Vol 63, No 7 (2023)
ОБЩИЕ ЧИСЛЕННЫЕ МЕТОДЫ
Stability Indicators of Nonnegative Matrices: Parametric and Sparse Cases
Abstract
Methods for the algorithmic construction of stability indicators of nonnegative matrices is described, and the application of these indicators to problems of modern mathematical biology and epidemiology is discussed. Specific features of such indicators when they are applied to problems about the parametric loss of stability of trivial equilibrium states of discrete dynamical systems are pointed out. Estimates of the efficiency of algorithms based on the proposed methods for the case of systems determined by sparse matrices are given. Examples of using the constructed algorithms for such systems are discussed.
ОПТИМАЛЬНОЕ УПРАВЛЕНИЕ
Sufficient Solvability Conditions for the Problem of Pursuit under an Impulse Action
Abstract
The article considers a linear differential pursuit game under the condition that an integral constraint is imposed on the evader’s control and the pursuer uses an impulse control. These impulse actions on the object are made at predetermined time points, and the corresponding control is represented using the Dirac delta function. The article studies linear conflicts described by a system of ordinary differential equations whose trajectories have jumps at certain times. The terminal set is represented by a cylinder in an n-dimensional Euclidean space. The problem is solved using the resolving function method. The fact that the lower bound is reached is proved using the theory of support functions. As a result, the quasi-strategy is replaced by an almost stroboscopic strategy and a method for building this strategy is presented. An example of a nonlinear right-hand side is given.
On the Existence of Optimal Control in the Problem of Optimizing the Lowest Coefficient of a Semilinear Evolutionary Equation
Abstract
The paper studies the problem of optimizing the lowest coefficient understood as a function with values in a Banach space, which enters linearly into an abstract semilinear pseudoparabolic evolutionary differential equation in a Banach space. For this problem, an existence theorem for an optimal control is proved. Due to the nonlinearity of the equation under study, the author uses his previous results on the total preservation of the unique global solvability (on the totally global solvability) and on the estimation of solutions for similar equations. This estimate turns out to be significant in the course of the study. As an example, the Oskolkov’s hydrodynamic system of equations is considered.
Conditional Gradient Method for Optimization Problems with a Constraint in the Form of the Intersection of a Convex Smooth Surface and a Convex Compact Set
Abstract
The conditional gradient method is generalized to nonconvex sets of constraints representing the set-theoretic intersection of a convex smooth surface and a convex compact set. Necessary optimality conditions are studied, and the convergence of the method is analyzed.
ОБЫКНОВЕННЫЕ ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ
Four Classes of Definite Integrals about Hyperbolic and Trigonometric Functions
Abstract
Четыре класса определенных интегралов с гиперболическими и тригонометрическими функциями
. Установлены рекуррентные соотношения для четырех классов определенных интегралов от степенных функций со множителями вида тригонометрических и гиперболических функций в знаменателе. Соотношения получены в терминах логарифмической функции, дзета-функции Римана и ее вариантов, а также бета-функции и функции Лежандра.
УРАВНЕНИЯ В ЧАСТНЫХ ПРОИЗВОДНЫХ
Local Solvability, Blow-Up, and Hölder Regularity of Solutions to Some Cauchy Problems for Nonlinear Plasma Wave Equations: III. Cauchy Problems
Abstract
Three Cauchy problems for Sobolev-type equations with a common linear part from the theory of ion acoustic and drift waves in a plasma are considered. The problems are reduced to equivalent integral equations. We prove the existence of unextendable solutions for two problems and the existence of a local-in-time solution for the third problem. For one of the problems, by applying a modified method of Kh.A. Levin, sufficient conditions for finite time blow-up of solutions are obtained and an upper bound for the solution blow-up time is found. For another problem, S.I. Pohozaev’s nonlinear capacity method is used to obtain a finite time blow-up result and two results concerning the nonexistence of even local solutions, and an upper bound for the solution blow-up time is obtained as well.
Parasitic Eigenvalues of Spectral Problems for the Laplacian with Third-Type Boundary Conditions
Abstract
Spectral problems for the Laplacian with Robin and Steklov (third-type) boundary conditions on a smooth boundary of a plane domain are considered. These conditions involve a small parameter and a coefficient of “wrong” sign, giving rise to negative eigenvalues, which are called parasitic. Such problems and eigenvalues arise in numerical schemes when regular variations in boundaries (small nonuniform shifts along the normal) are modeled by perturbations of differential operators in boundary conditions. Asymptotic expansions of some parasitic eigenvalues are constructed and justified, and a priori estimates are obtained, which help to determine their locations on the real axis and the effect exerted on the simulation errors.
Inverse Problems for the Helmholtz Equation on Finding the Right-Hand Side with Nonlocal Integral Observation
Abstract
The paper presents formulations of inverse problems for the Helmholtz equation on finding its right-hand side with a Samarskii–Ionkin-type additional integral condition and the justification of their well-posedness in the Hadamard sense in the class of regular solutions. The uniqueness of solutions of the formulated problems is proved on the basis of integral identities. Solutions of the problem are found in explicit form by the methods of separation of variables and integral equations.
Об одном методе решения смешанной краевой задачи для уравнения параболического типа с помощью модифицированных операторов синк-приближений
Abstract
Предложен новый метод получения обобщенного решения смешанной краевой задачи для уравнения параболического типа с граничными условиями третьего рода и непрерывным начальным условием. Обобщенные функции понимаются в смысле секвенциального подхода. В качестве промежуточного приближения используется модифицированный оператор синк-аппроксимаций. Решение получено в виде ряда, равномерно сходящегося внутри области определения решения. Библ. 49. Фиг. 1.
Solution Blow-Up and Global Solvability of the Cauchy Problem for the Equation of Moderately Long Longitudinal Waves in a Viscoelastic Rod
Abstract
The Cauchy problem for a nonlinear Sobolev-type differential equation modeling moderately long small-amplitude longitudinal waves in a viscoelastic rod is investigated in a space of continuous functions defined on the entire number line that have limits at infinity. Conditions for the existence of a global solution and for finite time solution blow-up are examined.
МАТЕМАТИЧЕСКАЯ ФИЗИКА
Laser-Induced Thermoelastic Response in an Isotropic Medium Having Variable Material Moduli
Abstract
Индуцированный лазером термоупругий отклик в изотропной среде с переменными модулями материала
. Исследованы температурные нагрузки и распределение напряжений в задаче о плоской деформации изотропной термоупругой среды с переменными модулями материала. Источники тепла индуцируются лазером. Для анализа тепловых характеристик использована модель теплопроводности гиперболического типа с тремя запаздываниями по фазе. Аналитическое решение задачи представлено с помощью интегрального преобразования Лапласа–Фурье. Представлены результаты численных расчетов и выделены характерные особенности полученных тепловых нагрузок.
Quasi-Gasdynamic Model and Numerical Algorithm for Describing Mixtures of Different Fluids
Abstract
An elegant and easy-to-implement numerical algorithm for simulating flows of homogeneous gas mixtures with component temperatures and velocities assumed to be equal is constructed and tested. The algorithm yields monotone density profiles for the components even if their specific heat ratios are widely different. The algorithm can be used to simulate some flows of gas–liquid mixtures.
Модифицированный метод секущих для энтропийных решеточных уравнений Больцмана
Abstract
Устойчивость решеточных уравнений Больцмана регулируется параметром, отвечающим за время релаксации неравновесной системы, который, в свою очередь, влияет на вязкость исследуемого течения. В энтропийном подходе время релаксации вычисляется из уравнения баланса энтропии таким образом, чтобы энтропия в каждый момент времени и в каждой пространственной точке не убывала. В настоящей статье рассматривается метод решения уравнения баланса энтропии на основе модифицированного метода секущих. Показано, что данный подход имеет хорошую точность. В качестве приложения предлагаемого метода рассмотрены численные решения задачи о двумерном двойном сдвиге. Проведено сравнение результатов расчетов с другими энтропийными методами.
On the Accuracy of Shock-Capturing Schemes Calculating Gas-Dynamic Shock Waves
Abstract
A comparative experimental accuracy study of three shock-capturing schemes (the second-order CABARET, third-order Rusanov, and fifth-order in space third-order in time A-WENO schemes) is carried out by numerically solving a Cauchy problem with smooth periodic initial data for the Euler equations of gas dynamics. In the studied example, the solution breaks down and shock waves emerge. It is shown that the CABARET and A-WENO schemes, which are constructed using nonlinear limiters as a stabilization mechanism, have approximately the same accuracy in the areas of shock wave influence, while the nonmonotone Rusanov scheme has significantly higher accuracy in these areas despite producing noticeable nonphysical oscillations in the immediate vicinities of shock waves. At the same time, the combined scheme obtained based on the Rusanov and CABARET schemes localizes shock wave fronts, which are captured in a non-oscillatory manner, and preserves higher accuracy in the areas of the shock influence.
Duality Method for Solving 3D Contact Problems with Friction
Abstract
The article studies a 3D contact problem with Coulomb friction for an elastic body resting on a rigid support. The solution of the quasi-variational formulation of the problem is defined as a fixed point of some mapping that associates the given force of the normal reaction of the support with the value of the normal stress in the contact zone. The fixed point is sought by the method of successive approximations, the convergence of which is proved using modified Lagrange functionals. The results of the numerical solution using finite element modeling and the proximal gradient method are presented.
A Modied Runge–Kutta Scheme for the Generalized Benjamin–Bona–Mahony–Burgers’ Equation
Abstract
Модифицированная схема Рунге–Кутты для обобщенного уравнения Бенджамина–Бона–Махони–Бюргерса
. Предложена модифицированная схема Рунге–Кутты (MRK) для обобщенного уравнения Бенджамина–Бона–Махони–Бюргерса, расщепляющая уравнение на две части, для которых применяют различные конечно-разностные схемы по пространству, а затем используют метод Рунге–Кутты четвертого порядка во времени. Схема условно устойчива и сходится с четвертым порядком по времени и вторым порядком по пространству. Сравнение существующих численных методов с предложенной схемой демонстрирует ее высокую точность и устойчивость.