A new numerical method is proposed for solving the generalized Buckley-Leverett problem, which describes the movement of two-phase mixtures of Bazhenov bed sediments in a class of discontinuous functions. To this end, we introduce an auxiliary problem that has advantages over the main problem, and using these advantages, an original finite difference method to solve of the auxiliary problem is developed. Using the suggested auxiliary problem, a solution which expresses exactly all physical characteristics of the problem is obtained.
{"title":"Numerical method to solution of generalized model Buckley-Leverett in a class of discontinuous functions","authors":"B. Sinsoysal, M. Rasulov, R. Iskenderova","doi":"10.31489/2023m1/131-140","DOIUrl":"https://doi.org/10.31489/2023m1/131-140","url":null,"abstract":"A new numerical method is proposed for solving the generalized Buckley-Leverett problem, which describes the movement of two-phase mixtures of Bazhenov bed sediments in a class of discontinuous functions. To this end, we introduce an auxiliary problem that has advantages over the main problem, and using these advantages, an original finite difference method to solve of the auxiliary problem is developed. Using the suggested auxiliary problem, a solution which expresses exactly all physical characteristics of the problem is obtained.","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":"48952926","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 present paper establishes a formula of Reidemeister torsion for Schottky representations. The theoretical results are applied to 3−manifolds with boundary consisting orientable surfaces with genus at least 2.
{"title":"A remark on Schottky representations and Reidemeister torsion","authors":"F. Hezenci, Y. Sozen","doi":"10.31489/2023m1/81-93","DOIUrl":"https://doi.org/10.31489/2023m1/81-93","url":null,"abstract":"The present paper establishes a formula of Reidemeister torsion for Schottky representations. The theoretical results are applied to 3−manifolds with boundary consisting orientable surfaces with genus at least 2.","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":"42515631","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 study, we present an investigation of the asymptotic behavior of solutions of sum-difference equations. Based on some mathematical inequalities, we have obtained our results. The obtained results can apply to some fractional type difference equations as well.
{"title":"Asymptotic behavior of solutions of sum-difference equations","authors":"H. Adiguzel, E. Can","doi":"10.31489/2023m1/14-23","DOIUrl":"https://doi.org/10.31489/2023m1/14-23","url":null,"abstract":"In this study, we present an investigation of the asymptotic behavior of solutions of sum-difference equations. Based on some mathematical inequalities, we have obtained our results. The obtained results can apply to some fractional type difference equations as well.","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":"47576558","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 deals with some differential inequalities for generalized fractional integro-differential equations by using the technique of upper and lower solutions. The fractional differential operator is taken in Caputo’s sense and the nonlinear term divided into two parts depends on the fractional integrals of an unknown function with two different fractional orders. The results are studied by employing a variety of coupled upper and lower solutions. These theorems have some potential for extending the iterative techniques to fractional order integro-differential equations and to coupled systems of integro-differential fractional equations to obtain the existence of solutions as well as approximate solutions for the considered problem.
{"title":"Extensions of some differential inequalities for fractional integro-differential equations via upper and lower solutions","authors":"A. Yakar, H. Kutlay","doi":"10.31489/2023m1/156-167","DOIUrl":"https://doi.org/10.31489/2023m1/156-167","url":null,"abstract":"This paper deals with some differential inequalities for generalized fractional integro-differential equations by using the technique of upper and lower solutions. The fractional differential operator is taken in Caputo’s sense and the nonlinear term divided into two parts depends on the fractional integrals of an unknown function with two different fractional orders. The results are studied by employing a variety of coupled upper and lower solutions. These theorems have some potential for extending the iterative techniques to fractional order integro-differential equations and to coupled systems of integro-differential fractional equations to obtain the existence of solutions as well as approximate solutions for the considered problem.","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":"45547319","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 the study is the approximation of the solution to the Dirichlet boundary value problem (DBVP) of the heat equation on a rectangle by developing a new difference method on a grid system of hexagons. It is proved that the given special scheme is unconditionally stable and converges to the exact solution on the grids with fourth order accuracy in space variables and second order accuracy in time variable. Secondly, an incomplete block factorization is given for symmetric positive definite block tridiagonal (SPD-BT) matrices utilizing a conservative iterative method that approximates the inverse of the pivoting diagonal blocks by preserving the symmetric positive definite property. Subsequently, by using this factorization block hybrid preconditioning of the conjugate gradient (BHP-CG) method is applied to solve the obtained algebraic system of equations at each time level.
{"title":"Solution of heat equation by a novel implicit scheme using block hybrid preconditioning of the conjugate gradient method","authors":"S. C. Buranay, N. Arshad","doi":"10.31489/2023m1/58-80","DOIUrl":"https://doi.org/10.31489/2023m1/58-80","url":null,"abstract":"The main goal of the study is the approximation of the solution to the Dirichlet boundary value problem (DBVP) of the heat equation on a rectangle by developing a new difference method on a grid system of hexagons. It is proved that the given special scheme is unconditionally stable and converges to the exact solution on the grids with fourth order accuracy in space variables and second order accuracy in time variable. Secondly, an incomplete block factorization is given for symmetric positive definite block tridiagonal (SPD-BT) matrices utilizing a conservative iterative method that approximates the inverse of the pivoting diagonal blocks by preserving the symmetric positive definite property. Subsequently, by using this factorization block hybrid preconditioning of the conjugate gradient (BHP-CG) method is applied to solve the obtained algebraic system of equations at each time level.","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":"46635529","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 second boundary value problem for the loaded heat equation in the first quadrant. The loaded term contains a fractional derivative in the Caputo sense of an order α, 2<α<3. The boundary value problem is reduced to an integro-differential equation with a difference kernel by inverting the differential part. It is proved that a homogeneous integro-differential equation has at least one non-zero solution. It is shown that the solution of the homogeneous boundary value problem corresponding to the original boundary value problem is not unique, and the load acts as a strong perturbation of the boundary value problem.
{"title":"On the non-uniqueness of the solution to a boundary value problem of heat conduction with a load in the form of a fractional derivative","authors":"M. Kosmakova, K.A. Izhanova, A.N. Khamzeyeva","doi":"10.31489/2022m4/98-106","DOIUrl":"https://doi.org/10.31489/2022m4/98-106","url":null,"abstract":"The paper deals with the second boundary value problem for the loaded heat equation in the first quadrant. The loaded term contains a fractional derivative in the Caputo sense of an order α, 2<α<3. The boundary value problem is reduced to an integro-differential equation with a difference kernel by inverting the differential part. It is proved that a homogeneous integro-differential equation has at least one non-zero solution. It is shown that the solution of the homogeneous boundary value problem corresponding to the original boundary value problem is not unique, and the load acts as a strong perturbation of the boundary value problem.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47629206","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. Kasimov, G. Yessenbayeva, B. Kasimov, O. Khabidolda
In the article, an analytical and numerical study based on one modified refined bending theory is presented. By the finite difference method, a general numerical calculation algorithm is developed. The solution obtained by the proposed method is compared with the results of known solutions, namely, with the solution of the classical theory, the exact solution, the solution in trigonometric series, as well as with experimental data. Comparison of the results obtained by the method given in the article with the solutions determined by other methods shows sufficient accuracy, which indicates the reliability of the proposed method based on one option of the modified refined bending theory. Classical theory is not applicable to such problems under consideration.
{"title":"Analytical and numerical research based on one modified refined bending theory","authors":"A. Kasimov, G. Yessenbayeva, B. Kasimov, O. Khabidolda","doi":"10.31489/2022m4/76-85","DOIUrl":"https://doi.org/10.31489/2022m4/76-85","url":null,"abstract":"In the article, an analytical and numerical study based on one modified refined bending theory is presented. By the finite difference method, a general numerical calculation algorithm is developed. The solution obtained by the proposed method is compared with the results of known solutions, namely, with the solution of the classical theory, the exact solution, the solution in trigonometric series, as well as with experimental data. Comparison of the results obtained by the method given in the article with the solutions determined by other methods shows sufficient accuracy, which indicates the reliability of the proposed method based on one option of the modified refined bending theory. Classical theory is not applicable to such problems under consideration.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46896665","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 study of Jonsson quasivarieties in a signature enriched with new predicate and constant symbols. New concepts of semantic Jonsson quasivariety and fragment-conservativeness of the center of the Jonsson theory are introduced. The cosemanticness classes of the Jonsson spectrum constructed for a semantic Jonsson quasvariety are considered. In this case, the Kaiser hull of the semantic Jonsson quasivariety is assumed to be existentially prime. By constructing a central type for classes of theories from the Jonsson spectrum, the following results are formulated and proved. In the first main result, the necessary and sufficient condition is given for the center of the cosemanticness class of an existentially prime semantic Jonsson quasivariety to be λ-stable. The second result is the criterion for the center of the class of theories to be ω-categorical in the enriched language. The obtained theorems can be useful in continuing studies of various Jonsson algebras, in particular, Jonsson quasivarieties.
{"title":"Existentially prime Jonsson quasivarieties and their Jonsson spectra","authors":"A. Yeshkeyev, I.O. Tungushbayeva, S.M. Amanbekov","doi":"10.31489/2022m4/117-124","DOIUrl":"https://doi.org/10.31489/2022m4/117-124","url":null,"abstract":"This article is devoted to the study of Jonsson quasivarieties in a signature enriched with new predicate and constant symbols. New concepts of semantic Jonsson quasivariety and fragment-conservativeness of the center of the Jonsson theory are introduced. The cosemanticness classes of the Jonsson spectrum constructed for a semantic Jonsson quasvariety are considered. In this case, the Kaiser hull of the semantic Jonsson quasivariety is assumed to be existentially prime. By constructing a central type for classes of theories from the Jonsson spectrum, the following results are formulated and proved. In the first main result, the necessary and sufficient condition is given for the center of the cosemanticness class of an existentially prime semantic Jonsson quasivariety to be λ-stable. The second result is the criterion for the center of the class of theories to be ω-categorical in the enriched language. The obtained theorems can be useful in continuing studies of various Jonsson algebras, in particular, Jonsson quasivarieties.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46460444","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 embedding of spaces of differentiable functions studies important relations of differential (smoothness) properties of functions in various metrics and has wide application in the theory of boundary value problems of mathematical physics, approximation theory and other fields of mathematics. In this article, we prove the theorems about traces and extensions for functions from Nikolsky-Besov spaces with generalized mixed smoothness and mixed metrics. The proofs of the obtained results is based on the inequality of different dimensions for trigonometric polynomials in Lebesgue spaces with mixed metrics and the embedding theorem of classical Nikolsky-Besov spaces in the space of continuous functions.
{"title":"The theorems about traces and extensions for functions from Nikolsky-Besov spaces with generalized mixed smoothness","authors":"K. Bekmaganbetov, K.Ye. Kervenev, Y. Toleugazy","doi":"10.31489/2022m4/42-50","DOIUrl":"https://doi.org/10.31489/2022m4/42-50","url":null,"abstract":"The theory of embedding of spaces of differentiable functions studies important relations of differential (smoothness) properties of functions in various metrics and has wide application in the theory of boundary value problems of mathematical physics, approximation theory and other fields of mathematics. In this article, we prove the theorems about traces and extensions for functions from Nikolsky-Besov spaces with generalized mixed smoothness and mixed metrics. The proofs of the obtained results is based on the inequality of different dimensions for trigonometric polynomials in Lebesgue spaces with mixed metrics and the embedding theorem of classical Nikolsky-Besov spaces in the space of continuous functions.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49620423","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 problem of the best approximation by linear methods of solutions to one Triebel-type equation. This problem was solved by using estimates of the linear widths of the unit ball in corresponding spaces of differentiable functions. According to the definition, linear widths give the best estimates for the approximation of compact sets in a given normed space by linear methods which are implemented through finite-dimensional operators. The problem includes answers to the questions about the solvability of the studied equation, the construction of the corresponding weighted space of differentiable functions, the development of a method for estimating linear widths of compact sets in weighted polynomial Sobolev space. In this work, conditions are obtained under which the considered operator has a bounded inverse. The weighted Sobolev space corresponding to the posed problem is determined. Upper estimates are obtained for the counting function for a sequence of linear widths, which correspond to the posed problem. One example is constructed in which two-sided estimates of linear widths are given. The method for solving this problem can be applied to the numerical solution of non-standard ordinary differential equations on an infinite axis.
{"title":"On the approximation of solutions of one singular differential equation on the axis","authors":"A. S. Kassym, L. Kussainova","doi":"10.31489/2022m/86-97","DOIUrl":"https://doi.org/10.31489/2022m/86-97","url":null,"abstract":"In this paper we study the problem of the best approximation by linear methods of solutions to one Triebel-type equation. This problem was solved by using estimates of the linear widths of the unit ball in corresponding spaces of differentiable functions. According to the definition, linear widths give the best estimates for the approximation of compact sets in a given normed space by linear methods which are implemented through finite-dimensional operators. The problem includes answers to the questions about the solvability of the studied equation, the construction of the corresponding weighted space of differentiable functions, the development of a method for estimating linear widths of compact sets in weighted polynomial Sobolev space. In this work, conditions are obtained under which the considered operator has a bounded inverse. The weighted Sobolev space corresponding to the posed problem is determined. Upper estimates are obtained for the counting function for a sequence of linear widths, which correspond to the posed problem. One example is constructed in which two-sided estimates of linear widths are given. The method for solving this problem can be applied to the numerical solution of non-standard ordinary differential equations on an infinite axis.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41855951","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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