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Bulletin of the Karaganda University-Mathematics最新文献

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Boundary value problem for a system of partial differential equations with the Dzhrbashyan–Nersesyan fractional differentiation operators Dzhrbashyan–Nersesyan分数微分算子的偏微分方程组的边值问题
IF 0.6
Bulletin of the Karaganda University-Mathematics Pub Date : 2022-06-30 DOI: 10.31489/2022m2/143-160
M. Mamchuev
{"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":null,"pages":null},"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}
引用次数: 1
On a second-order integro-differential equation with difference kernels and power nonlinearity 一类具有差分核和幂非线性的二阶积分微分方程
IF 0.6
Bulletin of the Karaganda University-Mathematics Pub Date : 2022-06-30 DOI: 10.31489/2022m2/38-48
S. Askhabov
{"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":null,"pages":null},"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}
引用次数: 0
Boundary control problem for a hyperbolic equation loaded along one of its characteristics 一类双曲型方程的边界控制问题
IF 0.6
Bulletin of the Karaganda University-Mathematics Pub Date : 2022-06-30 DOI: 10.31489/2022m2/49-58
A. Attaev
{"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":null,"pages":null},"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}
引用次数: 1
Steklov problem for a linear ordinary fractional delay differential equation with the Riemann-Liouville derivative 具有Riemann-Liouville导数的线性常分式时滞微分方程的Steklov问题
IF 0.6
Bulletin of the Karaganda University-Mathematics Pub Date : 2022-06-30 DOI: 10.31489/2022m2/161-171
M. G. Mazhgikhova
{"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":null,"pages":null},"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}
引用次数: 0
An analogue of the Lyapunov inequality for an ordinary second-order differential equation with a fractional derivative and a variable coefficient 具有分数阶导数和变系数的普通二阶微分方程的李雅普诺夫不等式的类似物
IF 0.6
Bulletin of the Karaganda University-Mathematics Pub Date : 2022-06-30 DOI: 10.31489/2022m2/83-92
B. Efendiev
{"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":null,"pages":null},"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}
引用次数: 0
Approximate solutions of the Riemann problem for a two-phase flow of immiscible liquids based on the Buckley–Leverett model 基于Buckley-Leverett模型的两相非混相流体Riemann问题的近似解
IF 0.6
Bulletin of the Karaganda University-Mathematics Pub Date : 2022-06-30 DOI: 10.31489/2022m2/4-17
Y.S. Aldanov, T.Zh. Toleuov, N. Tasbolatuly
{"title":"Approximate solutions of the Riemann problem for a two-phase flow of immiscible liquids based on the Buckley–Leverett model","authors":"Y.S. Aldanov, T.Zh. Toleuov, N. Tasbolatuly","doi":"10.31489/2022m2/4-17","DOIUrl":"https://doi.org/10.31489/2022m2/4-17","url":null,"abstract":"The article proposes an approximate method based on the \"vanishing viscosity\" method, which ensures the smoothness of the solution without taking into account the capillary pressure. We will consider the vanishing viscosity solution to the Riemann problem and to the boundary Riemann problem. It is not a weak solution, unless the system is conservative. One can prove that it is a viscosity solution actually meaning the extension of the semigroup of the vanishing viscosity solution to piecewise constant initial and boundary data. It is known that without taking into account the capillary pressure, the Buckley–Leverett model is the main one. Typically, from a computational point of view, approximate models are required for time slicing when creating computational algorithms. Analysis of the flow of a mixture of two immiscible liquids, the viscosity of which depends on pressure, leads to a further extension of the classical Buckley–Leverett model. Some two-phase flow models based on the expansion of Darcy’s law include the effect of capillary pressure. This is motivated by the fact that some fluids, e.g., crude oil, have a pressure-dependent viscosity and are noticeably sensitive to pressure fluctuations. Results confirm the insignificant influence of cross-coupling terms compared to the classical Darcy approach.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49012992","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}
引用次数: 0
Integro-differential equations with bounded operators in Banach spaces Banach空间中有界算子的积分微分方程
IF 0.6
Bulletin of the Karaganda University-Mathematics Pub Date : 2022-06-30 DOI: 10.31489/2022m2/93-107
V. Fedorov, A. D. Godova, B. T. Kien
{"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":null,"pages":null},"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}
引用次数: 0
Generalized boundary value problem for a linear ordinary differential equation with a discretely distributed fractional differentiation operator 一类具有离散分布分数阶微分算子的线性常微分方程的广义边值问题
IF 0.6
Bulletin of the Karaganda University-Mathematics Pub Date : 2022-06-30 DOI: 10.31489/2022m2/108-116
L. Gadzova
{"title":"Generalized boundary value problem for a linear ordinary differential equation with a discretely distributed fractional differentiation operator","authors":"L. Gadzova","doi":"10.31489/2022m2/108-116","DOIUrl":"https://doi.org/10.31489/2022m2/108-116","url":null,"abstract":"This paper formulates and solves a generalized boundary value problem for a linear ordinary differential equation with a discretely distributed fractional differentiation operator. The fractional derivative is understood as the Gerasimov–Caputo derivative. The boundary conditions are given in the form of linear functionals, which makes it possible to cover a wide class of linear local and non-local conditions. A representation of the solution is found in terms of special functions. A necessary and sufficient condition for the solvability of the problem under study is obtained, as well as conditions under which the solvability condition is certainly satisfied. The theorem of existence and uniqueness of the solution is proved.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42298389","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}
引用次数: 0
Inverse problems of determining coefficients of time type in a degenerate parabolic equation 退化抛物型方程中时间型系数的反演问题
IF 0.6
Bulletin of the Karaganda University-Mathematics Pub Date : 2022-06-30 DOI: 10.31489/2022m2/128-142
A. I. Kozhanov, U.U. Abulkayirov
{"title":"Inverse problems of determining coefficients of time type in a degenerate parabolic equation","authors":"A. I. Kozhanov, U.U. Abulkayirov","doi":"10.31489/2022m2/128-142","DOIUrl":"https://doi.org/10.31489/2022m2/128-142","url":null,"abstract":"The paper is devoted to the study of the solvability of inverse coefficient problems for degenerate parabolic equations of the second order. We study both linear inverse problems – the problems of determining an unknown right-hand side (external influence), and nonlinear problems of determining an unknown coefficient of the equation itself. The peculiarity of the studied work is that its unknown coefficients are functions of a time variable only. The work aims to prove the existence and uniqueness of regular solutions to the studied problems (having all the generalized in the sense of S.L. Sobolev derivatives entering the equation).","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42897037","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}
引用次数: 3
Examples of weakly compact sets in Orlicz spaces Orlicz空间中弱紧集的例子
IF 0.6
Bulletin of the Karaganda University-Mathematics Pub Date : 2022-06-30 DOI: 10.31489/2022m2/72-82
D. Dauitbek, Y. Nessipbayev, K. Tulenov
{"title":"Examples of weakly compact sets in Orlicz spaces","authors":"D. Dauitbek, Y. Nessipbayev, K. Tulenov","doi":"10.31489/2022m2/72-82","DOIUrl":"https://doi.org/10.31489/2022m2/72-82","url":null,"abstract":"This paper provides a number of examples of relatively weakly compact sets in Orlicz spaces. We show some results arising from these examples. Particularly, we provide a criterion which ensures that some Orlicz function is increasing more rapidly than another (in a sense of T. Ando). In addition, we point out that if a bounded subset K of the Orlicz space LΦ is not bounded by the modular Φ, then it is possible for a set K to remain unbounded under any modular Ψ increasing more rapidly than Φ.","PeriodicalId":29915,"journal":{"name":"Bulletin of the Karaganda University-Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69839849","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}
引用次数: 0
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