= Four-point midpoint third derivative
{title2=$\frac{u_{j+2}-3u_{j+1}+3u_j-u_{j-1}}{h^3}=u'''(x_{j+1/2})+O(h^2)$}
The antisymmetric four-node stencil at offsets $-3h/2,-h/2,h/2,3h/2$ about a midpoint approximates the third <derivative>. <Taylor expansion> gives leading error $h^2u^{(5)}/8$. For a third-order <boundary value problem> on nodes $0,\ldots,n+1$, use rows $j=1,\ldots,n-1$, prescribed endpoint values and one consistent one-sided <derivative> condition. This avoids a ghost node while providing the correct number of equations. For instance a known right-endpoint first <derivative> is approximated by $(3U_{n+1}-4U_n+U_{n-1})/(2h)$.
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