed by imposing $u_0=u_{M+1}=0$ on the displayed grid, the negative Dirichlet discrete Laplacian has positive frequencies squaredand each normal mode obeys $q_j^{\prime \prime }=(\alpha -{\kappa }_j)q_j$. For a fixed mesh, every solution is bounded for all time exactly whenEquality produces a linearly growing normal mode for nonzero modal initial velocity; above the threshold there is an exponentially growing normal mode. Since ${\kappa }_1(h)\sim {\pi }^{2}h\to 0$, any $0<\alpha <{\pi }^2/4$ gives bounded continuous solutions but unbounded literal discrete solutions on sufficiently fine meshes. For example, $\alpha =1$, $M=99$ gives ${\kappa }_1<1$, and a growing first sine normal mode. At $\alpha =0$, every fixed mesh is bounded for all time, but unit first-mode initial velocity has maximal displacement $1/\sqrt{{\kappa }_1(h)}$, which diverges on mesh refinement. For uniform-in-mesh, all-time displacement bounds for arbitrary bounded displacement/velocity data in the mesh-weighted L2 norm, the literal closed system therefore requires $\alpha <0$.

This differs from finite-time stability of a numerical method, governed here by displacement stability of a symmetric semidiscrete wave equation. For the conditional closure above, diagonalization of a matrix gives, with ${\alpha }_{+}=\max (\alpha ,0)$,Oscillatory modes use $|\sin (\omega t)/\omega |\le t$, including its value $t$ at $\omega =0$; growing modes use $\sinh (\beta t)/\beta \le t{e}^{\beta t}$. Thus the displacement map is mesh-uniformly bounded on each fixed finite time interval for every fixed real $\alpha$. This is a legitimate finite-time displacement stability estimate, but neither consistency of a numerical method nor the all-time conclusion of part (i). It does not assert a velocity bound in the unweighted displacement norm.

If the intended method instead uses $h^{-2}$ and the full interval, take interior indices $m=-M,\dots ,M$, $h=1/(M+1)$, with zeros at $m=\pm (M+1)$. Its first positive discrete eigenvalue isThe corrected, fixed-mesh all-time condition is $\alpha <{\lambda }_{1,h}$, with the same strictness at equality. For every fixed $\alpha <{\pi }^2/4$, this holds on all sufficiently fine meshes. This corrected interpretation is stated separately rather than silently replacing the PDF.