The main goal of this work is to check the hypothesis that the well-known N-particle statistical algorithm yields a solution estimate for the nonlinear Boltzmann equation with an (O(1{text{/}}N)) error. For this purpose, practically important optimal relations between (N) and the number (n) of sample estimate values are determined. Numerical results for a problem with a known solution confirm that the formulated estimates and conclusions are satisfactory.
The article examines the possibility of simultaneous reconstruction of sources equivalent in the external field and spectral characteristics of the useful signal. Examples of variational formulations are given for various versions of the method of linear integral representations, and the problem is formulated of finding the distribution density of gravitating or magnetic masses on several horizontal planes and the Fourier transform of an anomalous field element from the values of a disturbed signal known at points of a certain observation network.
A class of exact solutions to the Navier–Stokes equations for an axisymmetric rotational incompressible flow is obtained. Invariant manifolds of flows that are axisymmetric about a given axis in three-dimensional coordinate space are found, and the structure of solutions is described. It is established that typical invariant regions of such flows are figures of rotation homeomorphic to the torus, which form a topological stratification structure, for example, in a ball, cylinder, and general complexes made up of such figures. The results extend to similar solutions of the system of MHD equations and Maxwell’s electrodynamic equations, which have analogous properties in ({{mathbb{R}}_{3}}). Examples are given of axisymmetric vorticity vector fields and topological stratifications they generate on manifolds in ({{mathbb{R}}_{3}}) that are invariant under the dynamical systems specified by these fields.
Integral identities that are fulfilled for measure of difference between the exact solution of a parabolic-hyperbolic equation and any functions from a corresponding energy class are proved. These identities make it possible to derive two-sided a posteriori estimates for approximate solution to the corresponding Cauchy problem. The left-hand side of such an estimate is a natural measure of deviation from the solution, and the right-hand side depends on the problem data and the approximate solution itself and, therefore, it can be explicitly calculated. These estimates can be used to control the accuracy of approximate solutions and to compare solutions to Cauchy problems with different initial conditions. These estimates also allow one to quantitatively assess the effects occurring due to inaccuracies in the initial data and in the coefficients of the equation.
Investigation of nanofluid flow, mass and heat transfer, and impact of thermal relaxation time parameter against temperature in porous spinning rotating stretching disks was done. The behavior of heat and mass transfer, velocity, temperature and particles concentration volume fraction profiles against other parameters are investigated. A decrease in heat transfer in the lower disk and upper disk was observed as Eckert number, porosity parameter and Brownian diffusion parameter increased in value though with an increase in temperature ratio and thermal relaxation time parameter. The mass transfer rate at both lower and upper disk increased as Lewis number, Eckert number alongside porosity parameter increase in values, but decreases in value as Brownian diffusion parameter and temperature ratio increase in values. Increasing thermal relaxation time parameter caused decrease in temperature. The physical meaning is that because longer time is needed for transporting heat to nearby particles, the upper disk stretching parameter increases whenever radial velocity profile towards the lower disk is increased, which was exhibited by a negative sign towards the lower disk and positive sign in the neighbourhood of the upper disk. This implies that the fluid inside the two disks continues to flow in opposite directions of upward and downward directions. The partial differential equations which are non-linear are, transformed to non-linear coupled ordinary differential equations by app-lying Van Karman transformations which are then solved using MATLAP bvp4c with shooting technique.
Various iterative algorithms for solving the linear equation (ax = b) using a quantum computer operating on the principle of quantum annealing are studied. Assuming that the result produced by the computer is described by the Boltzmann distribution, conditions under which these algorithms converge are obtained and an estimate of their convergence rate is provided. Application of this approach for algorithms that use an infinite number of qubits and a small number of qubits is considered.
The Green’s function of the Riquier–Neumann problem for the polyharmonic equation in the unit ball is constructed, and an integral representation of solutions to the Riquier–Neumann problem is presented. Two examples are given.
A two-level time scheme for solving the simplified Fokker–Planck kinetic equation is created as applied to the transport of charged products of a thermonuclear reaction, which includes an interpolation procedure in a four-dimensional grid space. The instability of this scheme is discovered at low values of the particle velocity and a special choice of the particle deceleration rate in the ion field, which is included in the kinetic equation as a parameter. It is shown that the thermalization condition, which prohibits solving the kinetic equation for particles with an energy less than the average ion energy significantly limits the number of thermonuclear reactions in which instability can manifest itself. This scheme is tested on the problem of relaxation to a stationary state and on the problem with a given time dependence of the thermonuclear reaction rate, for which an exact solution to the kinetic equation can be found.
The paper presents a review of the current state of subgradient and accelerated convex optimization methods, including the cases with the presence of noise and access to various information about the objective function (function value, gradient, stochastic gradient, higher derivatives). For nonconvex problems, the Polyak–Lojasiewicz condition is considered and a review of the main results is given. The behavior of numerical methods in the presence of a sharp minimum is considered. The aim of this review is to show the influence of the works of B.T. Polyak (1935–2023) on gradient optimization methods and their surroundings on the modern development of numerical optimization methods.
The large deviations of linear control system trajectories from the equilibrium position during the stabilization process, known as peak-effect, represent a serious obstacle to the design of cascade control systems and to guidance stabilization. Evaluation of such deviations is an important problem not only for control theory but also for other branches of mathematics where similar phenomenon appears as, for example, in numerical analysis, in the study of hydrodynamic stability and transition to turbulence, in the dynamics of vehicular platoons, and many others. The aim of this paper is to improve one of the main results concerning upper bounds for the peak-effect in linear control systems in the work by B.T. Polyak and myself in 2016. As the lower bound for overshoot proved by R.N. Izmailov in 1987 shows, the asymptotic estimate for the upper bound obtained here cannot be improved.
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