论文标题
四阶线性边界值问题的有限差异方法
Finite difference approach to fourth-order linear boundary-value problems
论文作者
论文摘要
等式离散近似 \ begin {equination*} l_ {cont} u = u^{（4）} + d（x）u^{（3）} + a（x） x \在[0,1] \ end {方程*}中被考虑。这是Sturm-Liouville案例$ d（x）\ equiv H（x）\ equiv 0 $ [M. Ben-Artzi，J.-P。的扩展。 Croisille，D。Fishelov和R. Katzir，离散的四阶Sturm-Liouville问题，Ima J. Numer。肛门。 {\ bf 38}（2018），1485-1522。 doi：10.1093/imanum/drx038]到非自身的拥护者设置。 Sturm-Liouville情况下的“自然”边界条件是该功能及其导数的值。包括三阶离散导数的包含需要对基本离散功能演算的修订。该修订会迫使对第二，第三和第四阶导数的边界值的准确离散近似值进行评估。所得的功能性演算提供了基本sobolev性质的离散类似物 - 紧凑型和矫正性。它允许将离散近似值的一般收敛定理与精确解决方案获得。提出了一些代表性的数值示例。
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.
