Partial differential equations of the parabolic type with discontinuous coefficients and the heat equation degenerating in time, each separately, have been well studied by many authors. Conjugation problems for time-degenerate equations of the parabolic type with discontinuous coefficients are practically not studied. In this work, in an n-dimensional space, a conjugation problem is considered for a heat equation with discontinuous coefficients which degenerates at the initial moment of time. A fundamental solution to the set problem has been constructed and estimates of its derivatives have been found. With the help of these estimates, in the Sobolev classes, the estimate of the solution to the set problem was obtained.
{"title":"A priori estimate of the solution of the Cauchy problem in the Sobolev classes for discontinuous coefficients of degenerate heat equations","authors":"U.K. Koilyshov, K. Beisenbaeva, S.D. Zhapparova","doi":"10.31489/2022m3/59-69","DOIUrl":"https://doi.org/10.31489/2022m3/59-69","url":null,"abstract":"Partial differential equations of the parabolic type with discontinuous coefficients and the heat equation degenerating in time, each separately, have been well studied by many authors. Conjugation problems for time-degenerate equations of the parabolic type with discontinuous coefficients are practically not studied. In this work, in an n-dimensional space, a conjugation problem is considered for a heat equation with discontinuous coefficients which degenerates at the initial moment of time. A fundamental solution to the set problem has been constructed and estimates of its derivatives have been found. With the help of these estimates, in the Sobolev classes, the estimate of the solution to the set problem was obtained.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46214814","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 investigates the question of the existence of solutions to the semiperiodic Dirichlet problem for a class of singular differential equations of hyperbolic type. The problem of smoothness of solutions is also considered, i.e., maximum regularity of solutions. Such a problem will be interesting when the coefficients are strongly growing functions at infinity. For the first time, a weighted coercive estimate was obtained for solutions to a differential equation of hyperbolic type with strongly growing coefficients.
{"title":"Existence and smoothness of solutions of a singular differential equation of hyperbolic type","authors":"M. Muratbekov, Yerik Bayandiyev","doi":"10.31489/2022m3/98-104","DOIUrl":"https://doi.org/10.31489/2022m3/98-104","url":null,"abstract":"This paper investigates the question of the existence of solutions to the semiperiodic Dirichlet problem for a class of singular differential equations of hyperbolic type. The problem of smoothness of solutions is also considered, i.e., maximum regularity of solutions. Such a problem will be interesting when the coefficients are strongly growing functions at infinity. For the first time, a weighted coercive estimate was obtained for solutions to a differential equation of hyperbolic type with strongly growing coefficients.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43396388","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 is connected with the study of Jonsson spectrum notion of the fixed class of models of Sacts signature, assuming a group as a monoid of S-acts. The Jonsson spectrum notion is effective when describing theoretical-model properties of algebras classes whose theories admit joint embedding and amalgam properties. It is usually sufficient to consider universal-existential sentences true on models of this class. Up to the present paper, the Jonsson spectrum has tended to deal only with Jonsson theories. The authors of this study define the positive Jonsson spectrum notion, the elements of which can be, non-Jonsson theories. This happens because in the definition of the existentially positive Mustafin theories considered in a given paper involve not only isomorphic embeddings, but also immersions. In this connection, immersions are considered in the definition of amalgam and joint embedding properties. The resulting theories do not necessarily have to be Jonsson. We can observe that the above approach to the Jonsson spectrum study proves to be justified because even in the case of a non-Jonsson theory there exists regular method for finding such Jonsson theory that satisfies previously known notions and results, but that will also be directly related to the existentially positive Mustafin theory in question.
{"title":"Existentially positive Mustafin theories of S-acts over a group","authors":"A. Yeshkeyev, O. I. Ulbrikht, A. R. Yarullina","doi":"10.31489/2022m2/172-185","DOIUrl":"https://doi.org/10.31489/2022m2/172-185","url":null,"abstract":"The paper is connected with the study of Jonsson spectrum notion of the fixed class of models of Sacts signature, assuming a group as a monoid of S-acts. The Jonsson spectrum notion is effective when describing theoretical-model properties of algebras classes whose theories admit joint embedding and amalgam properties. It is usually sufficient to consider universal-existential sentences true on models of this class. Up to the present paper, the Jonsson spectrum has tended to deal only with Jonsson theories. The authors of this study define the positive Jonsson spectrum notion, the elements of which can be, non-Jonsson theories. This happens because in the definition of the existentially positive Mustafin theories considered in a given paper involve not only isomorphic embeddings, but also immersions. In this connection, immersions are considered in the definition of amalgam and joint embedding properties. The resulting theories do not necessarily have to be Jonsson. We can observe that the above approach to the Jonsson spectrum study proves to be justified because even in the case of a non-Jonsson theory there exists regular method for finding such Jonsson theory that satisfies previously known notions and results, but that will also be directly related to the existentially positive Mustafin theory in question.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46917815","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}
Initial boundary value problems with a time-nonlocal condition for a subdiffusion equation with the Riemann-Liouville time-fractional derivatives are considered. The elliptical part of the equation is the Laplace operator, defined in an arbitrary N−dimensional domain Ω with a sufficiently smooth boundary ∂Ω. The existence and uniqueness of the solution to the considered problems are proved. Inverse problems are studied for determining the right-hand side of the equation and a function in a time-nonlocal condition. The main research tool is the Fourier method, so the obtained results can be extended to subdiffusion equations with a more general elliptic operator.
{"title":"On the nonlocal problems in time for subdiffusion equations with the Riemann-Liouville derivatives","authors":"R. Ashurov, Y. Fayziev","doi":"10.31489/2022m2/18-37","DOIUrl":"https://doi.org/10.31489/2022m2/18-37","url":null,"abstract":"Initial boundary value problems with a time-nonlocal condition for a subdiffusion equation with the Riemann-Liouville time-fractional derivatives are considered. The elliptical part of the equation is the Laplace operator, defined in an arbitrary N−dimensional domain Ω with a sufficiently smooth boundary ∂Ω. The existence and uniqueness of the solution to the considered problems are proved. Inverse problems are studied for determining the right-hand side of the equation and a function in a time-nonlocal condition. The main research tool is the Fourier method, so the obtained results can be extended to subdiffusion equations with a more general elliptic operator.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42105254","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 boundary value problem in a rectangular domain for a system of partial differential equations with the Dzhrbashyan–Nersesyan fractional differentiation operators with constant coefficients is studied in the case when the matrix coefficients of the system have complex eigenvalues. Existence and uniqueness theorems for the solution to the boundary value problem under study are proved. The solution is constructed explicitly in terms of the Wright function of the matrix argument.
{"title":"Boundary value problem for a system of partial differential equations with the Dzhrbashyan–Nersesyan fractional differentiation operators","authors":"M. Mamchuev","doi":"10.31489/2022m2/143-160","DOIUrl":"https://doi.org/10.31489/2022m2/143-160","url":null,"abstract":"A boundary value problem in a rectangular domain for a system of partial differential equations with the Dzhrbashyan–Nersesyan fractional differentiation operators with constant coefficients is studied in the case when the matrix coefficients of the system have complex eigenvalues. Existence and uniqueness theorems for the solution to the boundary value problem under study are proved. The solution is constructed explicitly in terms of the Wright function of the matrix argument.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42481571","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 article studies a second-order integro-differential equation with difference kernels and power nonlinearity. A connection is established between this equation and an integral equation of the convolution type, which arises when describing the processes of liquid infiltration from a cylindrical reservoir into an isotropic homogeneous porous medium, the propagation of shock waves in pipes filled with gas and others. Since non-negative continuous solutions of this integral equation are of particular interest from an applied point of view, solutions of the corresponding integro-differential equation are sought in the cone of the space of continuously differentiable functions. Two-sided a priori estimates are obtained for any solution of the indicated integral equation, based on which the global theorem of existence and uniqueness of the solution is proved by the method of weighted metrics. It is shown that any solution of this integro-differential equation is simultaneously a solution of the integral equation and vice versa, under the additional condition on the kernel that any solution of this integral equation is a solution of this integro-differential equation. Using these results, a global theorem on the existence, uniqueness and method of finding a solution to an integrodifferential equation is proved. It is shown that this solution can be found by the method of successive approximations of the Picard type and an estimate for the rate of their convergence is established. Examples are given to illustrate the obtained results.
{"title":"On a second-order integro-differential equation with difference kernels and power nonlinearity","authors":"S. Askhabov","doi":"10.31489/2022m2/38-48","DOIUrl":"https://doi.org/10.31489/2022m2/38-48","url":null,"abstract":"The article studies a second-order integro-differential equation with difference kernels and power nonlinearity. A connection is established between this equation and an integral equation of the convolution type, which arises when describing the processes of liquid infiltration from a cylindrical reservoir into an isotropic homogeneous porous medium, the propagation of shock waves in pipes filled with gas and others. Since non-negative continuous solutions of this integral equation are of particular interest from an applied point of view, solutions of the corresponding integro-differential equation are sought in the cone of the space of continuously differentiable functions. Two-sided a priori estimates are obtained for any solution of the indicated integral equation, based on which the global theorem of existence and uniqueness of the solution is proved by the method of weighted metrics. It is shown that any solution of this integro-differential equation is simultaneously a solution of the integral equation and vice versa, under the additional condition on the kernel that any solution of this integral equation is a solution of this integro-differential equation. Using these results, a global theorem on the existence, uniqueness and method of finding a solution to an integrodifferential equation is proved. It is shown that this solution can be found by the method of successive approximations of the Picard type and an estimate for the rate of their convergence is established. Examples are given to illustrate the obtained results.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42125005","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 investigates the unique solvability of the boundary control problem for a one-dimensional wave equation loaded along one of its characteristic curves in terms of a regular solution. The solution method is based on an analogue of the d’Alembert formula constructed for this equation. We point out that the domain of definition for the solution of DE, when the initial and final Cauchy data given on intervals of the same length is a square. The side of the squire is equal to the interval length. The boundary controls are established by the components of an analogue of the d’Alembert formula, which, in turn, are uniquely established by the initial and final Cauchy data. It should be noted that the normalized distribution and centering are employed in the final formulas of sought boundary controls, which is not typical for initial and boundary value problems initiated by equations of hyperbolic type.
{"title":"Boundary control problem for a hyperbolic equation loaded along one of its characteristics","authors":"A. Attaev","doi":"10.31489/2022m2/49-58","DOIUrl":"https://doi.org/10.31489/2022m2/49-58","url":null,"abstract":"This paper investigates the unique solvability of the boundary control problem for a one-dimensional wave equation loaded along one of its characteristic curves in terms of a regular solution. The solution method is based on an analogue of the d’Alembert formula constructed for this equation. We point out that the domain of definition for the solution of DE, when the initial and final Cauchy data given on intervals of the same length is a square. The side of the squire is equal to the interval length. The boundary controls are established by the components of an analogue of the d’Alembert formula, which, in turn, are uniquely established by the initial and final Cauchy data. It should be noted that the normalized distribution and centering are employed in the final formulas of sought boundary controls, which is not typical for initial and boundary value problems initiated by equations of hyperbolic type.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47122358","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 studies a nonlocal boundary value problem with Steklov’s conditions of the first type for a linear ordinary delay differential equation of a fractional order with constant coefficients. The Green’s function of the problem with its properties is found. The solution to the problem is obtained explicitly in terms of the Green’s function. A condition for the unique solvability of the problem is found, as well as the conditions under which the solvability condition is satisfied. The existence and uniqueness theorem is proved using the representation of the Green’s function and its properties, as well as the representation of the fundamental solution to the equation and its properties. The question of eigenvalues is investigated. The theorem on the finiteness of the number of eigenvalues is proved using the notation of the solution in terms of the generalized Wright function, as well as the asymptotic properties of the generalized Wright function as λ → ∞ and λ → −∞.
{"title":"Steklov problem for a linear ordinary fractional delay differential equation with the Riemann-Liouville derivative","authors":"M. G. Mazhgikhova","doi":"10.31489/2022m2/161-171","DOIUrl":"https://doi.org/10.31489/2022m2/161-171","url":null,"abstract":"This paper studies a nonlocal boundary value problem with Steklov’s conditions of the first type for a linear ordinary delay differential equation of a fractional order with constant coefficients. The Green’s function of the problem with its properties is found. The solution to the problem is obtained explicitly in terms of the Green’s function. A condition for the unique solvability of the problem is found, as well as the conditions under which the solvability condition is satisfied. The existence and uniqueness theorem is proved using the representation of the Green’s function and its properties, as well as the representation of the fundamental solution to the equation and its properties. The question of eigenvalues is investigated. The theorem on the finiteness of the number of eigenvalues is proved using the notation of the solution in terms of the generalized Wright function, as well as the asymptotic properties of the generalized Wright function as λ → ∞ and λ → −∞.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45012682","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 studies an ordinary second-order differential equation with a fractional differentiation operator in the sense of Riemann-Liouville with a variable coefficient. We use the Green’s function’s method to find a representation of the solution of the Dirichlet problem for the equation under consideration when the solvability condition is satisfied. Green’s function to the problem is constructed in terms of the fundamental solution of the equation under study and its properties are proved. The necessary integral condition for the existence of a nontrivial solution to the homogeneous Dirichlet problem, called an analogue of the Lyapunov inequality, is found.
{"title":"An analogue of the Lyapunov inequality for an ordinary second-order differential equation with a fractional derivative and a variable coefficient","authors":"B. Efendiev","doi":"10.31489/2022m2/83-92","DOIUrl":"https://doi.org/10.31489/2022m2/83-92","url":null,"abstract":"This paper studies an ordinary second-order differential equation with a fractional differentiation operator in the sense of Riemann-Liouville with a variable coefficient. We use the Green’s function’s method to find a representation of the solution of the Dirichlet problem for the equation under consideration when the solvability condition is satisfied. Green’s function to the problem is constructed in terms of the fundamental solution of the equation under study and its properties are proved. The necessary integral condition for the existence of a nontrivial solution to the homogeneous Dirichlet problem, called an analogue of the Lyapunov inequality, is found.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46633657","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 integro-differential equations in Banach spaces with operators, which are a composition of convolution and differentiation operators. Depending on the order of action of these two operators, we talk about integro-differential operators of the Riemann—Liouville type, when the convolution operator acts first, and integro-differential operators of the Gerasimov type otherwise. Special cases of the operators under consideration are the fractional derivatives of Riemann—Liouville and Gerasimov, respectively. The classes of integro-differential operators under study also include those in which the convolution has an integral kernel without singularities. The conditions of the unique solvability of the Cauchy type problem for a linear integro-differential equation of the Riemann—Liouville type and the Cauchy problem for a linear integrodifferential equation of the Gerasimov type with a bounded operator at the unknown function are obtained. These results are used in the study of similar equations with a degenerate operator at an integro-differential operator under the condition of relative boundedness of the pair of operators from the equation. Abstract results are applied to the study of initial boundary value problems for partial differential equations with an integro-differential operator, the convolution in which is given by the Mittag-Leffler function multiplied by a power function.
{"title":"Integro-differential equations with bounded operators in Banach spaces","authors":"V. Fedorov, A. D. Godova, B. T. Kien","doi":"10.31489/2022m2/93-107","DOIUrl":"https://doi.org/10.31489/2022m2/93-107","url":null,"abstract":"The paper investigates integro-differential equations in Banach spaces with operators, which are a composition of convolution and differentiation operators. Depending on the order of action of these two operators, we talk about integro-differential operators of the Riemann—Liouville type, when the convolution operator acts first, and integro-differential operators of the Gerasimov type otherwise. Special cases of the operators under consideration are the fractional derivatives of Riemann—Liouville and Gerasimov, respectively. The classes of integro-differential operators under study also include those in which the convolution has an integral kernel without singularities. The conditions of the unique solvability of the Cauchy type problem for a linear integro-differential equation of the Riemann—Liouville type and the Cauchy problem for a linear integrodifferential equation of the Gerasimov type with a bounded operator at the unknown function are obtained. These results are used in the study of similar equations with a degenerate operator at an integro-differential operator under the condition of relative boundedness of the pair of operators from the equation. Abstract results are applied to the study of initial boundary value problems for partial differential equations with an integro-differential operator, the convolution in which is given by the Mittag-Leffler function multiplied by a power function.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47983354","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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