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Том 222, № 1 (2017)
- Год: 2017
- Статей: 7
- URL: https://ogarev-online.ru/1072-3374/issue/view/14816
Article
Weight Estimates of the Accuracy of Difference Schemes for the Sturm–Liouville Problem
Аннотация
It is shown that the rate of convergence of difference schemes used for the solution of the Sturm–Liouville problem for linear ordinary differential equations is higher in the immediate vicinity of the boundary. Moreover, we obtain the a priori estimates of accuracy that can be regarded as a quantitative confirmation of this effect. The results of numerical experiments corroborating this effect are also presented.
1-25
On a Three-Step Method with the Order of Convergence 1 + \( \sqrt{2} \) for the Solution of Systems of Nonlinear Operator Equations
Аннотация
We propose a three-step modification of a method with an order of convergence 1+ \( \sqrt{2} \) aimed at the solution of nonlinear operator equations. We prove that the method is convergent and estimate its error. We also perform the numerical investigation of this modification on test examples, compare the results with the base method, and make conclusions on the basis of these results. The results of verification of the method confirm the theoretical predictions.
26-34
Block-Diagonal Similarity and Semiscalar Equivalence of Matrices
Аннотация
We determine the canonical form of a complex matrix B with respect to the similarity B → S−1BS, where S is the direct sum of invertible upper triangular Toeplitz blocks. The conditions necessary and sufficient for the semiscalar equivalence of one type of polynomial matrices are established.
35-49
50-55
On Some Sequences of the Sets of Uniform Convergence for Two-Dimensional Continued Fractions
Аннотация
For two-dimensional continued fractions whose elements belong to some rectangular sets of a complex plane, we establish the truncation error bound of their figured approximants. It is shown that the two-dimensional continued fractions are uniformly convergent with respect to a sequence of these rectangular sets.
56-69
70-80
Practical Stability of the “Cross” Scheme in the Numerical Integration of Dynamic Equations for Flexible Thin-Walled Structural Elements Obeying the Hypotheses of the Timoshenko Theory
Аннотация
In the von Kármán approximation, we formulate the initial boundary-value problem of the dynamics of flexible isotropic and composite elastic beams-walls within the framework of two versions of the Timoshenko theory. We perform a qualitative analysis of the resolving system of equations of motion. It is shown that, in the geometrically linear statement, the dynamics of elastic beams is described by a hyperbolic system. At the same time, in the case of deformation of flexible beams, the system of resolving motion equations may change its type degenerating from a hyperbolic system into a system of mixed (composite) type. We develop finite-difference and variation-difference versions of the explicit (in time) “cross” scheme for the numerical integration of the posed initial boundary-value problems. On the basis of these numerical methods, we perform the numerical analyses of the dynamic flexural deformation of flexible metallic and composite beams under explosive-type loads. The result of these calculations demonstrate that, in almost cases, one can indicate the levels of loading of flexible beams under which the “cross” scheme becomes unstable, although the condition of stability obtained in the linear approximation is satisfied with a considerable margin. Thus, it is shown that, in the case of dynamic analysis of flexible beams, one can speak only about the practical stability of the “cross” scheme but not about its conditional stability.
81-102