 Solution

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.