Vol 222, No 1 (2017)

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.

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.

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.

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.