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.

中文翻译：

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四阶线性边值问题的有限差分法

方程的离散近似值 \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*} 被考虑。这是 Sturm-Liouville 案例 $D(x)\equiv H(x)\equiv 0$ [M. Ben-Artzi, J.-P. Croisille、D. Fishelov 和 R. Katzir，离散四阶 Sturm-Liouville 问题，IMA J. Numer。肛门。{\bf 38} (2018)，1485-1522。doi: 10.1093/imanum/drx038] 到非自伴随设置。Sturm-Liouville 案例中的“自然”边界条件是函数及其导数的值。包含三阶离散导数需要对基础离散泛函演算进行修订。此修订版强制评估第二个边界值的精确离散近似值，三阶和四阶导数。由此产生的函数演算提供了基本 Sobolev 属性的离散模拟——紧凑性和矫顽力。它允许获得精确解的离散近似的一般收敛定理。给出了一些有代表性的数值例子。