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 9 (2025)
General numerical methods
DIFFERENCE SCHEMES BASED ON EXPONENTIALLY CONVERGING QUADRATURES FOR THE CAUCHY INTEGRAL
Abstract
Traditional difference schemes are based on the interpolation of grid functions by a polynomial of finite degree. The error of such schemes decreases as a certain degree of step. In this paper, we propose a fundamentally new class of difference schemes with exponential convergence, which is dramatically faster than the traditional power-law one. The typical accuracy gain is 5–8 orders of magnitude or more. The proposed approach is uniformly applicable to various classes of mathematical physics problems and is demonstrated by the example of boundary value problems for ODEs. Examples illustrating the advantages of the proposed approach are given.
Computational Mathematics and Mathematical Physics. 2025;65(9):1469–1478
1469–1478
Partial Differential Equations
PRINCIPLES OF DUALISM IN THE THEORY OF SOLUTIONS OF INFINITE-DIMENSIONAL DIFFERENTIAL EQUATIONS DEPENDING ON EXISTING TYPES OF SYMMETRIES
Abstract
In the presented paper, in the case of a homogeneous medium, the dualism of spaces of soliton solutions and solutions of an induced point-type functional differential equation is described, and existence and uniqueness theorems for such dual solutions are formulated. Such dualism refers to a number of dualisms of various mathematical objects and, in particular, such as a topological linear space and its conjugate space. In the case of an inhomogeneous medium, a different type of dualism is described for spaces of quasi-soliton solutions and solutions of an induced one-parameter family of a point-type functional differential equation, and existence and uniqueness theorems for such dual solutions are formulated. The entire family of soliton (in the case of a homogeneous medium) and quasi-soliton (in the case of an inhomogeneous medium) solutions is constructed for the finite-difference analog of the wave equation with a nonlinear potential.
Computational Mathematics and Mathematical Physics. 2025;65(9):1479–1504
1479–1504
CONVERGENCE OF EIGENELEMENTS OF A STEKLOV-TYPE BOUNDARY VALUE PROBLEM FOR THE LAME OPERATOR IN A SEMI-CYLINDER WITH A SMALL CAVITY
Abstract
A Steklov-type boundary value problem for the Lame operator in a semi-cylinder containing a small cavity is investigated. The case is considered when an elastic, homogeneous isotropic medium filling a region with a small cavity is rigidly coupled to the lateral boundary of a semi-cylinder and the boundary of a small cavity, which corresponds to a homogeneous Dirichlet boundary condition, and the Steklov spectral condition is set on the basis of the semi-cylinder. The main result consists in proving a theorem on the convergence of the eigenelements of such a singularly perturbed boundary value problem to the eigenelements of the limiting problem (in a semi-cylinder without a cavity) with a small parameter ε > 0 tending to zero, characterizing the diameter of the cavity. To prove the theorem, a Hilbert space of infinitely differentiable vector functions with a finite Dirichlet integral over a semicylinder was introduced. In contrast to the situation with a limited domain, in the boundary value problem under study, the condition of finiteness of the Dirichlet integral is essential, since it generally ensures finiteness of the norm in the introduced space. The restriction on the finiteness of the Dirichlet integral made it possible to establish a priori estimates that guarantee the uniqueness of solutions to the limiting and perturbed boundary value problems and to establish the equivalence of norms necessary to prove the existence of a solution to the singularly perturbed boundary value problem under study.
Computational Mathematics and Mathematical Physics. 2025;65(9):1505-1517
1505-1517
ON THE EXISTENCE AND UNIQUENESS OF THE SOLUTION OF AN INTEGRO–DIFFERENTIAL EQUATION IN THE PROBLEM OF DIFFRACTION OF AN ELECTROMAGNETIC WAVE ON AN INHOMOGENEOUS DIEJECTRIC BODY COATED WITH GRAPHENE
Abstract
Boundary value problems for a system of Maxwell’s equations are fundamental in electrodynamics. Recently, there has been interest in problems with the presence of a thin graphene layer on the surface, which changes the coupling conditions. An integro-differential equation for the vector boundary value problem of electromagnetic wave diffraction on an inhomogeneous dielectric body coated with graphene is obtained. The existence and uniqueness of the solution of an integro-differential equation, which can be called a surface-volume equation, is proved.
Computational Mathematics and Mathematical Physics. 2025;65(9):1518-1524
1518-1524
Mathematical physics
High-precision difference boundary conditions for bicompact circuits split by transfer processes
Abstract
The splitting of a vector of Lax–Friedrichs and Rusanov type flows is considered, implemented in the form of splitting by physical processes: transfer processes. It is shown that it is a consequence of a single variable substitution. Two approaches to setting boundary conditions for problems with split flow vectors are proposed, ensuring zero splitting error. In accordance with these approaches, high-precision approximations of the boundary conditions of the first kind and the free exit for the quasi-linear transport equation, as well as the conditions of a rigid impermeable wall for the Eulerian equations, are constructed. A significant gain in accuracy from the use of new conditions in the application to bicompact schemes is demonstrated.
Computational Mathematics and Mathematical Physics. 2025;65(9):1525-1539
1525-1539
ANALYSIS OF PERTURBATION COEFFICIENTS IN THE PROBLEM OF FILTERING NONLINEAR DISTORTIONS IN FIBER OPTICS
Abstract
The article is devoted to the analysis of the perturbation coefficients of the nonlinear distortion compensation model in fiber-optic communication lines. The case of long-range signal transmission is considered, for which the effect of signal dispersion is in some sense more significant than nonlinear distortion. This makes it possible to use an approximation of the nonlinear Schrodinger equation based on perturbation theory with respect to a small parameter of nonlinearity to describe the signal propagation process. Using this approximation, analytical expressions are obtained for the coefficients of the first-order model in the case of a Gaussian pulse shape. A number of numerical experiments have been carried out to study the structure of the coefficient matrix. It has been found that this matrix is well approximated by a small rank in the absence of attenuation and amplification. In addition, it was found that when taking into account the effects of signal attenuation and amplification, the rank of the matrix approaching the original matrix with a fixed error is higher than in experiments without attenuation. Research confirms that taking into account the symmetry of the matrix and its approximation with a small rank can reduce the computational complexity of the nonlinear distortion filtering algorithm for a single symbol from O(N2) to O(RN ln N), where N is the size of the matrix and R is its rank.
Computational Mathematics and Mathematical Physics. 2025;65(9):1540-1555
1540-1555
1556-1559
A NOTE ON THE APPLICATION OF THE CHARACTERISTIC FUNCTION TO THE CALCULATION OF INERTIA INTEGRALS OF A RIGID BODY
Abstract
The analogue of the characteristic function, known from probability theory and mathematical statistics, is used to calculate moments of inertia (inertia integrals) of an arbitrary order. Bodies bounded by a rectangular parallelepiped, an ellipsoid, and an arbitrary tetrahedron are considered as examples.
Computational Mathematics and Mathematical Physics. 2025;65(9):1560-1565
1560-1565
Computer science
NUMERICAL-ANALYTICAL METHOD FOR ESTIMATING SIGNAL PARAMETERS ON A SET OF ALTERNATIVE GRIDS UNDER UNCERTAINTY CONDITIONS
Abstract
Using a variety of alternative computational grids, the problem of optimal estimation of signal parameters is solved in conditions where measurements contain various types of interference. A new method of forming the desired estimates is being developed, which ensures the decomposition of the computational procedure, a significant reduction in time and cost of its implementation, as well as a reduction in the estimation error for incorrect measurement conditions. Mathematical expressions are given for a comparative assessment of the effectiveness of the developed and known optimal estimation methods in conditions of uncertainty. Random and methodological errors are analyzed, as well as the computational effect achieved. An illustrative example is given.
Computational Mathematics and Mathematical Physics. 2025;65(9):1566–1580
1566–1580
SEPARABLE PHYSICS-INFORMED NEURAL NETWORKS FOR SOLVING ELASTICITY PROBLEMS
Abstract
Abstract –A method for solving elasticity problems based on separable physics-informed neural networks (SPINN) in conjunction with the deep energy method (DEM) is presented. Numerical experiments have been carried out for a number of problems showing that this method has a significantly higher convergence rate and accuracy than the vanilla physics-informed neural networks (PINN) and even SPINN based on a system of partial differential equations (PDEs). In addition, using the SPINN in the framework of DEM approach it is possible to solve problems of the linear theory of elasticity on complex geometries, which is unachievable with the help of PINNs in frames of partial differential equations. Considered problems are very close to the industrial problems in terms of geometry, loading, and material parameters. Bibl. 61. Fig. 6. Tabl. 8.
Computational Mathematics and Mathematical Physics. 2025;65(9):1581-1596
1581-1596
A DECOMPOSITION APPROACH ON THE BASE OF BROWNIAN ITERATION FOR THE LINEAR PROGRAMMING WHERE ALL BASIS MATRICES ARE M-MATRIX
Abstract
A new scheme for solving a problem for linear programming is proposed. The main property that distinguishes the considered problem is that the basis sub-matrices of its matrix are composed of only M-matrices. Based on the possibility created by this property, a matrix game with the same structure and size as its matrix is set against the given problem, and the possibility of constructing the optimal basis of the problem by partially executing the Brownian iteration leading to the optimal strategy of the second player is shown. Thus, we decompose the solution of the problem into the execution of a finite number of Brownian iterations. The areas of application of the solution scheme are shown. A numerical example illustrates the scheme. The possibility of replacing the game matrix with an integer-element matrix is also shown. This property allows Brownian iteration to be performed exactly. Bibl. 38.
Computational Mathematics and Mathematical Physics. 2025;65(9):1597-1606
1597-1606