Volume 63, Nº 5 (2023)

A method for symbolic computation of a condition of existence of a third- and fourth-order resonance for investigations of formal stability of an equilibrium state of a multiparameter Hamiltonian system with three degrees of freedom in the case of general position is proposed. This condition is formulated in the form of zeros of a quasi-homogeneous polynomial of the coefficients of the characteristic polynomial of the linear part of the Hamiltonian system. Computer algebra (Gröbner bases of elimination ideals) and power geometry (power transformations) are used to represent this condition for various resonance vectors in the form of rational algebraic curves. Given a linear approximation of the characteristic polynomial in the space of its coefficients, these curves are used to obtain a description of a partition of the stability domain into parts in which there are no strong resonances. An  example of a description of resonance sets for a two-parameter pendulum-type system is given. All computations are carried out in the computer algebra system Maple.

In this paper, we study the spectral properties of the periodized Radial Basis Function interpolation matrix as well as the related harmonic operators discretized using Radial Basis Functions. For Gaussian RBF, this procedure could be easily extended to an arbitrarily high dimensional space on a tensor-product grid as presented in the later parts of the paper. The experimental result of Boyd’s condition number [1] is analytically well predicted in the context of periodized RBF.

A method for synthesizing an optimal control that ensures the existence and stability of sliding modes for a system of nonlinear ordinary differential equations is proposed. This method uses an auxiliary optimal control problem. The solution gives a control in analytical form. It is proved that the trivial solution of the closed-loop system is Lyapunov stable. Application of the proposed method to linear and quasi-linear systems of equations is demonstrated, and an illustrative example is discussed.

We consider matrices with entries in some domain, i.e., in a ring, not necessarily commutative, not containing non-trivial zero divisors. The concepts of the row rank and the column rank are discussed. (Coefficients of linear dependencies belong to the domain ; left coefficients are used for rows, right coefficients for columns.) Assuming that the domain satisfies the Ore conditions, i.e., the existence of non-zero left and right common multiples for arbitrary non-zero elements, it is proven that these row and column ranks are equal, which allows us to speak about the rank of a matrix without specifying which rank (row or column) is meant. In fact, the existence of non-zero left and right common multiples for arbitrary non-zero elements of  is a necessary and sufficient condition for the equality of the row and column ranks of an arbitrary matrix over. An algorithm for calculating the rank of a given matrix is proposed. Our Maple implementation of this algorithm covers the domains of differential and (-)difference operators, both ordinary and with partial derivatives and differences.

This article considers delayed two-dimensional first order hyperbolic differential equations. The propagation of the discontinuity of the solution is also established. An alternating implicit finite difference method and backward Euler finite difference methods are presented. We proved that this method is first-order convergent. Illustrating numerical examples are given for validation. We also present an application of the proposed approach to the variable delay problem.

The paper considers an inverse problem of determining the coefficients in the model of small transverse vibrations of a homogeneous finite string one end of which is placed in a moving medium and the other is free. The vibrations are simulated by a hyperbolic equation on an interval. One boundary condition has a nonclassical form. Additional data for solving the inverse problem are the values of the solution of the forward problem with a known fixed value of the spatial argument. In the inverse problem, it is required to determine the function in the nonclassical boundary condition and a functional factor on the right-hand side of the equation. Uniqueness and existence theorems for the inverse problem are proved. For the forward problem, conditions for unique solvability are established in a form that simplifies the analysis of the inverse problem. For the numerical solution of the inverse problem, an algorithm is proposed for the stage-by-stage separate reconstruction of the sought-for functions using the method of successive approximations for integral equations.

The paper considers an inverse problem for a singularly perturbed integro-differential heat equation, which consists in determining the boundary condition from additional information on the solution of the initial-boundary value problem. It is proved that an approximate solution of the inverse problem can be obtained by using a finite number of terms in the expansion of the solution of the initial-boundary value problem in a small parameter.

A study of the developed turbulent boundary layer that emerges when incompressible viscous fluid flows around a plate at a zero angle of attack and with zero longitudinal pressure gradient is presented. The waveguide approach is used for describing the turbulent boundary layer; in this approach, turbulent fluctuations are related with Tollmien–Schlichting waves that are in three-wave resonance. To study the original nonlinear system of equations, an estimate of hydrodynamic quantities is proposed that does not violate the generally accepted approach in the boundary layer but leads to the appearance of a new small parameter—the ratio of the thickness of the boundary layer momentum loss to the damping length of the least damped mode of the Tollmien–Schlichting waves. Equations for the coherent and stochastic parts of fluctuations are obtained on the basis of the method of multiple scales. The dispersion characteristics of waves of the least damped mode on the profile of the average longitudinal velocity of the developed turbulent boundary layer are determined, and the conditions for the multiple three-wave resonance of this mode of the Tollmien–Schlichting waves are analyzed. For the coherent part of the fluctuations, the fluctuation characteristics are compared with the known numerical results.

The paper considers a boundary value problem for stationary equations of complex heat transfer with an undetermined boundary condition for the radiation intensity on a part of the boundary and an overdetermined condition on another part of the boundary. An optimization method for solving this problem is proposed, and an analysis of the corresponding problem of boundary optimal control is presented. It is shown that the sequence of solutions of extremum problems converges to the solution of a problem with Cauchy-type conditions. The efficiency of the algorithm is illustrated by numerical examples.