In this paper, we consider a boundary control problem for a parabolic equation in a segment. In the part of the domain’s bound it is a given value of the solution and it is required to find controls to get the average value of the solution. The given control problem is reduced to a system of Volterra integral equations of the first kind. By the mathematical-physics methods it is proved that like this control functions exist over some domain, the necessary estimates were found and obtained.
{"title":"Boundary control problem for the heat transfer equation associated with heating process of a rod","authors":"F. Dekhkonov","doi":"10.31489/2023m2/63-71","DOIUrl":"https://doi.org/10.31489/2023m2/63-71","url":null,"abstract":"In this paper, we consider a boundary control problem for a parabolic equation in a segment. In the part of the domain’s bound it is a given value of the solution and it is required to find controls to get the average value of the solution. The given control problem is reduced to a system of Volterra integral equations of the first kind. By the mathematical-physics methods it is proved that like this control functions exist over some domain, the necessary estimates were found and obtained.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42893843","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}
The existence of a finite identity basis for any finite lattice was established by R. McKenzie in 1970, but the analogous statement for quasi-identities is incorrect. So, there is a finite lattice that does not have a finite quasi-identity basis and, V.A. Gorbunov and D.M. Smirnov asked which finite lattices have finite quasiidentity bases. In 1984 V.I. Tumanov conjectured that a proper quasivariety generated by a finite modular lattice is not finitely based. He also found two conditions for quasivarieties which provide this conjecture. We construct a finite modular lattice that does not satisfy Tumanov’s conditions but quasivariety generated by this lattice is not finitely based.
R. McKenzie于1970年建立了任意有限格的有限恒等基的存在性，但拟恒等基的类似陈述是不正确的。所以，存在一个有限格它没有有限的拟恒等基，V.A. Gorbunov和D.M. Smirnov问哪些有限格有有限的拟恒等基。1984年，V.I. Tumanov推测由有限模格生成的适当拟变簇不是有限基的。他还发现了两个准变项的条件来支持这个猜想。构造了一个不满足图马诺夫条件的有限模格，但其生成的准变分不是有限基的。
{"title":"On quasi-identities of finite modular lattices. II","authors":"A. Basheyeva, S. Lutsak","doi":"10.31489/2023m2/45-52","DOIUrl":"https://doi.org/10.31489/2023m2/45-52","url":null,"abstract":"The existence of a finite identity basis for any finite lattice was established by R. McKenzie in 1970, but the analogous statement for quasi-identities is incorrect. So, there is a finite lattice that does not have a finite quasi-identity basis and, V.A. Gorbunov and D.M. Smirnov asked which finite lattices have finite quasiidentity bases. In 1984 V.I. Tumanov conjectured that a proper quasivariety generated by a finite modular lattice is not finitely based. He also found two conditions for quasivarieties which provide this conjecture. We construct a finite modular lattice that does not satisfy Tumanov’s conditions but quasivariety generated by this lattice is not finitely based.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45140136","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}
The paper investigates the issues of solvability and spectral properties of local and nonlocal problems for the fractional order diffusion-wave equation. The regular and strong solvability to problems stated in the domains, both with characteristic and non-characteristic boundaries are proved. Unambiguous solvability is established and theorems on the existence of eigenvalues or the Volterra property of the problems under consideration are proved.
{"title":"Spectral properties of local and nonlocal problems for the diffusion-wave equation of fractional order","authors":"N. Adil, A. Berdyshev","doi":"10.31489/2023m2/4-20","DOIUrl":"https://doi.org/10.31489/2023m2/4-20","url":null,"abstract":"The paper investigates the issues of solvability and spectral properties of local and nonlocal problems for the fractional order diffusion-wave equation. The regular and strong solvability to problems stated in the domains, both with characteristic and non-characteristic boundaries are proved. Unambiguous solvability is established and theorems on the existence of eigenvalues or the Volterra property of the problems under consideration are proved.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42883346","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}
The paper proposes a method for solving the problem of optimal performance for linear systems of ordinary differential equations in the presence of phase and integral restrictions, when the initial and final states of the system are elements of given convex closed sets, taking into account the control value restriction. The presented work refers to the mathematical theory of optimal processes from L.S. Pontryagin and his students and the theory of controllability of dynamic systems from R.E. Kalman. We study the problem of optimal speed for linear systems with boundary conditions from given sets close to the presence of phase and integral constraints, as well as constraints on the control value. A theory of the boundary value problem has been created and a method for solving it based on the study of solvability and the construction of a general solution to the Fredholm integral equation of the first kind has been developed. The main results are the distribution of all controls’ sets, each subject of which transfers the trajectory of the system from any initial state to any final state; reducing the initial boundary point to a special initial optimal control problem; constructing a system of algorithms for the gamma-algorithm study on the derivation of problems and rational execution with restrictions on the solution of the optimal speed’ problem with restrictions.
{"title":"Controllability and optimal speed-in-action of linear systems with boundary conditions","authors":"S. Aisagaliev, G.T. Korpebay","doi":"10.31489/2023m2/21-34","DOIUrl":"https://doi.org/10.31489/2023m2/21-34","url":null,"abstract":"The paper proposes a method for solving the problem of optimal performance for linear systems of ordinary differential equations in the presence of phase and integral restrictions, when the initial and final states of the system are elements of given convex closed sets, taking into account the control value restriction. The presented work refers to the mathematical theory of optimal processes from L.S. Pontryagin and his students and the theory of controllability of dynamic systems from R.E. Kalman. We study the problem of optimal speed for linear systems with boundary conditions from given sets close to the presence of phase and integral constraints, as well as constraints on the control value. A theory of the boundary value problem has been created and a method for solving it based on the study of solvability and the construction of a general solution to the Fredholm integral equation of the first kind has been developed. The main results are the distribution of all controls’ sets, each subject of which transfers the trajectory of the system from any initial state to any final state; reducing the initial boundary point to a special initial optimal control problem; constructing a system of algorithms for the gamma-algorithm study on the derivation of problems and rational execution with restrictions on the solution of the optimal speed’ problem with restrictions.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42117304","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}
The work is devoted to the problem of setting new boundary and internal boundary value problems for hyperbolic equations. The consideration of these settings is given on the example of a wave equation. The research involves the d’Alembert method, the mean value theorem and the method of successive approximations. The paper formulates and studies a number of non-local problems summarizing the classical Goursat and Dardu tasks. Some of them are marginal, and the other part is internal-marginal, and in both cases both characteristic and uncharacteristic displacements are considered. It should also be noted that a number of problems discussed below arose as a special case in the construction of the theory of correct problems for the model loaded equation of string oscillation.
{"title":"On Some Non-local Boundary Value and Internal Boundary Value Problems for the String Oscillation Equation","authors":"A. Attaev","doi":"10.31489/2023m2/35-44","DOIUrl":"https://doi.org/10.31489/2023m2/35-44","url":null,"abstract":"The work is devoted to the problem of setting new boundary and internal boundary value problems for hyperbolic equations. The consideration of these settings is given on the example of a wave equation. The research involves the d’Alembert method, the mean value theorem and the method of successive approximations. The paper formulates and studies a number of non-local problems summarizing the classical Goursat and Dardu tasks. Some of them are marginal, and the other part is internal-marginal, and in both cases both characteristic and uncharacteristic displacements are considered. It should also be noted that a number of problems discussed below arose as a special case in the construction of the theory of correct problems for the model loaded equation of string oscillation.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47227655","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}
The paper considers the space of generalized fractional-maximal function, constructed on the basis of a rearrangement-invariant space. Two types of cones generated by a nonincreasing rearrangement of a generalized fractional-maximal function and equipped with positive homogeneous functionals are constructed. The question of embedding the space of generalized fractional-maximal function in a rearrangement invariant space is investigated. This question reduces to the embedding of the considered cone in the corresponding rearrangement-invariant spaces. In addition, conditions for covering a cone generated by a generalized fractional-maximal function by the cone generated by generalized Riesz potentials are given. Cones from non-increasing rearrangements of generalized potentials were previously considered in the works of M. Goldman, E. Bakhtigareeva, G. Karshygina and others.
{"title":"Cones generated by a generalized fractional maximal function","authors":"N. Bokayev, A. Gogatishvili, А.N. Abek","doi":"10.31489/2023m2/53-62","DOIUrl":"https://doi.org/10.31489/2023m2/53-62","url":null,"abstract":"The paper considers the space of generalized fractional-maximal function, constructed on the basis of a rearrangement-invariant space. Two types of cones generated by a nonincreasing rearrangement of a generalized fractional-maximal function and equipped with positive homogeneous functionals are constructed. The question of embedding the space of generalized fractional-maximal function in a rearrangement invariant space is investigated. This question reduces to the embedding of the considered cone in the corresponding rearrangement-invariant spaces. In addition, conditions for covering a cone generated by a generalized fractional-maximal function by the cone generated by generalized Riesz potentials are given. Cones from non-increasing rearrangements of generalized potentials were previously considered in the works of M. Goldman, E. Bakhtigareeva, G. Karshygina and others.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42201443","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}
The paper investigates a non-local boundary value problem for the Benjamin-Bona-Mahony equation. This equation is a nonlinear pseudoparabolic equation of the third order with a mixed derivative. To find a solution to this problem, an algorithm for finding an approximate solution is proposed. Sufficient conditions for the feasibility and convergence of the proposed algorithm are established, as well as the existence of an isolated solution of a non-local boundary value problem for a nonlinear equation. Estimates are obtained between the exact and approximate solution of this problem.
{"title":"On one approximate solution of a nonlocal boundary value problem for the Benjamin-Bon-Mahoney equation","authors":"A.M. Manat, N. Orumbayeva","doi":"10.31489/2023m2/84-92","DOIUrl":"https://doi.org/10.31489/2023m2/84-92","url":null,"abstract":"The paper investigates a non-local boundary value problem for the Benjamin-Bona-Mahony equation. This equation is a nonlinear pseudoparabolic equation of the third order with a mixed derivative. To find a solution to this problem, an algorithm for finding an approximate solution is proposed. Sufficient conditions for the feasibility and convergence of the proposed algorithm are established, as well as the existence of an isolated solution of a non-local boundary value problem for a nonlinear equation. Estimates are obtained between the exact and approximate solution of this problem.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44221338","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}
In this paper, we have solved several extremal problems of the best mean-square approximation of functions f on the semiaxis with a power-law weight. In the Hilbert space L^2 with a power-law weight t^2α+1 we obtain Jackson–Stechkin type inequalities between the value of the E_σ(f)-best approximation of a function f(t) by partial Hankel integrals of an order not higher than σ over the Bessel functions of the first kind and the k-th order generalized modulus of smoothnes ω_k(B^r f, t), where B is a second–order differential operator.
{"title":"Generalized Hankel shifts and exact Jackson–Stechkin inequalities in L2","authors":"T. E. Tileubayev","doi":"10.31489/2023m2/142-159","DOIUrl":"https://doi.org/10.31489/2023m2/142-159","url":null,"abstract":"In this paper, we have solved several extremal problems of the best mean-square approximation of functions f on the semiaxis with a power-law weight. In the Hilbert space L^2 with a power-law weight t^2α+1 we obtain Jackson–Stechkin type inequalities between the value of the E_σ(f)-best approximation of a function f(t) by partial Hankel integrals of an order not higher than σ over the Bessel functions of the first kind and the k-th order generalized modulus of smoothnes ω_k(B^r f, t), where B is a second–order differential operator.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47870747","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}
In the paper, the boundary value problem for the loaded heat equation is solved, and the loaded term is represented as the Riemann-Liouville derivative with respect to the time variable. The domain of the unknown function is the cone. The order of the derivative in the loaded term is less than 1, and the load moves along the lateral surface of the cone, that is in the domain of the desired function. The boundary value problem is studied in the case of the isotropy property in an angular coordinate (case of axial symmetry). The problem is reduced to the Volterra integral equation, which is solved by the method of the Laplace integral transformation. It is also shown by direct verification that the resulting function satisfies the boundary value problem.
{"title":"A fractionally loaded boundary value problem two-dimensional in the spatial variable","authors":"M. Kosmakova, K.A. Izhanova, L. Kasymova","doi":"10.31489/2023m2/72-83","DOIUrl":"https://doi.org/10.31489/2023m2/72-83","url":null,"abstract":"In the paper, the boundary value problem for the loaded heat equation is solved, and the loaded term is represented as the Riemann-Liouville derivative with respect to the time variable. The domain of the unknown function is the cone. The order of the derivative in the loaded term is less than 1, and the load moves along the lateral surface of the cone, that is in the domain of the desired function. The boundary value problem is studied in the case of the isotropy property in an angular coordinate (case of axial symmetry). The problem is reduced to the Volterra integral equation, which is solved by the method of the Laplace integral transformation. It is also shown by direct verification that the resulting function satisfies the boundary value problem.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44310161","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}
This paper addresses the solution of a differential-difference type equation having an interior layer behaviour. A difference scheme is suggested to solve this equation using a non-standard finite difference method. Finite differences are derived from the first and second order derivatives. Using these approximations, the given equation is discretized. The discretized equation is solved using the algorithm for the tridiagonal system. The method is examined for convergence. Numerical examples are illustrated to validate the method. Maximum errors in the solution, in contrast to the other methods are organized to justify the method. The layer behaviour in the solution of the examples is depicted in graphs.
{"title":"Numerical solution of differential-difference equations having an interior layer using nonstandard finite differences","authors":"R. Omkar, M. Lalu, K. Phaneendra","doi":"10.31489/2023m2/104-115","DOIUrl":"https://doi.org/10.31489/2023m2/104-115","url":null,"abstract":"This paper addresses the solution of a differential-difference type equation having an interior layer behaviour. A difference scheme is suggested to solve this equation using a non-standard finite difference method. Finite differences are derived from the first and second order derivatives. Using these approximations, the given equation is discretized. The discretized equation is solved using the algorithm for the tridiagonal system. The method is examined for convergence. Numerical examples are illustrated to validate the method. Maximum errors in the solution, in contrast to the other methods are organized to justify the method. The layer behaviour in the solution of the examples is depicted in graphs.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69839631","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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