The classification of inverse problems of dynamics in the class of ordinary differential equations is given in the Galiullin’s monograph. The problem studied in this paper belongs to the main inverse problem of dynamics, but already in the class of second-order stochastic differential equations of the Ito type. Stochastic equations of the Lagrangian structure are constructed according to the given properties of motion under the assumption that the random perturbing forces belong to the class of processes with independent increments. The problem is solved as follows: First, a second-order Ito differential equation is constructed so that the properties of motion are the integral manifold of the constructed stochastic equation. At this stage, the quasi-inversion method, Erugin’s method and Ito’s rule of stochastic differentiation of a complex function are used. Then, by applying the constructed Ito equation, an equivalent stochastic equation of the Lagrangian structure is constructed. The necessary and sufficient conditions for the solvability of the problem of constructing the stochastic equation of the Lagrangian structure are illustrated by the example of the problem of constructing the Lagrange function from a motion property of an artificial Earth satellite under the action of gravitational forces and aerodynamic forces.
{"title":"Construction of the differential equations system of the program motion in Lagrangian variables in the presence of random perturbations","authors":"M. Tleubergenov, G. Vassilina, D. Azhymbaev","doi":"10.31489/2022m1/118-126","DOIUrl":"https://doi.org/10.31489/2022m1/118-126","url":null,"abstract":"The classification of inverse problems of dynamics in the class of ordinary differential equations is given in the Galiullin’s monograph. The problem studied in this paper belongs to the main inverse problem of dynamics, but already in the class of second-order stochastic differential equations of the Ito type. Stochastic equations of the Lagrangian structure are constructed according to the given properties of motion under the assumption that the random perturbing forces belong to the class of processes with independent increments. The problem is solved as follows: First, a second-order Ito differential equation is constructed so that the properties of motion are the integral manifold of the constructed stochastic equation. At this stage, the quasi-inversion method, Erugin’s method and Ito’s rule of stochastic differentiation of a complex function are used. Then, by applying the constructed Ito equation, an equivalent stochastic equation of the Lagrangian structure is constructed. The necessary and sufficient conditions for the solvability of the problem of constructing the stochastic equation of the Lagrangian structure are illustrated by the example of the problem of constructing the Lagrange function from a motion property of an artificial Earth satellite under the action of gravitational forces and aerodynamic forces.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48453781","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 study formal characters of simple modules with singular highest weights over classical Lie algebras of type B over an algebraically closed field of characteristic p ≥ h, where h is the Coxeter number. Assume that the highest weights of these simple modules are restricted. We have given a description of their formal characters. In particular, we have obtained some new examples of simple Weyl modules. In the restricted region, the representation theory of algebraic groups and its Lie algebras are equivalent. Therefore, we can use the tools of the representation theory of semisimple and simply-connected algebraic groups in positive characteristic. To describe the formal characters of simple modules, we construct Jantzen filtrations of Weyl modules of the corresponding highest weights.
{"title":"On simple modules with singular highest weights for so2l+1(K)","authors":"S. Ibraev, A. Seitmuratov, L. Kainbayeva","doi":"10.31489/2022m1/52-65","DOIUrl":"https://doi.org/10.31489/2022m1/52-65","url":null,"abstract":"In this paper, we study formal characters of simple modules with singular highest weights over classical Lie algebras of type B over an algebraically closed field of characteristic p ≥ h, where h is the Coxeter number. Assume that the highest weights of these simple modules are restricted. We have given a description of their formal characters. In particular, we have obtained some new examples of simple Weyl modules. In the restricted region, the representation theory of algebraic groups and its Lie algebras are equivalent. Therefore, we can use the tools of the representation theory of semisimple and simply-connected algebraic groups in positive characteristic. To describe the formal characters of simple modules, we construct Jantzen filtrations of Weyl modules of the corresponding highest weights.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45173834","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}
A. Assanova, S. Zhumatov, S. Mynbayeva, S. Karakenova
The paper proposes a constructive method to solve a nonlinear boundary value problem for a Fredholm integro-differential equation. Using D.S. Dzhumabaev parametrization method, the problem under consideration is transformed into an equivalent boundary value problem for a system of nonlinear integrodifferential equations with parameters on the subintervals. When applying the parametrization method to a nonlinear Fredholm integro-differential equation, the intermediate problem is a special Cauchy problem for a system of nonlinear integro-differential equations with parameters. By substitution the solution to the special Cauchy problem with parameters into the boundary condition and the continuity conditions of the solution to the original problem at the interior partition points, we construct a system of nonlinear algebraic equations in parameters. It is proved that the solvability of this system provides the existence of a solution to the original boundary value problem. The iterative methods are used to solve both the constructed system of algebraic equations in parameters and the special Cauchy problem. An algorithm for solving boundary value problem under consideration is provided.
{"title":"On solvability of boundary value problem for a nonlinear Fredholm integro-differential equation","authors":"A. Assanova, S. Zhumatov, S. Mynbayeva, S. Karakenova","doi":"10.31489/2022m1/25-34","DOIUrl":"https://doi.org/10.31489/2022m1/25-34","url":null,"abstract":"The paper proposes a constructive method to solve a nonlinear boundary value problem for a Fredholm integro-differential equation. Using D.S. Dzhumabaev parametrization method, the problem under consideration is transformed into an equivalent boundary value problem for a system of nonlinear integrodifferential equations with parameters on the subintervals. When applying the parametrization method to a nonlinear Fredholm integro-differential equation, the intermediate problem is a special Cauchy problem for a system of nonlinear integro-differential equations with parameters. By substitution the solution to the special Cauchy problem with parameters into the boundary condition and the continuity conditions of the solution to the original problem at the interior partition points, we construct a system of nonlinear algebraic equations in parameters. It is proved that the solvability of this system provides the existence of a solution to the original boundary value problem. The iterative methods are used to solve both the constructed system of algebraic equations in parameters and the special Cauchy problem. An algorithm for solving boundary value problem under consideration is provided.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":"20 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41291832","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}
A. Yeshkeyev, I.O. Tungushbayeva, M. T. Kassymetova
The paper aims to study the model-theoretic properties of differentially closed fields of zero and positive characteristics in framework of study of Jonsson theories. The main attention is paid to the amalgam and joint embedding properties of DCF theory as specific features of Jonsson theories, namely, the implication of JEP from AP. The necessity is identified and justified by importance of information about the mentioned properties for certain theories to obtain their detailed model-theoretic description. At the same time, the current apparatus for studying incomplete theories (Jonsson theories are generally incomplete) is not sufficiently developed. The following results have been obtained: The subclasses of Jonsson theories are determined from the point of view of joint embedding and amalgam properties. Within the exploration of one of these classes, namely the AP-theories, that the theories of differential and differentially closed fields of characteristic 0, differentially perfect and differentially closed fields of fixed positive characteristic are shown to be Jonsson and perfect. Along with this, the theory of differential fields of positive characteristic is considered as an example of an AP-theory that is not Jonsson, but has the model companion, which is perfect Jonsson theory, and the sufficient condition for the theory of differential fields is formulated in the context of being Jonsson.
{"title":"Connection between the amalgam and joint embedding properties","authors":"A. Yeshkeyev, I.O. Tungushbayeva, M. T. Kassymetova","doi":"10.31489/2022m1/127-135","DOIUrl":"https://doi.org/10.31489/2022m1/127-135","url":null,"abstract":"The paper aims to study the model-theoretic properties of differentially closed fields of zero and positive characteristics in framework of study of Jonsson theories. The main attention is paid to the amalgam and joint embedding properties of DCF theory as specific features of Jonsson theories, namely, the implication of JEP from AP. The necessity is identified and justified by importance of information about the mentioned properties for certain theories to obtain their detailed model-theoretic description. At the same time, the current apparatus for studying incomplete theories (Jonsson theories are generally incomplete) is not sufficiently developed. The following results have been obtained: The subclasses of Jonsson theories are determined from the point of view of joint embedding and amalgam properties. Within the exploration of one of these classes, namely the AP-theories, that the theories of differential and differentially closed fields of characteristic 0, differentially perfect and differentially closed fields of fixed positive characteristic are shown to be Jonsson and perfect. Along with this, the theory of differential fields of positive characteristic is considered as an example of an AP-theory that is not Jonsson, but has the model companion, which is perfect Jonsson theory, and the sufficient condition for the theory of differential fields is formulated in the context of being Jonsson.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45612436","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 obtain exact solutions of a new modification of the Schrödinger equation related to the Bessel q-operator. The theorem is proved on the existence of this solution in the Sobolev-type space W^2_q(R^+_q) in the q-calculus. The results on correctness in the corresponding spaces of the Sobolev-type are obtained. For simplicity, we give results involving fractional q-difference equations of real order a > 0 and given real numbers in q-calculus. Numerical treatment of fractional q-difference equations is also investigated. The obtained results can be used in this field and be supplement for studies in this field.
{"title":"The Schrödinger equations generated by q-Bessel operator in quantum calculus","authors":"S. Shaimardan, N. Tokmagambetov","doi":"10.31489/2022m1/102-108","DOIUrl":"https://doi.org/10.31489/2022m1/102-108","url":null,"abstract":"In this paper, we obtain exact solutions of a new modification of the Schrödinger equation related to the Bessel q-operator. The theorem is proved on the existence of this solution in the Sobolev-type space W^2_q(R^+_q) in the q-calculus. The results on correctness in the corresponding spaces of the Sobolev-type are obtained. For simplicity, we give results involving fractional q-difference equations of real order a > 0 and given real numbers in q-calculus. Numerical treatment of fractional q-difference equations is also investigated. The obtained results can be used in this field and be supplement for studies in this field.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44149752","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 considers a difference problem for a mixed-type equation, to which a problem of integral geometry for a family of curves satisfying certain regularity conditions is reduced. These problems are related to numerous applications, including interpretation problem of seismic data, problem of interpretation of Xray images, problems of computed tomography and technical diagnostics. The study of difference analogues of integral geometry problems has specific difficulties associated with the fact that for finite-difference analogues of partial derivatives, basic relations are performed with a certain shift in the discrete variable. In this regard, many relations obtained in a continuous formulation, when transitioned to a discrete analogue, have a more complex and cumbersome form, which requires additional studies of the resulting terms with a shift. Another important feature of the integral geometry problem is the absence of a theorem for existence of a solution in general case. Consequently, the paper uses the concept of correctness according to A.N.Tikhonov, particularly, it is assumed that there is a solution to the problem of integral geometry and its differential-difference analogue. The stability estimate of the difference analogue of the boundary value problem for a mixed-type equation obtained in this work is vital for understanding the effectiveness of numerical methods for solving problems of geotomography, medical tomography, flaw detection, etc. It also has a great practical significance in solving multidimensional inverse problems of acoustics, seismic exploration.
{"title":"On the stability of the difference analogue of the boundary value problem for a mixed type equation","authors":"G. Bakanov, S. Meldebekova","doi":"10.31489/2022m1/35-42","DOIUrl":"https://doi.org/10.31489/2022m1/35-42","url":null,"abstract":"This paper considers a difference problem for a mixed-type equation, to which a problem of integral geometry for a family of curves satisfying certain regularity conditions is reduced. These problems are related to numerous applications, including interpretation problem of seismic data, problem of interpretation of Xray images, problems of computed tomography and technical diagnostics. The study of difference analogues of integral geometry problems has specific difficulties associated with the fact that for finite-difference analogues of partial derivatives, basic relations are performed with a certain shift in the discrete variable. In this regard, many relations obtained in a continuous formulation, when transitioned to a discrete analogue, have a more complex and cumbersome form, which requires additional studies of the resulting terms with a shift. Another important feature of the integral geometry problem is the absence of a theorem for existence of a solution in general case. Consequently, the paper uses the concept of correctness according to A.N.Tikhonov, particularly, it is assumed that there is a solution to the problem of integral geometry and its differential-difference analogue. The stability estimate of the difference analogue of the boundary value problem for a mixed-type equation obtained in this work is vital for understanding the effectiveness of numerical methods for solving problems of geotomography, medical tomography, flaw detection, etc. It also has a great practical significance in solving multidimensional inverse problems of acoustics, seismic exploration.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48662393","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, in the space L_2(R^2), we study a third-order differential operator with continuous coefficients in R(−∞, +∞). Here, these coefficients can be unlimited functions at infinity. In addition under some restrictions on the coefficients, the bounded invertibility of the given operator is proved and a coercive estimate is obtained, i.e. separability is proved.
{"title":"Separability of the third-order differential operator given on the whole plane","authors":"A. O. Suleimbekova","doi":"10.31489/2022m1/109-117","DOIUrl":"https://doi.org/10.31489/2022m1/109-117","url":null,"abstract":"In this paper, in the space L_2(R^2), we study a third-order differential operator with continuous coefficients in R(−∞, +∞). Here, these coefficients can be unlimited functions at infinity. In addition under some restrictions on the coefficients, the bounded invertibility of the given operator is proved and a coercive estimate is obtained, i.e. separability is proved.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49114370","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 study the interpolation properties of the net spaces N_p,q(M), in the case when M is a sufficiently general arbitrary system of measurable subsets from R^n. The integral Urysohn operator is considered. This operator generalizes all linear, integral operators, and non-linear integral operators. The Urysohn operator is not a quasilinear or subadditive operator. Therefore, the classical interpolation theorems for these operators do not hold. A certain analogue of the Marcinkiewicz-type interpolation theorem for this class of operators is obtained. This theorem allows to obtain, in a sense, a strong estimate for Urysohn operators in net spaces from weak estimates for these operators in net spaces with local nets. For example, in order for the Urysohn integral operator in a net space, where the net is the set of all balls in R^n, it is sufficient for it to be of weak type for net spaces, where the net is concentric balls.
{"title":"Interpolation of nonlinear integral Urysohn operators in net spaces","authors":"A. H. Kalidolday, E. Nursultanov","doi":"10.31489/2022m1/66-73","DOIUrl":"https://doi.org/10.31489/2022m1/66-73","url":null,"abstract":"In this paper, we study the interpolation properties of the net spaces N_p,q(M), in the case when M is a sufficiently general arbitrary system of measurable subsets from R^n. The integral Urysohn operator is considered. This operator generalizes all linear, integral operators, and non-linear integral operators. The Urysohn operator is not a quasilinear or subadditive operator. Therefore, the classical interpolation theorems for these operators do not hold. A certain analogue of the Marcinkiewicz-type interpolation theorem for this class of operators is obtained. This theorem allows to obtain, in a sense, a strong estimate for Urysohn operators in net spaces from weak estimates for these operators in net spaces with local nets. For example, in order for the Urysohn integral operator in a net space, where the net is the set of all balls in R^n, it is sufficient for it to be of weak type for net spaces, where the net is concentric balls.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47968069","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 article is devoted to the solvability of degenerate nonlinear equations of pseudoparabolic type. Such problems appear naturally in physical and biological models. The article aims to study the solvability in the classes of regular solutions of (all derivatives generalized in the sense of S.L. Sobolev included in the equation) initial-boundary value problems for differential equations. For the problems under consideration, we have found conditions on parameters ensuring the existence of solutions and we have proved existence and uniqueness theorems. The main method for proving the solvability of boundary value problems is the regularization method.
{"title":"Solvability of an initial-boundary value problem for a nonlinear pseudoparabolic equation with degeneration","authors":"S. Aitzhanov, Zhandos T. Tileuberdi, G. Sanat","doi":"10.31489/2022m1/4-12","DOIUrl":"https://doi.org/10.31489/2022m1/4-12","url":null,"abstract":"This article is devoted to the solvability of degenerate nonlinear equations of pseudoparabolic type. Such problems appear naturally in physical and biological models. The article aims to study the solvability in the classes of regular solutions of (all derivatives generalized in the sense of S.L. Sobolev included in the equation) initial-boundary value problems for differential equations. For the problems under consideration, we have found conditions on parameters ensuring the existence of solutions and we have proved existence and uniqueness theorems. The main method for proving the solvability of boundary value problems is the regularization method.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47395942","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 theory of fractional calculus developed rapidly as the applications of this branch are extensive nowadays. There is no discipline of modern engineering and science that remains untouched by the techniques of fractional calculus. In fact, one could argue that real world processes are fractional order systems in general. In this article, we obtain the semi-integrals of certain algebraic functions in terms of difference of two complete elliptic integrals of different kinds by using series manipulation technique.
{"title":"Semi-integration of certain algebraic expressions","authors":"M. I. Qureshi, J. Majid","doi":"10.31489/2022m1/96-101","DOIUrl":"https://doi.org/10.31489/2022m1/96-101","url":null,"abstract":"The theory of fractional calculus developed rapidly as the applications of this branch are extensive nowadays. There is no discipline of modern engineering and science that remains untouched by the techniques of fractional calculus. In fact, one could argue that real world processes are fractional order systems in general. In this article, we obtain the semi-integrals of certain algebraic functions in terms of difference of two complete elliptic integrals of different kinds by using series manipulation technique.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47457920","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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