Computational Mathematics and Mathematical Physics
ISSN (print): 0044-4669
Founders: Russian Academy of Sciences, Federal Research Center IU named after. A. A. Dorodnitsyna RAS
Editor-in-Chief: Evgeniy Evgenievich Tyrtyshnikov, Academician of the Russian Academy of Sciences, Doctor of Physics and Mathematics sciences, professor
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Vol 65, No 11 (2025)
General numerical methods
RECONSTRUCTION OF POLYNOMIAL DEPENDENCIES FROM DATA WITH INTERVAL UNCERTAINTY
Abstract
A method for reconstructing linearly parameterized (in particular, polynomial) functional dependencies from data with interval uncertainty is developed. In many situations, it provides more adequate processing of imprecise measurement and observation results than traditional probability-theoretic approaches. The proposed method uses the mathematical apparatus of interval analysis and is based on the so-called maximum compatibility principle. It allows efficient construction of nonlinear functional dependencies in the form of generalized polynomials from interval data arising in both dependent and independent variables. As a practical example, the processing of real data from an aluminothermic process for industrial waste utilization is considered, where the new method demonstrates significant advantages compared to the traditional least squares method.
Computational Mathematics and Mathematical Physics. 2025;65(11):1761-1778
1761-1778
MULTIPOLE METHOD FOR SOLVING THE ZAREMBA PROBLEM IN COMPLEX DOMAINS WITH APPLICATION TO CONSTRUCTION OF CONFORMAL MAPPING
Abstract
This work, continuing the authors’ 2024 article, is devoted to the development of an analytical-numerical multipole method applied to the Zaremba problem, i.e., a mixed boundary value problem with Dirichlet–Neumann boundary conditions for the Laplace equation in planar simply connected domains of complex shape, whose boundary may contain singularities. The method allows obtaining not only the solution but also its derivatives on certain smooth parts of the boundary near singularities. The efficiency of the method was demonstrated by examples of constructing conformal mapping, and in previous works (with other co-authors) – by examples of constructing harmonic mapping of domains with complex curvilinear boundaries.
Computational Mathematics and Mathematical Physics. 2025;65(11):1779-1788
1779-1788
ORDER-OPTIMAL NUMERICAL METHODS FOR SOLVING SINGULAR INTEGRO-DIFFERENTIAL EQUATIONS
Abstract
A linear integro-differential equation with a singular differential operator in the principal part is studied. For its approximate solution in the space of generalized functions, special generalized versions of the methods of moments and subdomains are proposed and substantiated. Optimality of the methods in order of accuracy is established.
Computational Mathematics and Mathematical Physics. 2025;65(11):1789-1799
1789-1799
ESTIMATING THE SPECTRAL RADIUS OF THE JACOBIAN MATRIX IN EXPLICIT STABILIZED RUNGE–KUTTA METHODS
Abstract
When using explicit stabilized Runge–Kutta methods to solve stiff systems of ordinary differential equations, an estimate of the spectral radius of the Jacobian matrix is required. Such an estimate can be obtained by applying Gershgorin's theorem or the power method. This paper investigates estimation procedures based on the nonlinear power method that do not require computation of the Jacobian matrix. The proposed procedures are embedded in the integration method and allow estimating the spectral radius even when it changes during the solution process. Examples of solving test problems are provided.
Computational Mathematics and Mathematical Physics. 2025;65(11):1800-1812
1800-1812
Optimal control
ON THE REGULARIZATION OF THE LAGRANGE PRINCIPLE IN A NONLINEAR OPTIMAL CONTROL PROBLEM FOR A GOURSAT–DARBOUX SYSTEM WITH A POINTWISE STATE EQUALITY-CONSTRAINT
Abstract
Based on the conjugation of optimal control methods, nonlinear analysis and the theory of ill-posed problems, the regularization of the Lagrange principle (LP) in a non-differential form, in regular and irregular variants, in a nonlinear (non-convex) optimization problem of a general Goursat–Darboux system with a pointwise state equality-constraint is considered. This constraint is understood as an equality in the Hilbert space of square-summable functions and contains a parameter additively included in it, which makes it possible to use a “nonlinear version” of the perturbation method for studying the problem. The main purpose of both variants of the regularized LP is the stable generation of generalized minimizing sequences (GMS) in the problem under consideration, the existence of a solution to which is not assumed a priori. They can be interpreted as GMS-forming (regularizing) operators, which associate with each set of initial data of the problem a subminimal (minimal) of its regular augmented Lagrangian corresponding to this set, the dual variable in which is generated in accordance with the procedures specified in these variants. The construction of the augmented Lagrangian is completely determined by the form of “nonlinear” subdifferentials of a lower semicontinuous and, generally speaking, nonconvex function of values as a function of the problem parameter. The proximal subgradient and the Frechet subdifferential, well known in nonlinear analysis, are used as the latter. In the special case, when the problem is regular, in the sense of the existence of a generalized Kuhn–Tucker vector in it, and its initial data (the integrand of the quality functional and the right-hand side of the controlled system) depend affinely on the control, the limit passage in the relations of the regularized LP leads to classical optimality conditions in the form of the nondifferential Kuhn–Tucker theorem and the Pontryagin maximum principle.
Computational Mathematics and Mathematical Physics. 2025;65(11):1813-1833
1813-1833
ON OPTIMAL FEEDBACK CONTROL FOR OPERATOR EQUATIONS OF THE SECOND KIND
Abstract
The problem of correct construction of feedback control for operator equations of the second kind of general form is investigated. Correctness is understood as resolving the following three issues: 1) preservation of solvability of the controlled operator equation under variation of the control; 2) continuous dependence of the equation solution on the control; 3) existence of an optimal control for a given functional on the constructed class of controls. When solving the problem of correct construction of the class of feedback controls, the author's previous results on preserving the solvability of operator equations of the second kind, based on the concept of cone norm, are essentially used. As an example, a controlled ordinary differential equation in a Banach space is considered.
Computational Mathematics and Mathematical Physics. 2025;65(11):1834-1848
1834-1848
Ordinary differential equations
ALGORITHMS FOR SOLVING THE COULOMB TWO-CENTER PROBLEM
Abstract
New algorithms for solving the Coulomb two-center problem of discrete and continuous spectra in prolate spheroidal coordinates with separation of independent variables are presented. Energy eigenvalues and separation constants, as well as eigenfunctions of the discrete spectrum, are calculated using the secant method and the finite element method (FEM) on an appropriate grid with a real parameter – the distance between the Coulomb centers. At each step of the secant method, eigen solutions of the discrete spectrum are computed using the KANTBP 5M program implementing FEM in the Maple system. For the continuous spectrum problem (at a fixed energy eigenvalue), it is sufficient to solve the eigenvalue problem for the quasianqular equation with respect to the separation constant and use it when solving the boundary value problem for the quasiradial equation with respect to the unknown phase shift and eigenfunction using the KANTBP 5M program. The results of test calculations agree with reference calculations performed by programs implementing alternative methods in FORTRAN with the required accuracy.
Computational Mathematics and Mathematical Physics. 2025;65(11):1849-1864
1849-1864
Partial Differential Equations
ON THE POINCARÉ—STEKLOV OPERATOR FOR AN INCOMPRESSIBLE ELASTIC STRIP
Abstract
For an incompressible stratified elastic strip, we consider the Poincaré—Steklov operator that maps normal stresses into normal displacements on a part of the boundary. To construct the transfer function (TF) of this operator, a variational formulation of the boundary value problem for displacement transforms is used. A definition is given and the existence and uniqueness are proved for a generalized solution of the variational problem. This problem is approximated by the finite element method. The leading term of the asymptotic expansion of the TF for small and the three-term asymptotic expansion of the TF for large values of the Fourier transform parameter are obtained. Padé approximations of the obtained asymptotic series are constructed. To reduce computational costs a combined approach to calculating the TF has been developed using its asymptotic expansions and Padé approximations.
Computational Mathematics and Mathematical Physics. 2025;65(11):1865-1880
1865-1880
BLOW-UP AND GLOBAL SOLVABILITY OF THE CAUCHY PROBLEM FOR A NONLINEAR TIMOSHENKO BEAM VIBRATION EQUATION
Abstract
For a nonlinear fourth-order partial differential equation modeling the propagation of flexural waves in a Timoshenko beam, the Cauchy problem is studied in the space of continuous functions defined on the entire real axis and having limits at infinity. The time interval of existence and uniqueness of the classical solution to an auxiliary Cauchy problem related to the original one is established, and an estimate for the norm of this local solution is provided. Conditions ensuring the connection between local classical solutions of the original and auxiliary Cauchy problems on a specific time interval are found. Sufficient conditions for extending the local classical solution of the Cauchy problem to a global one and for the blow-up of the solution to the nonlinear Timoshenko equation on a finite time interval are considered.
Computational Mathematics and Mathematical Physics. 2025;65(11):1881-1898
1881-1898
Mathematical physics
ON THE FULFILLMENT OF THE H-THEOREM FOR THE S-MODEL KINETIC EQUATION
Abstract
The H-function, specifically defined for the S-model kinetic equation, is studied for various physical situations. Spatially homogeneous relaxation is considered. A fairly broad class of initial conditions is investigated. It is shown numerically that the H-theorem is valid for them.
Computational Mathematics and Mathematical Physics. 2025;65(11):1899-1907
1899-1907
VISCOUS-FLUID FLOW THROUGH A NEAR-WALL STATIONARY GRANULAR LAYER IN THE FORM OF INFINITE RECTANGULAR BARRIER
Abstract
The problem of viscous fluid flow along a flat solid surface with a stationary granular layer in the form of an infinite rectangular barrier is considered. The granular layer consists of an infinite number of identical spherical granules statistically uniformly distributed in the layer. The problem is solved based on the previously developed self-consistent field method, which allows studying the effects of hydrodynamic interaction of a large number of spherical particles in viscous fluid flows, including in the presence of external boundaries, and obtaining averaged dynamic characteristics of such flows. The solution to the problem, describing the averaged field of fluid velocity both outside and inside the granular layer, is obtained in the final analytical form in the first approximation by the volume fraction of granules in the layer. As a result, a characteristic feature of the fluid flow in the form of a large-scale stationary vortex is obtained, inside which the entire layer is located.
Computational Mathematics and Mathematical Physics. 2025;65(11):1908-1919
1908-1919
SEPARATION OF GAS MIXTURES WITH SIMILAR MOLECULAR WEIGHTS BASED ON THE RADIOMETRIC EFFECT
Abstract
Numerical modeling of the separation of a binary gas mixture with similar molecular weights in a thermal micropump based on the radiometric effect is carried out. The simulation method is based on the direct solution of the Boltzmann kinetic equation using a splitting scheme. Relaxation problems are solved using a conservative projection method. Transport equations are solved using first- and second-order schemes. A design of an installation that can be used for separating mixtures of similar masses is proposed. Based on the modeling, an assessment of the efficiency of this device is performed.
Computational Mathematics and Mathematical Physics. 2025;65(11):1920-1931
1920-1931
Lattice Boltzmann Equations Based on the Shakhov Model and Applications to Modeling Rarefied Poiseuille Flow at High Subsonic Velocity
Abstract
This work investigates the applicability of lattice Boltzmann equations based on the Shakhov kinetic equation to modeling rarefied flows under significant external force. As a benchmark problem, a two-dimensional plane Poiseuille flow at different Knudsen numbers and different amplitudes of external force is considered, resulting in characteristic flow velocities corresponding to Mach numbers in the range of 0.4 to 0.6. It is shown that two-dimensional lattice models with 37 velocities can describe nonequilibrium effects beyond the applicability of the continuum medium approximation, in the slip flow regime. Profiles of longitudinal and transverse heat fluxes, as well as velocity and temperature profiles of the rarefied gas, are examined.
Computational Mathematics and Mathematical Physics. 2025;65(11):1932-1942
1932-1942
APPLICATION OF A GLOBAL ADAPTIVE VELOCITY SPACE GRID FOR REDUCING OSCILLATIONS CAUSED BY THE “RAY EFFECT”
Abstract
The occurrence of oscillations in the numerical solution of unsteady rarefied gas flow problems with discontinuous boundary conditions using the discrete velocity method is a problem known in the literature as the “ray effect”. This effect is a significant obstacle in the numerical integration of kinetic equations under strong non-equilibrium conditions with low collision frequency. The application of global adaptation in velocity space allows in many cases to reduce oscillations in macroparameters. The results of using this algorithm are demonstrated by solving a two-dimensional evaporation problem during laser-matter interaction.
Computational Mathematics and Mathematical Physics. 2025;65(11):1943-1954
1943-1954
GRID-CHARACTERISTIC METHOD WITH IMPLICIT SCHEMES AND CONTACT CONDITIONS AT THE MATERIAL INTERFACE
Abstract
The paper presents a modified grid-characteristic method for modeling the propagation of elastic waves in heterogeneous media with explicit material boundaries. The proposed approach is based on the use of implicit and explicit numerical solution schemes, which together ensure stability and accuracy at large time steps and small space steps in areas with elongated shapes. The correct formulation of contact conditions between materials was implemented both in the form of reflected and refracted waves, and by modifying the lines of the system of equations for the implicit scheme. Numerical experiments for one-dimensional and two-dimensional problems, including wave modeling in multilayer structures and glass composites, are presented. The convergence order has been estimated for various schemes. The results show the possibility of the developed method to provide high modeling accuracy and the ability to describe complex wave patterns in heterogeneous media.
Computational Mathematics and Mathematical Physics. 2025;65(11):1955-1970
1955-1970