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Explicit Numerically Implementable Formulas for Poincaré–Steklov Operators 普因卡雷-斯特克洛夫算子的明确数值可实现公式
IF 0.7 4区 数学
Computational Mathematics and Mathematical Physics Pub Date : 2024-04-01 DOI: 10.1134/s0965542524020040
A. S. Demidov, A. S. Samokhin
{"title":"Explicit Numerically Implementable Formulas for Poincaré–Steklov Operators","authors":"A. S. Demidov, A. S. Samokhin","doi":"10.1134/s0965542524020040","DOIUrl":"https://doi.org/10.1134/s0965542524020040","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper presents explicit numerically implementable formulas for the Poincaré–Steklov operators, such as the Dirichlet–Neumann, Dirichlet–Robin, Robin1–Robin2, and Grinberg–Mayergoiz operators, related to the two-dimensional Laplace equation. These formulas are based on the lemma about a univalent isometric mapping of a closed analytic curve onto a circle. Numerical results for domains with very complex geometries were obtained for several test harmonic functions for the Dirichlet–Neumann and Dirichlet–Robin operators.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Smooth Lyapunov Manifolds for Autonomous Systems of Nonlinear Ordinary Differential Equations and Their Application to Solving Singular Boundary Value Problems 非线性常微分方程自治系统的光滑 Lyapunov Manifolds 及其在解决奇异边值问题中的应用
IF 0.7 4区 数学
Computational Mathematics and Mathematical Physics Pub Date : 2024-04-01 DOI: 10.1134/s0965542524020064
N. B. Konyukhova
{"title":"Smooth Lyapunov Manifolds for Autonomous Systems of Nonlinear Ordinary Differential Equations and Their Application to Solving Singular Boundary Value Problems","authors":"N. B. Konyukhova","doi":"10.1134/s0965542524020064","DOIUrl":"https://doi.org/10.1134/s0965542524020064","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>For an autonomous system of <span>(N)</span> nonlinear ordinary differential equations considered on a semi-infinite interval <span>({{T}_{0}} leqslant t &lt; infty )</span> and having a (pseudo)hyperbolic equilibrium point, the paper considers an <span>(n)</span>-dimensional <span>((0 &lt; n &lt; N))</span> stable solution manifold, or a manifold of conditional Lyapunov stability, which, for each sufficiently large <span>(t)</span>, exists in the phase space of the system’s variables in the neighborhood of its saddle point. A smooth separatrix saddle surface for such a system is described by solving a singular Lyapunov-type problem for a system of quasilinear first-order partial differential equations with degeneracy in the initial data. An application of the results to the correct formulation of boundary conditions at infinity and their transfer to the end point for an autonomous system of nonlinear equations is given, and the use of this approach in some applied problems is indicated.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Estimation of QTT Ranks of Regular Functions on a Uniform Square Grid 估计均匀正方形网格上正则函数的 QTT 等级
IF 0.7 4区 数学
Computational Mathematics and Mathematical Physics Pub Date : 2024-04-01 DOI: 10.1134/s0965542524020143
A. Zyl’, N. Zamarashkin
{"title":"Estimation of QTT Ranks of Regular Functions on a Uniform Square Grid","authors":"A. Zyl’, N. Zamarashkin","doi":"10.1134/s0965542524020143","DOIUrl":"https://doi.org/10.1134/s0965542524020143","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper proves estimates of <span>(varepsilon )</span>-ranks for TT decompositions of tensors obtained by tensorizing the values of a regular function of one complex variable on a uniform square grid in the complex plane. A relation between the approximation accuracy and the geometry of the domain of regularity of the function is established.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Synthesis of an Optimal Stable Affine System 合成最佳稳定仿射系统
IF 0.7 4区 数学
Computational Mathematics and Mathematical Physics Pub Date : 2024-04-01 DOI: 10.1134/s0965542524020039
L. T. Ashchepkov
{"title":"Synthesis of an Optimal Stable Affine System","authors":"L. T. Ashchepkov","doi":"10.1134/s0965542524020039","DOIUrl":"https://doi.org/10.1134/s0965542524020039","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A method for constructing a feedback that ensures the attraction of trajectories of an affine system to an equilibrium state and to a given manifold is proposed. The feedback is found in an analytical form as a solution to an auxiliary optimal control problem. Sufficient conditions for the existence of the optimal control are given. Application of the proposed method to some classes of linear and nonlinear systems is discussed.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Improving the Accuracy of Exponentially Converging Quadratures 提高指数收敛四则运算的精度
IF 0.7 4区 数学
Computational Mathematics and Mathematical Physics Pub Date : 2024-03-21 DOI: 10.1134/s0965542524010020
A. A. Belov, V. S. Khokhlachev
{"title":"Improving the Accuracy of Exponentially Converging Quadratures","authors":"A. A. Belov, V. S. Khokhlachev","doi":"10.1134/s0965542524010020","DOIUrl":"https://doi.org/10.1134/s0965542524010020","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Evaluation of one-dimensional integrals arises in many problems in physics and technology. This is most often done using simple quadratures of midpoints, trapezoids and Simpson on a uniform grid. For integrals of periodic functions over the full period, the convergence of these quadratures drastically accelerates and depends on the number of grid steps according to an exponential law. In this paper, new asymptotically accurate estimates of the error of such quadratures are obtained. They take into account the location and multiplicity of the poles of the integrand in the complex plane. A generalization of these estimates is constructed for the case when there is no a priori information about the poles of the integrand. An error extrapolation procedure is described that drastically accelerates the convergence of quadratures.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Convergence of Some Difference Schemes of the Support Operator Method for Repeated Rotational Operations 重复旋转操作支持算子法某些差分方案的收敛性
IF 0.7 4区 数学
Computational Mathematics and Mathematical Physics Pub Date : 2024-03-21 DOI: 10.1134/s0965542524010123
Yu. A. Poveshchenko, A. Yu. Krukovskii, V. O. Podryga, P. I. Rahimly
{"title":"Convergence of Some Difference Schemes of the Support Operator Method for Repeated Rotational Operations","authors":"Yu. A. Poveshchenko, A. Yu. Krukovskii, V. O. Podryga, P. I. Rahimly","doi":"10.1134/s0965542524010123","DOIUrl":"https://doi.org/10.1134/s0965542524010123","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>An approach for describing the metric properties of a difference mesh for discretizing repeated rotational operations of vector analysis as applied to modeling electromagnetic fields is proposed. Based on the support operator method, integral-consistent operations (gradient, divergence and curl) are constructed, which are necessary to obtain estimates of the convergence of difference schemes for repeated rotational operations designed to solve specific problems of magnetohydrodynamics. Using smooth solutions of a model magnetostatic problem with first-order accuracy, the convergence of the difference schemes constructed in this work with a zero eigenvalue of the spectral problem is proved. In this case, no restrictions are imposed on the difference tetrahedral mesh, except for its nondegeneracy. Calculation of electromagnetic fields for a three-dimensional problem of magnetic hydrodynamics in a two-temperature approximation with the full set of spatial components of velocity and electromagnetic fields is presented. The dynamics of electromagnetic fields is developed against the background of rotational diffusion of the magnetic field vector.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196218","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Numerical Analysis of the Blow-Up of One-Dimensional Polymer Fluid Flow with a Front 带前沿的一维聚合物流体爆破的数值分析
IF 0.7 4区 数学
Computational Mathematics and Mathematical Physics Pub Date : 2024-03-21 DOI: 10.1134/s0965542524010068
L. S. Bryndin, B. V. Semisalov, V. A. Beliaev, V. P. Shapeev
{"title":"Numerical Analysis of the Blow-Up of One-Dimensional Polymer Fluid Flow with a Front","authors":"L. S. Bryndin, B. V. Semisalov, V. A. Beliaev, V. P. Shapeev","doi":"10.1134/s0965542524010068","DOIUrl":"https://doi.org/10.1134/s0965542524010068","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>One-dimensional flows of an incompressible viscoelastic polymer fluid that are qualitatively similar to the solutions of Burgers’ equation are described on the basis of mesoscopic approach for the first time. The corresponding initial boundary-value problem is posed for the system of quasilinear differential equations. The numerical algorithm for solving it is designed and verified. The algorithm uses the explicit fifth-order scheme to approximate unknown functions with respect to time variable and the rational barycentric interpolations with respect to space variable. A method for localization of singular points of the solution in the complex plain and for adaptation of the spatial grid to them is implemented using the Chebyshev-Padé approximations. Two regimes of evolution of the solution to the problem are discovered and characterized while using the algorithm: regime 1—a smooth solution exists in a sufficiently large time interval (the singular point moves parallel to the real axis in the complex plane); regime 2—the smooth solution blows up at the beginning of evolution (the singular point reaches the segment of the real axis where the problem is posed). We study the influence of the rheological parameters of fluid on the realizability of these regimes and on the length of time interval where the smooth solution exists. The obtained results are important for the analysis of laminar-turbulent transitions in viscoelastic polymer continua.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140205362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Simulation of Domain Walls: Simple Waves in the Magnetodynamics Equation 域壁模拟：磁动力学方程中的简单波
IF 0.7 4区 数学
Computational Mathematics and Mathematical Physics Pub Date : 2024-03-21 DOI: 10.1134/s0965542524010093
L. A. Kalyakin, E. G. Ekomasov
{"title":"Simulation of Domain Walls: Simple Waves in the Magnetodynamics Equation","authors":"L. A. Kalyakin, E. G. Ekomasov","doi":"10.1134/s0965542524010093","DOIUrl":"https://doi.org/10.1134/s0965542524010093","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A partial differential equation modeling the motion of a domain wall taking into account external magnetic fields and damping is considered. In the case of constant coefficients, this equation has a set of trivial solutions—equilibria. Solutions in the form of simple (traveling) waves that correspond to a dynamic transition from one equilibrium to another are studied. Possible types of waves that are stable in linear approximation are listed. A method for calculating the velocity of such waves is given.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Multiplicative Control Problem for a Nonlinear Reaction–Diffusion Model 非线性反应-扩散模型的乘法控制问题
IF 0.7 4区 数学
Computational Mathematics and Mathematical Physics Pub Date : 2024-03-21 DOI: 10.1134/s0965542524010056
R. V. Brizitskii, A. A. Donchak
{"title":"Multiplicative Control Problem for a Nonlinear Reaction–Diffusion Model","authors":"R. V. Brizitskii, A. A. Donchak","doi":"10.1134/s0965542524010056","DOIUrl":"https://doi.org/10.1134/s0965542524010056","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper studies a multiplicative control problem for the reaction–diffusion equation in which the reaction coefficient nonlinearly depends on the substance concentration, as well as on spatial variables. The role of multiplicative controls is played by the coefficients of diffusion and mass transfer. The solvability of the extremum problem is proved, and optimality systems are derived for a specific reaction coefficient. Based on the analysis of these systems, the relay property of multiplicative and distributed controls is established, and estimates of the local stability of optimal solutions to small perturbations of both the quality functionals and one of the given functions of the boundary value problem are derived.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Actual Accuracy of Linear Schemes of High-Order Approximation in Gasdynamic Simulations 气体动力学模拟中高阶逼近线性方案的实际精度
IF 0.7 4区 数学
Computational Mathematics and Mathematical Physics Pub Date : 2024-03-21 DOI: 10.1134/s0965542524010044
M. D. Bragin
{"title":"Actual Accuracy of Linear Schemes of High-Order Approximation in Gasdynamic Simulations","authors":"M. D. Bragin","doi":"10.1134/s0965542524010044","DOIUrl":"https://doi.org/10.1134/s0965542524010044","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A new test problem for one-dimensional gas dynamics equations is considered. Initial data in the problem is a periodic smooth wave. Shock waves are formed in the gas flow over a finite time. The convergence under mesh refinement is analyzed for two third-order accurate linear schemes, namely, a bicompact scheme and Rusanov’s scheme. It is demonstrated that both schemes have only the first order of integral convergence in the shock influence area. However, when applied to equations of isentropic gas dynamics, the schemes converge with at least the second order.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196408","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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