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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Том 65, № 12 (2025)
General numerical methods
ESTIMATION OF THE REMAINDER TERM OF THE APPEL HYPERGEOMETRIC SERIES F2
Аннотация
Integral representations and estimates for the remainder term for the summation of the double hypergeometric Appell series F2 are constructed. The resulting formulas have applications in developing algorithms for computing the Appell functions F1 and F3 in C2 using analytic continuation formulas. The results have applications to problems in mathematical physics and computational function theory, including the construction of conformal mappings of complicated polygons based on the Christoffel-Schwarz integral.
Computational Mathematics and Mathematical Physics. 2025;65(12):1973-1994
1973-1994
Optimal control
CALCULATION OF EXTREMALS IN AN OPTIMAL CONTROL PROBLEM WITH A HIGHER-ORDER STATE CONSTRAINT
Аннотация
The problem of controlling the k-th derivative of an object state under a linear state constraint, where k is an arbitrary natural number, is studied. According to the existing terminology in literature, this is a so-called state-constrained control problem of order k (the term 'of depth k' is also used). This paper applies Pontryagin's maximum principle to the problem under study and conducts a theoretical analysis of the resulting optimality conditions. Based on this analysis, a computational scheme for finding extremals is proposed.
Computational Mathematics and Mathematical Physics. 2025;65(12):1995-2008
1995-2008
ACCELERATED ITERATIVE LEARNING CONTROL ALGORITHMS FOR DISCRETE SYSTEMS UNDER RANDOM DISTURBANCES
Аннотация
Iterative learning control (ILC) algorithms emerged in connection with the tasks of increasing the accuracy of repetitive operations performed by robots. They use information from past repetitions to adjust the control signal for the current repetition. In the ILC literature, these repetitions are called trial steps, trials, or passes. A critical indicator of the efficiency of such algorithms is the rate of convergence of the learning error to a given value, ideally to zero. To increase the convergence rate of ILC algorithms, the authors in their recent works proposed a combination of the heavy ball method and the vector Lyapunov function method for repetitive processes that they had developed earlier. It turns out that this approach allows one to implicitly predict the gradient direction of the cost function, which allows one to significantly increase the convergence rate. In the examples, convergence to computer zero was achieved in just a few trial steps. However, this did not take into account the inevitably present random disturbances affecting the system and measurement noise, which reduce the achievable accuracy. In this paper, the specified approach is extended to the case of discrete systems, taking into account the aforementioned random factors. The results of modeling, confirming the theoretical results, are presented using the example of a laboratory portal robot.
Computational Mathematics and Mathematical Physics. 2025;65(12):2010-2021
2010-2021
OPTIMAL CONTROL OF COMPLEX HEAT TRANSFER EQUATIONS WITH CAUCHY BOUNDARY CONDITIONS
Аннотация
This paper presents an analysis of the optimal control problem for a steady- state diffusion model of complex heat transfer, including the P1 - approximation of the radiative heat transfer equation. A formulation is considered in which the boundary values of the temperature and its normal derivative are known, while the boundary conditions for the radiative intensity are not specified. The solvability of the control problem is established, and the necessary optimality conditions are obtained. A sufficient condition for the uniqueness of a solution to the optimal control problem is presented.
Computational Mathematics and Mathematical Physics. 2025;65(12):2024–2030
2024–2030
Ordinary differential equations
EXPLICIT FORM OF ASYMPTOIC COEFFICIENTS AT THE ENTERING CORNER FOR CONFORMAL MAPPING OF THE L-SHAPED DOMAIN
Аннотация
For the conformal mapping of an L - shaped domain with an arbitrary length A and width h of its shelves and an entering corner of πβ, explicit analytical formulas are found for the coefficients cn of the mapping function expansion near the vertex w1 of the entering corner. The formulas for the quantities cn, called the Stress Intensity Factors (SIF), are obtained as a series in the powers of the small parameter δ:=exp(-πA/h) with coefficients determined only by the exponent of the angle β using explicit formulas. A brief bibliographic review on the problem of calculating stress intensity factors is also included.
Computational Mathematics and Mathematical Physics. 2025;65(12):2031-2044
2031-2044
Partial Differential Equations
CONSTRUCTION OF A HARMONIC MAPPING OF ONE CLASS DOMAINS WITH A CURVILINEAR BOUNDARY BY USING THE MULTIPOLE METHOD
Аннотация
We present an algorithm based on the multipole method for harmonic mapping of a class of domains g with a curvilinear boundary containing incoming arc angles and narrow isthmuses. Results of a numerical implementation of this algorithm are given for two such domains. The use of several hundred approximation functions (multipoles) ensured an accuracy of the order of 10−4 in the C( ) norm. In a previous work, the authors presented a similar conformal mapping algorithm for the same domains, based on this method, along with a corresponding numerical implementation that demonstrated the same accuracy. A comparison of these previous results with those obtained in this work provides material for analyzing the quality of computational grids obtained using conformal and harmonic mappings.
Computational Mathematics and Mathematical Physics. 2025;65(12):2045-2053
2045-2053
CONSTRUCTION OF BARRIERS FOR SINGULARLY PERTURBED PARABOLIC PROBLEMS WITH CUBIC NONLINEARITIES TAKING INTO ACCOUNT THE INFLECTION POINT
Аннотация
An initial boundary value problem for singularly perturbed parabolic equations with cubic nonlinearities is considered in a rectangle. The inflection point of the cubic parabola is assumed to be located to the left of the root of the degenerate equation. The nonlinear method of angular parabolic functions is used to construct a complete asymptotic expansion of the solution to the problem and prove its uniformity in a closed rectangle with respect to a small parameter. This work completes the study of singularly perturbed parabolic problems with cubic nonlinearities.
Computational Mathematics and Mathematical Physics. 2025;65(12):2054-2063
2054-2063
Mathematical physics
ABOUT THE PARTICLE-IN-CELL METHOD AND SOLUTIONS THAT LOSE SMOOTHNESS
Аннотация
To test various versions of the particle-in-cell method, a benchmark problem simulating the breaking effect of relativistic plasma oscillations is proposed. It is shown that the breaking effect can be realized using the TSC version of the method, but not using the CIC version. Unsuccessful modeling attempts lead to the observation of a gradient catastrophe directly on the symmetry axis of the domain, which contradicts the corresponding theoretical results. The cause of these failures is the resonant growth of the error over time at one of the algorithm stages. The proposed testing method is suitable for an arbitrary one-dimensional version of the particle-in-cell method.
Computational Mathematics and Mathematical Physics. 2025;65(12):2064-2076
2064-2076
SURFACE CHARGE DENSITY OF A CONDUCTING ELLIPSOID IN A COAXIAL ELECTRIC FIELD
Аннотация
An effective method for determining the surface charge density of ellipsoidal conductors in an external axisymmetric electric field is proposed. The method is based on solving a one-dimensional Fredholm integral equation of the first kind using the Galerkin method. Numerical experiments conducted to solve model problems illustrate a significant reduction in computational error using the proposed method compared to errors obtained in previous studies on this topic.
Computational Mathematics and Mathematical Physics. 2025;65(12):2077-2083
2077-2083
ON APPROXIMATE SOLUTIONS OF MAGNETIC BOUNDARY VALUE PROBLEMS BY THE METHOD OF A SYSTEM OF INTEGRAL EQUATIONS
Аннотация
A justification for the collocation method for a system of integral equations of magnetic boundary value problems for the Helmholtz vector equation is given. At certain selected points, the system of integral equations is replaced by a system of algebraic equations, and the existence and uniqueness of a solution to the system of algebraic equations is established. Convergence of the solution of the system of algebraic equations to the exact solution of the system of integral equations is proved, and the rate of convergence of the method is indicated. Furthermore, a sequence converging to the exact solution of magnetic boundary value problems is constructed.
Computational Mathematics and Mathematical Physics. 2025;65(12):2084–2096
2084–2096
ALGORITHM FOR CALCULATION OF NONLINEAR WAVE PROCESSES IN A MICROWAVE GENERATOR WITH MAGNETIC INSULATION IN THE THREE-DIMENSIONAL CASE
Аннотация
This paper considers the pressing problem of numerical modeling of nonlinear wave processes in a magnetically insulated microwave generator used to generate relativistic electron beams and the radiation they generate. A distinctive feature of the problem formulation is the realistic three-dimensional geometry of the generator. A three-dimensional mathematical model, including Maxwell's equations and the equations of motion for relativistic charged particles, was used for the numerical analysis. The model takes into account the field emission of electrons from the cathode surface and the presence of relativistic plasma. A new numerical algorithm is proposed for solving the problem, combining explicit time-dependent schemes, the finite volume method on irregular grids, and the cloud particle method. The software implementation is designed for parallel computing. To test the three-dimensional numerical approach, two relevant problems of generating relativistic electron beams are considered in one- and two-dimensional settings. Analysis of the calculation results confirmed the reproduction of one- and two-dimensional solutions in the three-dimensional code.
Computational Mathematics and Mathematical Physics. 2025;65(12):2097-2106
2097-2106