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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.
fourth-order ordinary differential equation, boundary conditions, high-order discrete approximation
© The Author(s) 2020. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://dbpia.nl.go.kr/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
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