Том 63, № 7 (2023)

Methods for the algorithmic construction of stability indicators of nonnegative matrices is described, and the application of these indicators to problems of modern mathematical biology and epidemiology is discussed. Specific features of such indicators when they are applied to problems about the parametric loss of stability of trivial equilibrium states of discrete dynamical systems are pointed out. Estimates of the efficiency of algorithms based on the proposed methods for the case of systems determined by sparse matrices are given. Examples of using the constructed algorithms for such systems are discussed.

The article considers a linear differential pursuit game under the condition that an integral constraint is imposed on the evader’s control and the pursuer uses an impulse control. These impulse actions on the object are made at predetermined time points, and the corresponding control is represented using the Dirac delta function. The article studies linear conflicts described by a system of ordinary differential equations whose trajectories have jumps at certain times. The terminal set is represented by a cylinder in an n-dimensional Euclidean space. The problem is solved using the resolving function method. The fact that the lower bound is reached is proved using the theory of support functions. As a result, the quasi-strategy is replaced by an almost stroboscopic strategy and a method for building this strategy is presented. An example of a nonlinear right-hand side is given.

The conditional gradient method is generalized to nonconvex sets of constraints representing the set-theoretic intersection of a convex smooth surface and a convex compact set. Necessary optimality conditions are studied, and the convergence of the method is analyzed.

By establishing recurrence relations and then determining boundary values, we examine four classes of definite integrals of xm over higher powers of cosh x, sinh x, cos x and sin x in denominators. They are explicitly evaluated in terms of the logarithm function, the Riemann zeta function and its variants, such as Dirichlet beta function and Legendre’s chi-function.

Three Cauchy problems for Sobolev-type equations with a common linear part from the theory of ion acoustic and drift waves in a plasma are considered. The problems are reduced to equivalent integral equations. We prove the existence of unextendable solutions for two problems and the existence of a local-in-time solution for the third problem. For one of the problems, by applying a modified method of Kh.A. Levin, sufficient conditions for finite time blow-up of solutions are obtained and an upper bound for the solution blow-up time is found. For another problem, S.I. Pohozaev’s nonlinear capacity method is used to obtain a finite time blow-up result and two results concerning the nonexistence of even local solutions, and an upper bound for the solution blow-up time is obtained as well.

The paper presents formulations of inverse problems for the Helmholtz equation on finding its right-hand side with a Samarskii–Ionkin-type additional integral condition and the justification of their well-posedness in the Hadamard sense in the class of regular solutions. The uniqueness of solutions of the formulated problems is proved on the basis of integral identities. Solutions of the problem are found in explicit form by the methods of separation of variables and integral equations.

The Cauchy problem for a nonlinear Sobolev-type differential equation modeling moderately long small-amplitude longitudinal waves in a viscoelastic rod is investigated in a space of continuous functions defined on the entire number line that have limits at infinity. Conditions for the existence of a global solution and for finite time solution blow-up are examined.

Thermal displacement and stress distribution are analysed in the domain of plane-strain geometry of an isotropic thermoelastic medium having variable material moduli. In this context of the analysis, laser induced heat sources are planted on the reference plane. To analyse the thermal responses due to the injected heat sources into the medium, hyperbolic type heat conduction model with three phase-lags incorporated. Analytical solutions are expressed in-terms of Laplace–Fourier integral transform domain and the physical behaviour of displacement and stresses are depicted through a discretised form of inverse-integral transformation technique. Finally, parameterized characteristic analysis are presented graphically and the salient features of thermal displacements are highlighted.

Stability of lattice Boltzmann equations is governed by a parameter that is responsible for the relaxation time of the nonequilibrium system which, in turn, affects the viscosity of the flow under examination. In the entropic approach, the relaxation time is evaluated from the entropy balance equation in such a way that the entropy does not decrease at each time and spatial point. In this paper, a technique for solving the entropy balance equation using a modified secant method is proposed. It is shown that this approach provides high accuracy. As an application of the proposed method, numerical solutions of the two-dimensional double shear problem are considered. The simulation results are compared with the results obtained by other entropic methods.

The article studies a 3D contact problem with Coulomb friction for an elastic body resting on a rigid support. The solution of the quasi-variational formulation of the problem is defined as a fixed point of some mapping that associates the given force of the normal reaction of the support with the value of the normal stress in the contact zone. The fixed point is sought by the method of successive approximations, the convergence of which is proved using modified Lagrange functionals. The results of the numerical solution using finite element modeling and the proximal gradient method are presented.