Pub Date : 2020-12-15DOI: 10.22363/2658-4670-2020-28-1-17-34
A. Baddour, M. Malykh, A. A. Panin, L. Sevastianov
We consider moving singular points of systems of ordinary differential equations. A review of Painlevé’s results on the algebraicity of these points and their relation to the Marchuk problem of determining the position and order of moving singularities by means of finite difference method is carried out. We present an implementation of a numerical method for solving this problem, proposed by N. N. Kalitkin and A. Al’shina (2005) based on the Rosenbrock complex scheme in the Sage computer algebra system, the package CROS for Sage. The main functions of this package are described and numerical examples of usage are presented for each of them. To verify the method, computer experiments are executed (1) with equations possessing the Painlevé property, for which the orders are expected to be integer; (2) dynamic Calogero system. This system, well-known as a nontrivial example of a completely integrable Hamiltonian system, in the present context is interesting due to the fact that coordinates and momenta are algebraic functions of time, and the orders of moving branching points can be calculated explicitly. Numerical experiments revealed that the applicability conditions of the method require additional stipulations related to the elimination of superconvergence points.
考虑常微分方程系统的运动奇异点。对painlev关于这些点的代数性的结果及其与用有限差分法确定运动奇异点的位置和阶数的Marchuk问题的关系进行了评述。我们提出了一种解决该问题的数值方法的实现，该方法由N. N. Kalitkin和a . Al 'shina(2005)提出，基于Sage计算机代数系统(Sage的软件包CROS)中的Rosenbrock复格式。描述了该包的主要功能，并给出了每个功能的数值示例。为了验证该方法，对具有painlev性质的方程进行了计算机实验(1)，其中阶数期望为整数;(2)动态Calogero系统。这个系统，众所周知是一个完全可积哈密顿系统的非平凡例子，在目前的情况下是有趣的，因为坐标和动量是时间的代数函数，并且运动分支点的阶数可以显式计算。数值实验表明，该方法的适用条件需要附加有关消除超收敛点的规定。
{"title":"Numerical determination of the singularity order of a system of differential equations","authors":"A. Baddour, M. Malykh, A. A. Panin, L. Sevastianov","doi":"10.22363/2658-4670-2020-28-1-17-34","DOIUrl":"https://doi.org/10.22363/2658-4670-2020-28-1-17-34","url":null,"abstract":"We consider moving singular points of systems of ordinary differential equations. A review of Painlevé’s results on the algebraicity of these points and their relation to the Marchuk problem of determining the position and order of moving singularities by means of finite difference method is carried out. We present an implementation of a numerical method for solving this problem, proposed by N. N. Kalitkin and A. Al’shina (2005) based on the Rosenbrock complex scheme in the Sage computer algebra system, the package CROS for Sage. The main functions of this package are described and numerical examples of usage are presented for each of them. To verify the method, computer experiments are executed (1) with equations possessing the Painlevé property, for which the orders are expected to be integer; (2) dynamic Calogero system. This system, well-known as a nontrivial example of a completely integrable Hamiltonian system, in the present context is interesting due to the fact that coordinates and momenta are algebraic functions of time, and the orders of moving branching points can be calculated explicitly. Numerical experiments revealed that the applicability conditions of the method require additional stipulations related to the elimination of superconvergence points.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44434370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-15DOI: 10.22363/2658-4670-2020-28-4-327-345
Yu Ying, Ин Юй, M. Malykh, М Д Малых
We implement several explicit Runge-Kutta schemes that preserve quadratic invariants of autonomous dynamical systems in Sage. In this paper, we want to present our package ex.sage and the results of our numerical experiments. In the package, the functions rrk_solve, idt_solve and project_1 are constructed for the case when only one given quadratic invariant will be exactly preserved. The function phi_solve_1 allows us to preserve two specified quadratic invariants simultaneously. To solve the equations with respect to parameters determined by the conservation law we use the elimination technique based on Grbner basis implemented in Sage. An elliptic oscillator is used as a test example of the presented package. This dynamical system has two quadratic invariants. Numerical results of the comparing of standard explicit Runge-Kutta method RK(4,4) with rrk_solve are presented. In addition, for the functions rrk_solve and idt_solve, that preserve only one given invariant, we investigated the change of the second quadratic invariant of the elliptic oscillator. In conclusion, the drawbacks of using these schemes are discussed.
{"title":"On the realization of explicit Runge-Kutta schemes preserving quadratic invariants of dynamical systems","authors":"Yu Ying, Ин Юй, M. Malykh, М Д Малых","doi":"10.22363/2658-4670-2020-28-4-327-345","DOIUrl":"https://doi.org/10.22363/2658-4670-2020-28-4-327-345","url":null,"abstract":"We implement several explicit Runge-Kutta schemes that preserve quadratic invariants of autonomous dynamical systems in Sage. In this paper, we want to present our package ex.sage and the results of our numerical experiments. In the package, the functions rrk_solve, idt_solve and project_1 are constructed for the case when only one given quadratic invariant will be exactly preserved. The function phi_solve_1 allows us to preserve two specified quadratic invariants simultaneously. To solve the equations with respect to parameters determined by the conservation law we use the elimination technique based on Grbner basis implemented in Sage. An elliptic oscillator is used as a test example of the presented package. This dynamical system has two quadratic invariants. Numerical results of the comparing of standard explicit Runge-Kutta method RK(4,4) with rrk_solve are presented. In addition, for the functions rrk_solve and idt_solve, that preserve only one given invariant, we investigated the change of the second quadratic invariant of the elliptic oscillator. In conclusion, the drawbacks of using these schemes are discussed.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45620224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-15DOI: 10.22363/2658-4670-2020-28-4-398-405
I. Amirkhanov, N. Sarker, I. Sarkhadov
In this paper, we report a numerical simulation of laser ablation of a material by ultrashort laser pulses. The thermal mechanism of laser ablation is described in terms of a one-dimensional nonstationary heat conduction equation in a coordinate system associated with a moving evaporation front. The laser action is taken into account through the functions of the source in the thermal conductivity equation that determine the coordinate and time dependence of the laser source. For a given dose of irradiation of the sample, the profiles of the sample temperature at different times, the dynamics of the displacement of the sample boundary due to evaporation, the velocity of this boundary, and the temperature of the sample at the moving boundary are obtained. The dependence of the maximum temperature on the sample surface and the thickness of the ablation layer on the radiation dose of the incident laser pulse is obtained. Numerical calculations were performed using the finite difference method. The obtained results agree with the results of other works obtained by their authors.
{"title":"Numerical modeling of laser ablation of materials","authors":"I. Amirkhanov, N. Sarker, I. Sarkhadov","doi":"10.22363/2658-4670-2020-28-4-398-405","DOIUrl":"https://doi.org/10.22363/2658-4670-2020-28-4-398-405","url":null,"abstract":"In this paper, we report a numerical simulation of laser ablation of a material by ultrashort laser pulses. The thermal mechanism of laser ablation is described in terms of a one-dimensional nonstationary heat conduction equation in a coordinate system associated with a moving evaporation front. The laser action is taken into account through the functions of the source in the thermal conductivity equation that determine the coordinate and time dependence of the laser source. For a given dose of irradiation of the sample, the profiles of the sample temperature at different times, the dynamics of the displacement of the sample boundary due to evaporation, the velocity of this boundary, and the temperature of the sample at the moving boundary are obtained. The dependence of the maximum temperature on the sample surface and the thickness of the ablation layer on the radiation dose of the incident laser pulse is obtained. Numerical calculations were performed using the finite difference method. The obtained results agree with the results of other works obtained by their authors.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48636109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-15DOI: 10.22363/2658-4670-2020-28-4-361-377
A. Sevastianov
This paper investigates the waveguide propagation of polarized electromagnetic radiation in a thin-film integral optical waveguide. To describe this propagation, the adiabatic approximation of solutions of Maxwells equations is used. The construction of a reduced model for adiabatic waveguide modes that retains all the properties of the corresponding approximate solutions of the Maxwell system of equations was carried out by the author in a previous publication in DCM ACS, 2020, No 3. In this work, for a special case when the geometry of the waveguide and the electromagnetic field are invariant in the transverse direction. In this case, there are separate nontrivial TEand TM-polarized solutions of this reduced model. The paper describes the parametrically dependent on longitudinal coordinates solutions of problems for eigenvalues and eigenfunctions - adiabatic waveguide TE and TM polarizations. In this work, we present a statement of the problem of finding solutions to the model of adiabatic waveguide modes that describe the stationary propagation of electromagnetic radiation. The paper presents solutions for the single-mode propagation of TE and TM polarized adiabatic waveguide waves.
本文研究了极化电磁辐射在薄膜积分光波导中的波导传播。为了描述这种传播，使用了麦克斯韦方程组解的绝热近似。作者在之前发表于DCM ACS, 2020, No . 3的文章中构建了绝热波导模式的简化模型，该模型保留了麦克斯韦方程组相应近似解的所有性质。在本工作中，对于波导几何形状和电磁场在横向上不变的特殊情况。在这种情况下，该简化模型存在独立的非平凡tev和tm极化解。本文描述了绝热波导TE和TM极化问题中特征值和特征函数的参数依赖于纵向坐标的解。在这项工作中，我们提出了一个关于寻找描述电磁辐射稳态传播的绝热波导模式模型解的问题的陈述。本文提出了TE和TM极化绝热波导单模传播的解决方案。
{"title":"Single-mode propagation of adiabatic guided modes in smoothly irregular integral optical waveguides","authors":"A. Sevastianov","doi":"10.22363/2658-4670-2020-28-4-361-377","DOIUrl":"https://doi.org/10.22363/2658-4670-2020-28-4-361-377","url":null,"abstract":"This paper investigates the waveguide propagation of polarized electromagnetic radiation in a thin-film integral optical waveguide. To describe this propagation, the adiabatic approximation of solutions of Maxwells equations is used. The construction of a reduced model for adiabatic waveguide modes that retains all the properties of the corresponding approximate solutions of the Maxwell system of equations was carried out by the author in a previous publication in DCM ACS, 2020, No 3. In this work, for a special case when the geometry of the waveguide and the electromagnetic field are invariant in the transverse direction. In this case, there are separate nontrivial TEand TM-polarized solutions of this reduced model. The paper describes the parametrically dependent on longitudinal coordinates solutions of problems for eigenvalues and eigenfunctions - adiabatic waveguide TE and TM polarizations. In this work, we present a statement of the problem of finding solutions to the model of adiabatic waveguide modes that describe the stationary propagation of electromagnetic radiation. The paper presents solutions for the single-mode propagation of TE and TM polarized adiabatic waveguide waves.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47576594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-15DOI: 10.22363/2658-4670-2020-28-4-319-326
Mohamed Boualem
In this investigation, we consider an M/G/1 queue with general retrial times allowing balking and server subject to breakdowns and repairs. In addition, the customer whose service is interrupted can stay at the server waiting for repair or leave and return while the server is being repaired. The server is not allowed to begin service on other customers until the current customer has completed service, even if current customer is temporarily absent. This model has a potential application in various fields, such as in the cognitive radio network and the manufacturing systems, etc. The methodology is strongly based on the general theory of stochastic orders. Particularly, we derive insensitive bounds for the stationary distribution of the embedded Markov chain of the considered system.
{"title":"Stochastic analysis of a single server unreliable queue with balking and general retrial time","authors":"Mohamed Boualem","doi":"10.22363/2658-4670-2020-28-4-319-326","DOIUrl":"https://doi.org/10.22363/2658-4670-2020-28-4-319-326","url":null,"abstract":"In this investigation, we consider an M/G/1 queue with general retrial times allowing balking and server subject to breakdowns and repairs. In addition, the customer whose service is interrupted can stay at the server waiting for repair or leave and return while the server is being repaired. The server is not allowed to begin service on other customers until the current customer has completed service, even if current customer is temporarily absent. This model has a potential application in various fields, such as in the cognitive radio network and the manufacturing systems, etc. The methodology is strongly based on the general theory of stochastic orders. Particularly, we derive insensitive bounds for the stationary distribution of the embedded Markov chain of the considered system.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48480611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-15DOI: 10.22363/2658-4670-2019-27-4-316-324
V. Duzhin, В С Дужин
Robinson-Schensted-Knuth (RSK) correspondence occurs in different contexts of algebra and combinatorics. Recently, this topic has been actively investigated by many researchers. At the same time, many investigations require conducting the computer experiments involving very large Young tableaux. The article is devoted to such experiments. RSK algorithm establishes a bijection between sequences of elements of linearly ordered set and the pairs of Young tableaux of the same shape called insertion tableau and recording tableau . In this paper we study the dynamics of tableau and the dynamics of different concrete values in tableau during the iterations of RSK algorithm. Particularly, we examine the paths within tableaux called bumping routes along which the elements of an input sequence pass. The results of computer experiments with Young tableaux of sizes up to 108 were presented. These experiments were made using the software package for dealing with 2D and 3D Young diagrams and tableaux.
{"title":"Investigation of insertion tableau evolution in the Robinson-Schensted-Knuth correspondence","authors":"V. Duzhin, В С Дужин","doi":"10.22363/2658-4670-2019-27-4-316-324","DOIUrl":"https://doi.org/10.22363/2658-4670-2019-27-4-316-324","url":null,"abstract":"Robinson-Schensted-Knuth (RSK) correspondence occurs in different contexts of algebra and combinatorics. Recently, this topic has been actively investigated by many researchers. At the same time, many investigations require conducting the computer experiments involving very large Young tableaux. The article is devoted to such experiments. RSK algorithm establishes a bijection between sequences of elements of linearly ordered set and the pairs of Young tableaux of the same shape called insertion tableau and recording tableau . In this paper we study the dynamics of tableau and the dynamics of different concrete values in tableau during the iterations of RSK algorithm. Particularly, we examine the paths within tableaux called bumping routes along which the elements of an input sequence pass. The results of computer experiments with Young tableaux of sizes up to 108 were presented. These experiments were made using the software package for dealing with 2D and 3D Young diagrams and tableaux.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42608090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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