Consider the boundary value problem \[ u''(x) = f(x),\quad \text {for }0 < x < 1, \] \[ u(0) = 0,\quad u(1) = 0, \]
where \(f\) is a smooth function on \([0, 1]\), and the finite difference scheme \[ \frac {U_{j-1}-2U_j+U_{j+1}}{h^2} = f(x_j), \qquad j = 1,2,\ldots ,m, \] for finding approximations \(U_j\) to \(u(x_j)\). Here, \(x_j = jh\) and \((m+1)h = 1\).
Analyze the accuracy, stability, and convergence properties of the scheme.