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Volume 65, Nº 10 (2025)
Optimal control
LORENZ MAJORIZATION AND PIGOU—DALTON TRANSFERS IN THE RAMSEY—BEWLEY MODEL
Resumo
In this paper, a model of the evolution of the Lorenz curve, describing the distribution of income between economic agents, is proposed. It is proved that the evolution of income distribution is consistent with Lorenz majorization in the Ramsey—Bewley model. A Pigou—Dalton transfer (tax and subsidy) system, which generates a stationary income distribution chosen by the welfare state, is constructed. Numerical calculations allow us to formulate a conjecture about the stability of the Lorenz curve corresponding to the selected income distribution.
Computational Mathematics and Mathematical Physics. 2025;65(10):1608–1624
1608–1624
ON THE CONSTRUCTION OF A GRADIENT QUADRATIC OPTIMIZATION METHOD, OPTIMAL IN TERMS OF MINIMIZING THE DISTANCE TO THE EXACT SOLUTION
Resumo
Quadratic optimization problems in Hilbert space often arise when solving ill-posed problems for differential equations. At the same time, the target value of the functional is known. In addition, the functional structure makes it possible to calculate the gradient by solving correct problems, which allows applying first-order methods. This article is devoted to the construction of the m-moment method of minimal errors, an effective method that minimizes the distance to an accurate solution. The convergence and optimality of the constructed method are proved, as well as the impossibility of uniform convergence of methods operating in Krylov subspaces. Numerical experiments are being conducted to demonstrate the effectiveness of applying the m-moment minimum error method to solving various incorrect problems: the initial boundary value problem for the Helmholtz equation, the retrospective Cauchy problem for the heat equation, and the inverse thermoacoustics problem. Куа.
Computational Mathematics and Mathematical Physics. 2025;65(10):1625-1648
1625-1648
Ordinary differential equations
LOCALIZATION OF MOVABLE SINGULARITIES OF THE BLASIUS EQUATION
Resumo
We study movable singularities of the Blasius equation in the complex plane. Numerical algorithms of their localization are given that allow to find singularities with high accuracy. All these singularities are equivalent and may be represented by one of them. We obtain an asymptotic expansion in the neighborhood of the singularity in explicit form and compute its coefficients. This power-logarithmic expansion is shown to be convergent and giving a local parametrization of the Riemann surface of the Blasius function.
Computational Mathematics and Mathematical Physics. 2025;65(10):1649–1661
1649–1661
Partial Differential Equations
ON THE FIRST INITIAL BOUNDARY VALUE PROBLEM FOR PARABOLIC SYSTEMS IN A BOUNDED DOMAIN WITH CURVED LATERAL BOUNDARIES
Resumo
The first initial boundary value problem for a second-order Petrovsky parabolic system with coefficients satisfying the double Dini condition in a bounded domain on the plane is considered. The lateral boundaries of the region are defined by continuously differentiable functions. It is established that if the right-hand parts of the boundary conditions of the first kind are continuously differentiable, and the initial function is continuous and bounded together with its first and second derivatives, so the solution to the problem belongs to the space of functions that are continuous and bounded together with their higher derivatives in the closure of the domain. The corresponding estimates have been proved. An integral representation of the solution is obtained. If the lateral boundaries of the domain admit the presence of angles, and the boundary functions have piecewise continuous derivatives, then in this case it is established that the higher derivatives of the solution are continuous everywhere in the closure of the domain, with the exception of corner points, and thus bounded.
Computational Mathematics and Mathematical Physics. 2025;65(10):1662–1674
1662–1674
Solvability of the Boundary Value Problem for the Stationary Boussinesq Magnetic Hydrodynamics Model with Variable Leading Coefficients
Resumo
A boundary value problem for a stationary model of magnetic hydrodynamics of a viscous heat-conducting liquid with variable leading coefficients is investigated. The model under consideration consists of the Navier–Stokes equations, Maxwell's equations, generalized Ohm's law for a moving fluid, and the convection–diffusion equation for temperature, which are nonlinearly interconnected. Sufficient conditions are established for variable coefficients and other data to ensure the global solvability of the specified problem and the local uniqueness of its solution.
Computational Mathematics and Mathematical Physics. 2025;65(10):1675-1689
1675-1689
LOCALIZATION OF WAVES AND THEIR BRAKING ZONES IN A THIN-WALLED HOLLOW QUANTUM WAVEGUIDE WITH PERIODIC BAFFLES
Resumo
An asymptotic analysis of the Dirichlet spectral problem in a thin-walled cylinder with a periodic family of partitions perpendicular to the generators of the cylindrical surface establishes the presence of open gaps in the waveguide spectrum, determining the size and position of the spectral segments. At different values of the relative thickness of the baffles various types of localization of the eigenfunctions of the model problem with the Floquet parameter on the periodicity cell are observed. Accordingly, the passing waves are localized on the baffles (large thicknesses) or near their edges (small thicknesses). The results were obtained using various procedures of dimensionality reduction and analysis of the boundary layer phenomenon near the junction zone of the baffles to the cylinder, which (layer) is described by the Dirichlet problem in a flat T-shaped joint of a single strip and a strip of varying thickness perpendicular to it. The localization method is determined by the presence or absence of a discrete spectrum for the latter task.
Computational Mathematics and Mathematical Physics. 2025;65(10):1690–1706
1690–1706
Mathematical physics
TURBULENT POISEUILLE FLOW IN A CIRCULAR TUBE AS A SUPERPOSITION OF STEADY SOLUTION AND PERTURBATIONS
Resumo
The turbulent flow of a viscous incompressible fluid in a circular tube caused by a pressure drop is investigated. It is assumed that the characteristic Reynolds number, calculated from the maximum velocity of the averaged flow and the length of the pipe, is large, and the radius of the pipe is small compared to its length. To find solutions to the Navier–Stokes equations, an asymptotic method of many scales is used, in which velocities and pressures are represented as series consisting of the sum of steady and perturbed terms, instead of the traditional decomposition of the solution into time-averaged values and their fluctuations. The paper finds a viscous self-sustaining steady flow that occurs in a pipe against the background of fast turbulent fluctuations. The connection of such a solution with Prigogine's theory of dissipative structures for open nonlinear systems of parabolic type is indicated. A solution has been found for the radial steady velocity, which describes the self-induced outflow of fluid from the core of the flow to a solid/permeable wall. As a result, solutions for the longitudinal velocity have been obtained that differ markedly from the laminar regimes. A qualitative comparison with well-known experiments and works on the direct numerical simulation (DNS) has been performed.
Computational Mathematics and Mathematical Physics. 2025;65(10):1707–1719
1707–1719
TWO-DIMENSIONAL MODIFICATION OF THE GODUNOV METHOD OF THE 4TH ORDER IN SPACE AND THE 3RD ORDER IN TIME
Resumo
A modification of the Godunov method for two-dimensional unsteady equations of gas dynamics is presented, which has a 4th order of approximation in space and a 3rd order in time. The difference scheme of the method is based on the joint discretization of equations in space and time without the use of Runge–Kutta stages, i.e. it is completely discrete. The flows are calculated as the result of solving the Riemann problem with corrections to its arguments. New versions of TVD limiters of central differences are proposed, applied to derivatives above the second order of accuracy. The results of an experimental verification of the approximation order of the method on two-dimensional smooth solutions inside Riemann and Prandtl–Meyer expansion fans are presented. A comparison has been made with other methods, both in terms of accuracy and performance.
Computational Mathematics and Mathematical Physics. 2025;65(10):1720-1734
1720-1734
ON THE APPROACH OF THE GRAVITATIONAL FIELD OF A SMALL CELESTIAL BODY BY THE FIELD OF ATTRACTION OF EQUIMOMENTAL FOUR MATERIAL POINTS
Resumo
The problem of constructing a system of four point masses, the totality of which is equal to a predetermined solid, is solved. A family of such systems has been built, depending on six parameters. The freedom to choose parameters makes it possible to set the task of finding the set of masses that best approximates the momentums of the distribution of masses of the third order of the body. The problem in this formulation is solved in relation to the nucleus of comet 67P/Churyumov–Gerasimenko. The criterion for the quality of the coincidence of the momentums of the third-order mass distribution is the standard deviation of the momentums of the point mass system from the corresponding momentums of the comet nucleus. A system of four material points is constructed, minimizing the value of the root-mean-square error. This value turned out to be less than the similar values obtained in previous studies. It is noteworthy that the masses of the found points turned out to be different from each other and all are located inside the core, so that the three smallest of them are located in a larger fraction of the core, and the fourth, which concentrates ≈ 28.5% of the total mass of the core, is located inside a smaller fraction. This mass distribution is in good agreement with the well-known estimate of 27% of the volume of a smaller fraction of the comet’s core from the total volume.
Computational Mathematics and Mathematical Physics. 2025;65(10):1735-1745
1735-1745
A NUMERICAL METHOD FOR SOLVING THE MICROWAVE TOMOGRAPHY PROBLEM OF RESTORING INHOMOGENETTES IN A CYLINDRICAL BODY
Resumo
In this paper, a vector three-dimensional inverse diffraction problem on a cylindrical body is solved based on a two-step method. The diffuser is filled with an inhomogeneous nonmagnetic dielectric material. The initial boundary value problem for the Maxwell system of equations is reduced to a system of integro-differential equations. A numerical method for solving a first-order equation in special classes of functions is described. A distinctive feature of the proposed numerical method is its non-iteration, in addition, a two-step method for solving the inverse problem does not require a good initial approximation. The calculation results are presented. It is shown that the two-step method is an effective approach to solving vector problems of microwave tomography.
Computational Mathematics and Mathematical Physics. 2025;65(10):1746-1758
1746-1758