Finite difference approach to fourth-order linear boundary-value problems

Abstract

Abstract Discrete approximations to the equation $$\begin{equation*}L_\textrm{cont}u = u^{(4)} + D(x) u^{(3)} + A(x) u^{(2)} + (A^{\prime}(x)+H(x)) u^{(1)} + B(x) u = f, \quad x\in[0,1]\end{equation*}$$are considered. This is an extension of the Sturm–Liouville case $D(x)\equiv H(x)\equiv 0$ (Ben-Artzi et al. (2018) Discrete fourth-order Sturm–Liouville problems. IMA J. Numer. Anal., 38, 1485–1522) 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 of compactness and coercivity. It allows us to obtain a general convergence theorem of the discrete approximations to the exact solution. Some representative numerical examples are presented.