  Discrete approximations to the equation
  \begin{equation*} L_{cont}u = u^{(4)} + D(x) u^{(3)} + A(x) u^{(2)} +
(A'(x)+H(x)) u^{(1)} + B(x) u = f, \; x\in[0,1] \end{equation*} are considered.
This is an extension of the Sturm-Liouville case $D(x)\equiv H(x)\equiv 0$ [ M.
Ben-Artzi, J.-P. Croisille, D. Fishelov and R. Katzir, Discrete fourth-order
Sturm-Liouville problems, IMA J. Numer. Anal. {\bf 38} (2018), 1485-1522. doi:
10.1093/imanum/drx038] to the non-self-adjoint setting. The "natural" boundary
conditions in the Sturm-Liouville case are the values of the function and its
derivative. The inclusion of a third-order discrete derivative entails a
revision of the underlying discrete functional calculus. This revision forces
evaluations of accurate discrete approximations to the boundary values of the
second, third and fourth order derivatives. The resulting functional calculus
provides the discrete analogs of the fundamental Sobolev
properties--compactness and coercivity. It allows to obtain a general
convergence theorem of the discrete approximations to the exact solution. Some
representative numerical examples are presented.
