Vol 59, No 2 (2019)

The paper is concerned with initial-boundary value problems for a Sobolev-type equation with a Gerasimov–Caputo fractional derivative with memory effect. A priori estimates of the solutions are obtained in the differential and difference forms, which imply their uniqueness and stability with respect to the initial data and the right-hand side, as well as the convergence of the solution of the difference problem to the solution of the differential problem.

A new elementwise bound on the cross approximation error used for approximating multi-index arrays (tensors) in the format of a tensor train is obtained. The new bound is the first known error bound that differs from the best bound by a factor that depends only on the rank of the approximation \(r\) and on the dimensionality of the tensor \(d\), and the dependence on the dimensionality at a fixed rank has only the order \({{d}^{{{\text{const}}}}}\) rather than const^d. Thus, this bound justifies the use of the cross method even for high dimensional tensors.

An algorithm consisting of gradient and quasilinearization iterations is constructed for obtaining a high-accuracy numerical solution of a boundary value problem. An “ideal” solution of a multiobjective optimal control problem is produced by applying primal and dual algorithms, which ensure an efficient search for both scalarization coefficients and an optimal control. The efficiency of the proposed multimethod algorithms is demonstrated by soling application problems.

The problems of the existence and construction of solutions of a nonlocal boundary value problem for the homogeneous second-order Fredholm integrodifferential equation with a degenerate kernel and with two spectral parameters are considered. The singularities arising from the definition of arbitrary (unknown) constants are studied. The values of the spectral parameters are calculated and the solvability of the boundary value problem is established. The corresponding theorems are proven. Meaningful examples are provided.

Schemes of the Samarskii alternating triangular method are based on splitting the problem operator into two operators that are conjugate to each other. When the Cauchy problem for a first-order evolution equation is solved approximately, this makes it possible to construct unconditionally stable two-component factorized splitting schemes. Explicit schemes are constructed for parabolic problems based on the alternating triangular method. The approximation properties can be improved by using three-level schemes. The main possibilities are indicated for constructing alternating triangular schemes for second-order evolution equations. New schemes are constructed based on the regularization of the standard alternating triangular schemes. The features of constructing alternating triangular schemes are pointed out for problems with many operator terms and for second-order evolution equations involving operator terms for the first time derivative. The study is based on the general stability (well-posedness) theory for operator-difference schemes.

This paper is devoted to the simulation of the mean curvature flow on a surface of revolution. The surface is discretized using the icosahedral decomposition, and the flow equation is discretized by the finite element method. The scheme stability is improved using spline interpolation.

A numerical method for finding the potential of a multiatomic system in the real space is proposed. A distinctive feature of this method is the decomposition of the electron density \(\rho \) and the potential \(\varphi \) into two parts \(\rho = {{\rho }_{0}} + \hat {\rho }\) and \(\varphi = {{\varphi }_{0}} + \hat {\varphi }\), where \({{\rho }_{0}}\) is the sum of the spherical atom densities and the potential \({{\varphi }_{0}}\) is generated by the density \({{\rho }_{0}}\). The potential \(\hat {\varphi }\) is found by solving Poisson’s equation. The boundary conditions are obtained by expanding the reciprocal distance between two points in a series in Legendre polynomials. To improve the accuracy of the method, the computation domain is decomposed into Voronoi polyhedra, and asymptotic estimates of iterations are used when the characteristic function is replaced by its smooth approximations. Poisson’s equation is numerically solved using the two-grid method and the Fourier transform. An estimate \(O({{h}^{{\gamma - 1}}})\), where \(h\) is the grid size and \(\gamma \) is a fixed number greater than one, is obtained for the accuracy of the method. The error of the method is analyzed using a two-atom problem as an example.

A closed waveguide of a constant cross section \(S\) with perfectly conducting walls is considered. It is assumed that its filling is described by function \(\varepsilon \) and \(\mu \) invariable along the waveguide axis and piecewise continuous over the waveguide cross section. The aim of the paper is to show that, in such a system, it is possible to make a change of variables that makes it possible to work only with continuous functions. Instead of discontinuous transverse components of the electromagnetic field \({\mathbf{E}}\), it is proposed to use potentials \({{u}_{e}}\) and \({{v}_{e}}\) related to the field as \({{{\mathbf{E}}}_{ \bot }} = \nabla {{u}_{e}} + \tfrac{1}{\varepsilon }\nabla {\kern 1pt} '{{v}_{e}}\) and, instead of discontinuous transverse components of the magnetic field \({\mathbf{H}}\), to use the potentials \({{u}_{h}}\) and \({{v}_{h}}\) related to the field as \({{{\mathbf{H}}}_{ \bot }} = \nabla {{v}_{h}} + \tfrac{1}{\mu }\nabla {\kern 1pt} '{{u}_{h}}\). It is proven that any field in the waveguide admits the representation in this form if the potentials \({{u}_{e}},{{u}_{h}}\) are elements of the Sobolev space \(\mathop {W_{2}^{1}}\limits^0 (S)\) and \({{v}_{e}},{{v}_{h}}\) are elements of the space \(W_{2}^{1}(S)\).