M. Aripov, D. Utebaev, M.M. Kazimbetova, R.Sh. Yarlashov
Difference schemes of the finite difference method and the finite element method of high-order accuracy in time and space are proposed and investigated for a pseudo-parabolic Sobolev type equation. The order of accuracy in space is improved in two ways using the finite difference method and the finite element method. The order of accuracy of the scheme in time is improved by a special discretization of the time variable. The corresponding a priori estimates are determined and, on their basis, the accuracy estimates of the proposed difference schemes are obtained with sufficient smoothness of the solution to the original differential problem. Algorithms for the implementation of the constructed difference schemes are proposed.
{"title":"On convergence of difference schemes of high accuracy for one pseudo-parabolic Sobolev type equation","authors":"M. Aripov, D. Utebaev, M.M. Kazimbetova, R.Sh. Yarlashov","doi":"10.31489/2023m1/24-37","DOIUrl":"https://doi.org/10.31489/2023m1/24-37","url":null,"abstract":"Difference schemes of the finite difference method and the finite element method of high-order accuracy in time and space are proposed and investigated for a pseudo-parabolic Sobolev type equation. The order of accuracy in space is improved in two ways using the finite difference method and the finite element method. The order of accuracy of the scheme in time is improved by a special discretization of the time variable. The corresponding a priori estimates are determined and, on their basis, the accuracy estimates of the proposed difference schemes are obtained with sufficient smoothness of the solution to the original differential problem. Algorithms for the implementation of the constructed difference schemes are proposed.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42708812","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}
Using the natural cubic spline function, this paper finds the numerical solution of Volterra-Fredholm integral equations of the second kind. The proposed method is based on employing the natural cubic spline function of the unknown function at an arbitrary point and using the integration method to turn the VolterraFredholm integral equation into a system of linear equations concerning to the unknown function. An approximate solution can be easily established by solving the given system. This is accomplished with the help of a computer program that runs on Python 3.9.
{"title":"Solving Volterra-Fredholm integral equations by natural cubic spline function","authors":"S. Salim, K. Jwamer, R. Saeed","doi":"10.31489/2023m1/124-130","DOIUrl":"https://doi.org/10.31489/2023m1/124-130","url":null,"abstract":"Using the natural cubic spline function, this paper finds the numerical solution of Volterra-Fredholm integral equations of the second kind. The proposed method is based on employing the natural cubic spline function of the unknown function at an arbitrary point and using the integration method to turn the VolterraFredholm integral equation into a system of linear equations concerning to the unknown function. An approximate solution can be easily established by solving the given system. This is accomplished with the help of a computer program that runs on Python 3.9.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46850678","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 main goal of this study is to adapt the classical differential transformation method to solve new types of boundary value problems. The advantage of this method lies in its simplicity, since there is no need for discretization, perturbation or linearization of the differential equation being solved. It is an efficient technique for obtaining series solution for both linear and nonlinear differential equations and differs from the classical Taylor’s series method, which requires the calculation of the values of higher derivatives of given function. It is known that the differential transformation method is designed for solving single interval problems and it is not clear how to apply it to many-interval problems. In this paper we have adapted the classical differential transformation method for solving boundary value problems for two-interval differential equations. To substantiate the proposed new technique, a boundary value problem was solved for the Weber equation given on two non-intersecting segments with a common end, on which the left and right solutions were connected by two additional transmission conditions.
{"title":"Generalized differential transformation method for solving two-interval Weber equation subject to transmission conditions","authors":"M. Yücel, F. Muhtarov, O. Mukhtarov","doi":"10.31489/2023m1/168-176","DOIUrl":"https://doi.org/10.31489/2023m1/168-176","url":null,"abstract":"The main goal of this study is to adapt the classical differential transformation method to solve new types of boundary value problems. The advantage of this method lies in its simplicity, since there is no need for discretization, perturbation or linearization of the differential equation being solved. It is an efficient technique for obtaining series solution for both linear and nonlinear differential equations and differs from the classical Taylor’s series method, which requires the calculation of the values of higher derivatives of given function. It is known that the differential transformation method is designed for solving single interval problems and it is not clear how to apply it to many-interval problems. In this paper we have adapted the classical differential transformation method for solving boundary value problems for two-interval differential equations. To substantiate the proposed new technique, a boundary value problem was solved for the Weber equation given on two non-intersecting segments with a common end, on which the left and right solutions were connected by two additional transmission conditions.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43503669","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 consider an optimal control problem with a «pure», integral boundary condition. The Green’s function is constructed. Using contracting Banach mappings, a sufficient condition for the existence and uniqueness of a solution to one class of integral boundary value problems for fixed admissible controls is established. Using the functional increment method, the Pontryagin‘s maximum principle is proved. The first and second variations of the functional are calculated. Further, various necessary conditions for optimality of the second order are obtained by using variations of controls.
{"title":"An optimal control problem for the systems with integral boundary conditions","authors":"M. Mardanov, Y. Sharifov","doi":"10.31489/2023m1/110-123","DOIUrl":"https://doi.org/10.31489/2023m1/110-123","url":null,"abstract":"In this paper, we consider an optimal control problem with a «pure», integral boundary condition. The Green’s function is constructed. Using contracting Banach mappings, a sufficient condition for the existence and uniqueness of a solution to one class of integral boundary value problems for fixed admissible controls is established. Using the functional increment method, the Pontryagin‘s maximum principle is proved. The first and second variations of the functional are calculated. Further, various necessary conditions for optimality of the second order are obtained by using variations of controls.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47257017","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 solvability of the nonlinear boundary optimization problem has been investigated for the oscillation processes, described by the integro-differential equation in partial derivatives with Fredholm integral operator. It has been established that the components of the boundary vector control are defined as a solution to a system of nonlinear integral equations of a specific form, and the equations of this system have the property of equal relations. An algorithm for constructing a solution to the problem of nonlinear optimization has been developed.
{"title":"On the solvability of a nonlinear optimization problem with boundary vector control of oscillatory processes","authors":"E. Abdyldaeva, A. Kerimbekov, M.T. Zhaparov","doi":"10.31489/2023m1/5-13","DOIUrl":"https://doi.org/10.31489/2023m1/5-13","url":null,"abstract":"In the paper, the solvability of the nonlinear boundary optimization problem has been investigated for the oscillation processes, described by the integro-differential equation in partial derivatives with Fredholm integral operator. It has been established that the components of the boundary vector control are defined as a solution to a system of nonlinear integral equations of a specific form, and the equations of this system have the property of equal relations. An algorithm for constructing a solution to the problem of nonlinear optimization has been developed.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42719481","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 deals with the construction of minimizing sequences for the problem of minimizing a weakly nonlinearly perturbed quadratic performance index on trajectories of a weakly nonlinear system with threetempo state variables. For this purpose, the so-called direct scheme for constructing an asymptotic solution is used, which consists in immediate substituting the postulated asymptotic expansion of the solution into the problem conditions and constructing a series of optimal control problems (in the case under consideration, linear-quadratic ones), the solutions of which are terms of the asymptotic expansion of the solution of the original nonlinear control problem. An estimate is obtained for the proximity of the optimal trajectory to the trajectory of the equation of state when some asymptotic approximation to the optimal control is used as a control. An example is given that illustrates in detail the proposed scheme for constructing minimizing sequences.
{"title":"Minimizing sequences for a linear-quadratic control problem with three-tempo variables under weak nonlinear perturbations","authors":"G. Kurina, M. Kalashnikova","doi":"10.31489/2023m1/94-109","DOIUrl":"https://doi.org/10.31489/2023m1/94-109","url":null,"abstract":"The paper deals with the construction of minimizing sequences for the problem of minimizing a weakly nonlinearly perturbed quadratic performance index on trajectories of a weakly nonlinear system with threetempo state variables. For this purpose, the so-called direct scheme for constructing an asymptotic solution is used, which consists in immediate substituting the postulated asymptotic expansion of the solution into the problem conditions and constructing a series of optimal control problems (in the case under consideration, linear-quadratic ones), the solutions of which are terms of the asymptotic expansion of the solution of the original nonlinear control problem. An estimate is obtained for the proximity of the optimal trajectory to the trajectory of the equation of state when some asymptotic approximation to the optimal control is used as a control. An example is given that illustrates in detail the proposed scheme for constructing minimizing sequences.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42242678","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 work, we study two boundary value problems for involutary parabolic equation with the first and second kind conditions. We propose absolute stable difference schemes for numerical solutions of these boundary value problems. Actually the stability estimates for solutions of difference schemes are proved. Later error analysis for the numerical solution of both difference schemes are illustrated by test examples.
{"title":"Numerical solution of the boundary value problems for the parabolic equation with involution","authors":"A. Ashyralyev, C. Ashyralyyev, A. Ahmed","doi":"10.31489/2023m1/48-57","DOIUrl":"https://doi.org/10.31489/2023m1/48-57","url":null,"abstract":"In this work, we study two boundary value problems for involutary parabolic equation with the first and second kind conditions. We propose absolute stable difference schemes for numerical solutions of these boundary value problems. Actually the stability estimates for solutions of difference schemes are proved. Later error analysis for the numerical solution of both difference schemes are illustrated by test examples.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49633293","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 a rectangular domain, we consider a boundary value problem periodic in one variable for a system of partial differential equations of hyperbolic type. Introducing a new unknown function, this problem is reduced to an equivalent boundary value problem for an ordinary differential equation with an integral condition. Based on the parametrization method, new approaches to finding an approximate solution to an equivalent problem are proposed and its convergence is proved. This made it possible to establish conditions for the existence of a unique solution of a semiperiodic boundary value problem for a system of second-order hyperbolic equations.
{"title":"On one solution of a periodic boundary value problem for a hyperbolic equations","authors":"T. Tokmagambetova, N. Orumbayeva","doi":"10.31489/2023m1/141-155","DOIUrl":"https://doi.org/10.31489/2023m1/141-155","url":null,"abstract":"In a rectangular domain, we consider a boundary value problem periodic in one variable for a system of partial differential equations of hyperbolic type. Introducing a new unknown function, this problem is reduced to an equivalent boundary value problem for an ordinary differential equation with an integral condition. Based on the parametrization method, new approaches to finding an approximate solution to an equivalent problem are proposed and its convergence is proved. This made it possible to establish conditions for the existence of a unique solution of a semiperiodic boundary value problem for a system of second-order hyperbolic equations.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44676680","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 present paper, the initial value problem for the hyperbolic type involutory in t second order linear partial differential equation is studied. The initial value problem for the fourth order partial differential equations equivalent to this problem is obtained. The stability estimates for the solution and its first and second order derivatives of this problem are established.
{"title":"On the hyperbolic type differential equation with time involution","authors":"A. Ashyralyev, A. Ashyralyyev, B. Abdalmohammed","doi":"10.31489/2023m1/38-47","DOIUrl":"https://doi.org/10.31489/2023m1/38-47","url":null,"abstract":"In the present paper, the initial value problem for the hyperbolic type involutory in t second order linear partial differential equation is studied. The initial value problem for the fourth order partial differential equations equivalent to this problem is obtained. The stability estimates for the solution and its first and second order derivatives of this problem are established.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46030272","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 issue is a collection of 13 selected papers. These papers are presented at the Sixth International Conference on Analysis and Applied Mathematics (ICAAM 2022) organized by Bahcesehir University, Turkey, Institute of Mathematics and Mathematical Modelling, Kazakhstan, and Analysis & PDE Center, Ghent University, Belgium. The meeting was held on October 31 – November 6, 2022, in Antalya, Turkey. The conference is organized biannually. Previous conferences were held in Gumushane, Turkey in 2012; in Shymkent, Kazakhstan in 2014; in Almaty, Kazakhstan in 2016; in Cyprus, Turkey in 2018 and 2020; in Antalya, Turkey in 2022. The proceedings of ICAAM 2012, ICAAM 2014, ICAAM 2016, ICAAM 2018, and ICAAM 2020 were published in AIP Conference Proceedings (American Institute of Physics) and in some rating scientific journals. Proceedings of ICAAM 2022 will be published in the world-renowned AIP Conference Proceeding Series. The main aim of the International Conferences on Analysis and Applied Mathematics (ICAAM) is to bring mathematicians working in the area of analysis and applied mathematics together to share new trends of applications of mathematics. In mathematics, the developments in the field of applied mathematics open new research areas in analysis and vice versa. That is why, we planned to found the conference series to provide a forum for researches and scientists to communicate their recent developments and to present their original results in various fields of analysis and applied mathematics. This issue presents papers by authors from different countries: Azerbaijan, Iraq, Russian Federation, Cyprus, Turkey, Kazakhstan, Turkmenistan, Uzbekistan, Kyrgyzstan. Especially we are pleased with the fact that many articles are written by co-authors who work in different countries. We are confident that such international integration provides an opportunity for a significant increase in the quality and quantity of scientific publications. Special thanks to Charyyar Ashyralyyev (Turkey) for their valuable assistance. Finally, but not least, we would like to thank the Editorial board of the «Bulletin of the Karaganda University. Mathematics series», who kindly provided an opportunity for the formation of this special issue.
{"title":"About the conference ICAAM 2022. Preface","authors":"A. Ashyralyev, M. Sadybekov","doi":"10.31489/2023m1/4","DOIUrl":"https://doi.org/10.31489/2023m1/4","url":null,"abstract":"This issue is a collection of 13 selected papers. These papers are presented at the Sixth International Conference on Analysis and Applied Mathematics (ICAAM 2022) organized by Bahcesehir University, Turkey, Institute of Mathematics and Mathematical Modelling, Kazakhstan, and Analysis & PDE Center, Ghent University, Belgium. The meeting was held on October 31 – November 6, 2022, in Antalya, Turkey. The conference is organized biannually. Previous conferences were held in Gumushane, Turkey in 2012; in Shymkent, Kazakhstan in 2014; in Almaty, Kazakhstan in 2016; in Cyprus, Turkey in 2018 and 2020; in Antalya, Turkey in 2022. The proceedings of ICAAM 2012, ICAAM 2014, ICAAM 2016, ICAAM 2018, and ICAAM 2020 were published in AIP Conference Proceedings (American Institute of Physics) and in some rating scientific journals. Proceedings of ICAAM 2022 will be published in the world-renowned AIP Conference Proceeding Series. The main aim of the International Conferences on Analysis and Applied Mathematics (ICAAM) is to bring mathematicians working in the area of analysis and applied mathematics together to share new trends of applications of mathematics. In mathematics, the developments in the field of applied mathematics open new research areas in analysis and vice versa. That is why, we planned to found the conference series to provide a forum for researches and scientists to communicate their recent developments and to present their original results in various fields of analysis and applied mathematics. This issue presents papers by authors from different countries: Azerbaijan, Iraq, Russian Federation, Cyprus, Turkey, Kazakhstan, Turkmenistan, Uzbekistan, Kyrgyzstan. Especially we are pleased with the fact that many articles are written by co-authors who work in different countries. We are confident that such international integration provides an opportunity for a significant increase in the quality and quantity of scientific publications. Special thanks to Charyyar Ashyralyyev (Turkey) for their valuable assistance. Finally, but not least, we would like to thank the Editorial board of the «Bulletin of the Karaganda University. Mathematics series», who kindly provided an opportunity for the formation of this special issue.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44023918","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：