Pub Date : 2024-07-18DOI: 10.1134/s0965542524700428
F. Hıra
Abstract
The modified or hybrid method is created by combining the differential transform method (DTM), Padé approximant, and Laplace transform. This method has been used frequently in many fields, especially in solving problems created by nonlinear equations. In this study, we examined the (q)-analog of this method. We define the method as the (q)-modified differential transformation method ((q)-MDTM). By creating (q)-analogs of some problems solved by Laplace transform and DTM in the literature, we are solving them with (q)-MDTM. In this sense, the study is an example of creating nonlinear (q)-differential equations and solving them by applying the (q)-modified method.
{"title":"q-Modified Differential Transform Method","authors":"F. Hıra","doi":"10.1134/s0965542524700428","DOIUrl":"https://doi.org/10.1134/s0965542524700428","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The modified or hybrid method is created by combining the differential transform method (DTM), Padé approximant, and Laplace transform. This method has been used frequently in many fields, especially in solving problems created by nonlinear equations. In this study, we examined the <span>(q)</span>-analog of this method. We define the method as the <span>(q)</span>-modified differential transformation method (<span>(q)</span>-MDTM). By creating <span>(q)</span>-analogs of some problems solved by Laplace transform and DTM in the literature, we are solving them with <span>(q)</span>-MDTM. In this sense, the study is an example of creating nonlinear <span>(q)</span>-differential equations and solving them by applying the <span>(q)</span>-modified method.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738205","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}
Pub Date : 2024-07-18DOI: 10.1134/s0965542524700544
R. K. Gaydukov, V. G. Danilov
Abstract
Mathematical modeling of the ice–water phase transition during liquid flow inside a pipe with a small ice buildup on the wall at high Reynolds numbers is considered. As a mathematical model describing the dynamics of the phase transition, a double-deck boundary layer model and a phase field system are used. Results of numerical simulation are presented.
{"title":"Modeling the Ice–Water Phase Transition in a Tube with Small Ice Buildups on the Wall","authors":"R. K. Gaydukov, V. G. Danilov","doi":"10.1134/s0965542524700544","DOIUrl":"https://doi.org/10.1134/s0965542524700544","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Mathematical modeling of the ice–water phase transition during liquid flow inside a pipe with a small ice buildup on the wall at high Reynolds numbers is considered. As a mathematical model describing the dynamics of the phase transition, a double-deck boundary layer model and a phase field system are used. Results of numerical simulation are presented.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738283","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}
Pub Date : 2024-07-18DOI: 10.1134/s0965542524700441
Yu. Yu. Ogorodnikov, R. A. Rudakov, D. M. Khachai, M. Yu. Khachai
Abstract
The design of fault-tolerant production and delivery systems is one of the priority areas in modern operations research. The traditional approach to modeling such systems is based on the use of stochastic models that describe the choice of a possible scenario of actions in the event of problems in a production or transportation network. Along with a number of advantages, this approach has a known drawback. The occurrence of problems of an unknown nature that can jeopardize the performance of the entire simulated system significantly complicates its use. This paper introduces the minimax problem of constructing fault-tolerant production plans (reliable production process design problem, RPPDP), the purpose of which is to ensure the uninterrupted operation of a distributed production system with minimal guaranteed cost. It is shown that the RPPDP is NP-hard in the strong sense and remains intractable under quite specific conditions. To find exact and approximate solutions with accuracy guarantees for this problem, branch-and-bound methods are developed based on the proposed compact model of the mixed integer linear program (MILP) and novel heuristic of adaptive search in large neighborhoods (adaptive large neighborhood search, ALNS) as part of extensions of the well-known Gurobi MIP solver. The high performance and complementarity of the proposed algorithms is confirmed by the results of numerical experiments carried out on a public library of benchmarking instances developed by the authors based on instances from the PCGTSPLIB library.
{"title":"Fault-Tolerant Families of Production Plans: Mathematical Model, Computational Complexity, and Branch-and-Bound Algorithms","authors":"Yu. Yu. Ogorodnikov, R. A. Rudakov, D. M. Khachai, M. Yu. Khachai","doi":"10.1134/s0965542524700441","DOIUrl":"https://doi.org/10.1134/s0965542524700441","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The design of fault-tolerant production and delivery systems is one of the priority areas in modern operations research. The traditional approach to modeling such systems is based on the use of stochastic models that describe the choice of a possible scenario of actions in the event of problems in a production or transportation network. Along with a number of advantages, this approach has a known drawback. The occurrence of problems of an unknown nature that can jeopardize the performance of the entire simulated system significantly complicates its use. This paper introduces the minimax problem of constructing fault-tolerant production plans (reliable production process design problem, RPPDP), the purpose of which is to ensure the uninterrupted operation of a distributed production system with minimal guaranteed cost. It is shown that the RPPDP is NP-hard in the strong sense and remains intractable under quite specific conditions. To find exact and approximate solutions with accuracy guarantees for this problem, branch-and-bound methods are developed based on the proposed compact model of the mixed integer linear program (MILP) and novel heuristic of adaptive search in large neighborhoods (adaptive large neighborhood search, ALNS) as part of extensions of the well-known Gurobi MIP solver. The high performance and complementarity of the proposed algorithms is confirmed by the results of numerical experiments carried out on a public library of benchmarking instances developed by the authors based on instances from the PCGTSPLIB library.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738208","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}
Pub Date : 2024-07-18DOI: 10.1134/s0965542524700490
Ya. M. Dymarskii
Abstract
The functional of eigenvalues on the manifold of periodic potentials is described analytically and topologically.
摘要 对周期势流形上的特征值函数进行了分析和拓扑描述。
{"title":"Functionals of Eigenvalues on the Manifold of Potentials","authors":"Ya. M. Dymarskii","doi":"10.1134/s0965542524700490","DOIUrl":"https://doi.org/10.1134/s0965542524700490","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The functional of eigenvalues on the manifold of periodic potentials is described analytically and topologically.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738214","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}
Pub Date : 2024-07-18DOI: 10.1134/s0965542524700507
S. I. Sakharov
Abstract
Initial-boundary value problems are considered for homogeneous parabolic systems with Dini-continuous coefficients and zero initial conditions in a semibounded plane domain with a nonsmooth lateral boundary admitting cusps, on which general boundary conditions with variable coefficients are given. A theorem on unique classical solvability of these problems in the space of functions that are continuous and bounded together with their first spatial derivatives in the closure of the domain is proved by applying the boundary integral equation method. A representation of the resulting solutions in the form of vector single-layer potentials is given.
{"title":"Initial-Boundary Value Problems for Parabolic Systems in a Semibounded Plane Domain with General Boundary Conditions","authors":"S. I. Sakharov","doi":"10.1134/s0965542524700507","DOIUrl":"https://doi.org/10.1134/s0965542524700507","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Initial-boundary value problems are considered for homogeneous parabolic systems with Dini-continuous coefficients and zero initial conditions in a semibounded plane domain with a nonsmooth lateral boundary admitting cusps, on which general boundary conditions with variable coefficients are given. A theorem on unique classical solvability of these problems in the space of functions that are continuous and bounded together with their first spatial derivatives in the closure of the domain is proved by applying the boundary integral equation method. A representation of the resulting solutions in the form of vector single-layer potentials is given.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738284","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}
Pub Date : 2024-07-18DOI: 10.1134/s0965542524700489
O. V. Baskov, D. K. Potapov
Abstract
We examine the existence of solutions to the Sturm–Liouville problem with a non-self-adjoint differential operator and discontinuous nonlinearity in the phase variable. For positive values of the spectral parameter, theorems on the existence of nontrivial (positive and negative) solutions of the problem are proved. Examples illustrating the theorems are given.
{"title":"Existence of Solutions to the Non-Self-Adjoint Sturm–Liouville Problem with Discontinuous Nonlinearity","authors":"O. V. Baskov, D. K. Potapov","doi":"10.1134/s0965542524700489","DOIUrl":"https://doi.org/10.1134/s0965542524700489","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>We examine the existence of solutions to the Sturm–Liouville problem with a non-self-adjoint differential operator and discontinuous nonlinearity in the phase variable. For positive values of the spectral parameter, theorems on the existence of nontrivial (positive and negative) solutions of the problem are proved. Examples illustrating the theorems are given.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738213","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}
Pub Date : 2024-07-18DOI: 10.1134/s0965542524700453
V. K. Tolstykh
Abstract
The problem of controllability for problems of optimal control and optimization of distributed parameter systems governed by partial differential equations is considered. The concept of controllability understood as Tikhonov correctness for solving optimization problems is introduced. A theorem formulating controllability conditions for directly solving optimization problems (direct minimization of the objective functional) is presented. A test example of the numerical solution of the optimization problem for a nonlinear hyperbolic system describing the unsteady flow of water in an open channel is considered. The analysis of controllability is demonstrated that ensures the correctness of the problem solution and high accuracy of optimization of the distributed friction coefficient in the flow equations.
{"title":"Controllability of Distributed Parameter Systems","authors":"V. K. Tolstykh","doi":"10.1134/s0965542524700453","DOIUrl":"https://doi.org/10.1134/s0965542524700453","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The problem of controllability for problems of optimal control and optimization of distributed parameter systems governed by partial differential equations is considered. The concept of controllability understood as Tikhonov correctness for solving optimization problems is introduced. A theorem formulating controllability conditions for directly solving optimization problems (direct minimization of the objective functional) is presented. A test example of the numerical solution of the optimization problem for a nonlinear hyperbolic system describing the unsteady flow of water in an open channel is considered. The analysis of controllability is demonstrated that ensures the correctness of the problem solution and high accuracy of optimization of the distributed friction coefficient in the flow equations.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738210","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}
Pub Date : 2024-07-18DOI: 10.1134/s096554252470057x
H. Rouah, Y. Joundy, A. Taik
Abstract
The influence of certain parameters on the stability conditions of the reaction front in a porous medium is studied in this article. The mathematical model includes the heat equation, the concentration equation and the equations of motion under the Boussinesq–Darcy approximation. An asymptotic analysis was carried out using the method of Zeldovich and Frank-Kamentskii to obtain the interface problem. A stability analysis was performed to determine a linearized problem that will be solved numerically using the finite difference method with an implicit scheme. This will allow to conclude the effect of each parameter on the stability of the front, in particular the amplitude and the frequency of the vibrations.
{"title":"Asymptotic and Stability Analysis of Reaction Fronts","authors":"H. Rouah, Y. Joundy, A. Taik","doi":"10.1134/s096554252470057x","DOIUrl":"https://doi.org/10.1134/s096554252470057x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The influence of certain parameters on the stability conditions of the reaction front in a porous medium is studied in this article. The mathematical model includes the heat equation, the concentration equation and the equations of motion under the Boussinesq–Darcy approximation. An asymptotic analysis was carried out using the method of Zeldovich and Frank-Kamentskii to obtain the interface problem. A stability analysis was performed to determine a linearized problem that will be solved numerically using the finite difference method with an implicit scheme. This will allow to conclude the effect of each parameter on the stability of the front, in particular the amplitude and the frequency of the vibrations.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738292","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}
Pub Date : 2024-06-13DOI: 10.1134/s0965542524700283
D. A. Maslov
Abstract
The paper proposes a new method for numerically solving nonlinear stiff problems, based on the numerical implementation of the method of holomorphic regularization of the Cauchy problem for singularly perturbed nonlinear differential equations.
{"title":"About One Method for Numerical Solution of the Cauchy Problem for Singularly Perturbed Differential Equations","authors":"D. A. Maslov","doi":"10.1134/s0965542524700283","DOIUrl":"https://doi.org/10.1134/s0965542524700283","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper proposes a new method for numerically solving nonlinear stiff problems, based on the numerical implementation of the method of holomorphic regularization of the Cauchy problem for singularly perturbed nonlinear differential equations.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523611","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}
Pub Date : 2024-06-13DOI: 10.1134/s0965542524700337
Y. Alagöz, G. Özyurt
Abstract
The main aim of this paper is to introduce generalized quaternions with hyper-number coefficients. For this, firstly, a new number system is defined, which is the generalization of bicomplex numbers, hyper-double numbers and hyper-dual numbers. And any element of this generalization is called a hyper-number. Then, real matrix representation and vector representation of a hyper-number are given. Secondly, hyper-number generalized quaternions and their algebraic properties are introduced. For a hyper-number generalized quaternion, (4 times 4) real generalized quaternion matrix representation is presented. Next, because of lack of commutativity, for a hyper-number generalized quaternion, two different hyper-number matrix representations are calculated. Moreover, real matrix representations of a hyper-number generalized quaternion is expressed by matrix representation of a hyper-number. Finally, vector representations of a hyper-number generalized quaternion are given and properties of this representations are investigated.
{"title":"Hyper-Number Generalized Quaternions","authors":"Y. Alagöz, G. Özyurt","doi":"10.1134/s0965542524700337","DOIUrl":"https://doi.org/10.1134/s0965542524700337","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The main aim of this paper is to introduce generalized quaternions with hyper-number coefficients. For this, firstly, a new number system is defined, which is the generalization of bicomplex numbers, hyper-double numbers and hyper-dual numbers. And any element of this generalization is called a hyper-number. Then, real matrix representation and vector representation of a hyper-number are given. Secondly, hyper-number generalized quaternions and their algebraic properties are introduced. For a hyper-number generalized quaternion, <span>(4 times 4)</span> real generalized quaternion matrix representation is presented. Next, because of lack of commutativity, for a hyper-number generalized quaternion, two different hyper-number matrix representations are calculated. Moreover, real matrix representations of a hyper-number generalized quaternion is expressed by matrix representation of a hyper-number. Finally, vector representations of a hyper-number generalized quaternion are given and properties of this representations are investigated.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501879","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}
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