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Contemporary Mathematics. Fundamental Directions最新文献

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The second-order accuracy difference schemes for integral-type time-nonlocal parabolic problems 积分型非局部抛物型问题的二阶精度差分格式
Contemporary Mathematics. Fundamental Directions Pub Date : 2023-03-31 DOI: 10.22363/2413-3639-2023-69-1-32-49
A. Ashyralyev, C. Ashyralyyev
{"title":"The second-order accuracy difference schemes for integral-type time-nonlocal parabolic problems","authors":"A. Ashyralyev, C. Ashyralyyev","doi":"10.22363/2413-3639-2023-69-1-32-49","DOIUrl":"https://doi.org/10.22363/2413-3639-2023-69-1-32-49","url":null,"abstract":"This is a discussion on the second-order accuracy difference schemes for approximate solution of the integral-type time-nonlocal parabolic problems. The theorems on the stability of r-modified Crank-Nicolson difference schemes and second-order accuracy implicit difference scheme for approximate solution of the integral-type time-nonlocal parabolic problems in a Hilbert space with self-adjoint positive definite operator are established. In practice, stability estimates for the solutions of the second-order accuracy in t difference schemes for the one and multidimensional time-nonlocal parabolic problems are obtained. Numerical results are given.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115707634","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
L2-estimates of error in homogenization of parabolic equations with correctors taken into account l2 -考虑校正的抛物方程均匀化误差估计
Contemporary Mathematics. Fundamental Directions Pub Date : 2023-03-31 DOI: 10.22363/2413-3639-2023-69-1-134-151
S. Pastukhova
{"title":"L2-estimates of error in homogenization of parabolic equations with correctors taken into account","authors":"S. Pastukhova","doi":"10.22363/2413-3639-2023-69-1-134-151","DOIUrl":"https://doi.org/10.22363/2413-3639-2023-69-1-134-151","url":null,"abstract":"We consider second-order parabolic equations with bounded measurable (varepsilon)-periodic coefficients. To solve the Cauchy problem in the layer ( R^d times(0,T) ) with the nonhomogeneous equation, we obtain approximations in the norm (|cdot|_{L^2(R^dtimes(0,T))}) with remainder of order (varepsilon^2) as (varepsilon to 0.)","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129253984","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 singular problems for quasilinear equations involving mixed reaction-diffusion 拟线性混合反应扩散方程的边界奇异问题
Contemporary Mathematics. Fundamental Directions Pub Date : 2022-12-15 DOI: 10.22363/2413-3639-2022-68-4-564-574
L. Véron
{"title":"Boundary singular problems for quasilinear equations involving mixed reaction-diffusion","authors":"L. Véron","doi":"10.22363/2413-3639-2022-68-4-564-574","DOIUrl":"https://doi.org/10.22363/2413-3639-2022-68-4-564-574","url":null,"abstract":"We study the existence of solutions to the problem \u0000[label{eng_A1} \u0000begin{array}{rl} \u0000-Delta u+u^p-M|nabla u|^q=0 text{in };Omega, \u0000u=mu text{on };partialOmega \u0000end{array}] \u0000in a bounded domain (Omega), where (p1), (1q2), (M0), (mu) is a nonnegative Radon measure in (partialOmega), and the associated problem with a boundary isolated singularity at (ainpartialOmega,) \u0000[label{eng_A2} \u0000begin{array}{rl} \u0000-Delta u+u^p-M|nabla u|^q=0 text{in };Omega, \u0000u=0 text{on };partialOmegasetminus{a}. \u0000end{array}] \u0000The difficulty lies in the opposition between the two nonlinear terms which are not on the same nature. Existence of solutions to[eng_A1] is obtained under a capacitary condition [mu(K)leq \u0000cminleft{cap^{partialOmega}_{frac{2}{p},p'},cap^{partialOmega}_{frac{2-q}{q},q'}right}quadtext{for \u0000all compacts }KsubsetpartialOmega.] Problem[eng_A2] depends on several critical exponents on (p) and (q) as well as the position of (q) with respect to (dfrac{2p}{p+1}).","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121281226","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
In uence of numerical diffusion on the growth rate of viscous ngers in the numerical implementation of the Peaceman model by the finite volume method 用有限体积法对Peaceman模型进行数值求解时，数值扩散对粘性颗粒生长速率的影响
Contemporary Mathematics. Fundamental Directions Pub Date : 2022-12-15 DOI: 10.22363/2413-3639-2022-68-4-553-563
D. Apushkinskaya, G. Lazareva, V. A. Okishev
{"title":"In uence of numerical diffusion on the growth rate of viscous ngers in the numerical implementation of the Peaceman model by the finite volume method","authors":"D. Apushkinskaya, G. Lazareva, V. A. Okishev","doi":"10.22363/2413-3639-2022-68-4-553-563","DOIUrl":"https://doi.org/10.22363/2413-3639-2022-68-4-553-563","url":null,"abstract":"A numerical model of oil displacement by a mixture of water and polymer based on the Peaceman model is considered. Numerical experiments were carried out using the DuMux package, which is a software library designed for modeling nonstationary hydrodynamic problems in porous media. The software package uses the vertex-centered variant of finite volume method. The effect of diffusion on the growth rate of ''viscous fingers'' has been studied. The dependencies of the leading front velocity on the value of model diffusion are obtained for three viscosity models. It is shown that the effect of numerical diffusion on the growth rate of ''viscous fingers'' imposes limitations on calculations for small values of model diffusion.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"88 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125812781","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
Nonautonomous dynamics: classification, invariants, and implementation 非自治动态:分类、不变量和实现
Contemporary Mathematics. Fundamental Directions Pub Date : 2022-12-15 DOI: 10.22363/2413-3639-2022-68-4-596-620
V. Grines, L. Lerman
{"title":"Nonautonomous dynamics: classification, invariants, and implementation","authors":"V. Grines, L. Lerman","doi":"10.22363/2413-3639-2022-68-4-596-620","DOIUrl":"https://doi.org/10.22363/2413-3639-2022-68-4-596-620","url":null,"abstract":"The work is a brief review of the results obtained in nonautonomous dynamics based on the concept of uniform equivalence of nonautonomous systems. This approach to the study of nonautonomous systems was proposed in [10] and further developed in the works of the second author, and recently - jointly by both authors. Such an approach seems to be fruitful and promising, since it allows one to develop a nonautonomous analogue of the theory of dynamical systems for the indicated classes of systems and give a classi cation of some natural classes of nonautonomous systems using combinatorial type invariants. We show this for classes of nonautonomous gradient-like vector elds on closed manifolds of dimensions one, two, and three. In the latter case, a new equivalence invariant appears, the wild embedding type for stable and unstable manifolds [14,17], as shown in a recent paper by the authors [5].","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"14 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113939217","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
Homogenization of a parabolic equation in a perforated domain with a unilateral dynamic boundary condition: critical case 具有单侧动态边界条件的穿孔区域抛物方程的均匀化:临界情况
Contemporary Mathematics. Fundamental Directions Pub Date : 2022-12-15 DOI: 10.22363/2413-3639-2022-68-4-671-685
A. V. Podolskiy, T. Shaposhnikova
{"title":"Homogenization of a parabolic equation in a perforated domain with a unilateral dynamic boundary condition: critical case","authors":"A. V. Podolskiy, T. Shaposhnikova","doi":"10.22363/2413-3639-2022-68-4-671-685","DOIUrl":"https://doi.org/10.22363/2413-3639-2022-68-4-671-685","url":null,"abstract":"In this paper, we study the homogenization of a parabolic equation given in a domain perforated by ''tiny'' balls. On the boundary of these perforations, a unilateral dynamic boundary constraints are specified. We address the so-called ''critical'' case that is characterized by a relation between the coefficient in the boundary condition, the period of the structure and the size of the holes. In this case, the homogenized equation contains a nonlocal ''strange'' term. This term is obtained as a solution of the variational problem involving ordinary differential operator.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124068006","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
A model of string system deformations on a star graph with nonlinear condition at the node 节点处具有非线性条件的星图上的弦系统变形模型
Contemporary Mathematics. Fundamental Directions Pub Date : 2022-12-15 DOI: 10.22363/2413-3639-2022-68-4-635-652
M. Zvereva
{"title":"A model of string system deformations on a star graph with nonlinear condition at the node","authors":"M. Zvereva","doi":"10.22363/2413-3639-2022-68-4-635-652","DOIUrl":"https://doi.org/10.22363/2413-3639-2022-68-4-635-652","url":null,"abstract":"In this paper, a model of deformations of Stieltjes strings system located along a geometric star graph with a nonlinear condition at the node is studied. This kind of condition arises due to the presence of a limiter for the movement of strings in the node under the in uence of an external load. In the present paper, the necessary and sufficient conditions for the extremum of the energy functional are established; existence and uniqueness theorems for the solution are proved; the critical loads at which the strings come into contact with the limiter are analyzed; the dependence of the solution on the limiter length is studied.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127283238","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
Existence of solution of a free boundary problem for reaction-diffusion systems 反应扩散系统自由边界问题解的存在性
Contemporary Mathematics. Fundamental Directions Pub Date : 2022-12-15 DOI: 10.22363/2413-3639-2022-68-4-716-731
G. A. Younes, N. El Khatib, V. Volpert
{"title":"Existence of solution of a free boundary problem for reaction-diffusion systems","authors":"G. A. Younes, N. El Khatib, V. Volpert","doi":"10.22363/2413-3639-2022-68-4-716-731","DOIUrl":"https://doi.org/10.22363/2413-3639-2022-68-4-716-731","url":null,"abstract":"In this paper, we prove the existence of solution of a novel free boundary problem for reaction-diffusion systems describing growth of biological tissues due to cell influx and proliferation. For this aim, we transform it into a problem with fixed boundary, through a change of variables. The new problem thus obtained has space and time dependent coeffcients with nonlinear terms. We then prove the existence of solution for the corresponding linear problem, and deduce the existence of solution for the nonlinear problem using the xed point theorem. Finally, we return to the problem with free boundary to conclude the existence of its solution.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133991093","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
On a system of differential equations with random parameters 关于一个具有随机参数的微分方程组
Contemporary Mathematics. Fundamental Directions Pub Date : 2022-12-15 DOI: 10.22363/2413-3639-2022-68-4-621-634
V. Zadorozhniy, G. Tikhomirov
{"title":"On a system of differential equations with random parameters","authors":"V. Zadorozhniy, G. Tikhomirov","doi":"10.22363/2413-3639-2022-68-4-621-634","DOIUrl":"https://doi.org/10.22363/2413-3639-2022-68-4-621-634","url":null,"abstract":"An explicit formula for the mathematical expectation and second moment functions of a solution to a linear system of ordinary differential equations with a random parameter and a vector random righthand side is obtained. The problem is reduced to the deterministic Cauchy problem for systems of first-order linear partial differential equations. We obtain an explicit formula for a solution of linear systems of partial differential equations of the first order with constant coeffcients. An example is given showing that random factors can have a stabilizing effect on a linear system of differential equations.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"62 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115317639","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
Construction of the planar vector fields with nonsimple critical point of prescribed topological structure 具有规定拓扑结构非简单临界点的平面矢量场的构造
Contemporary Mathematics. Fundamental Directions Pub Date : 2022-12-15 DOI: 10.22363/2413-3639-2022-68-4-575-595
S. Volkov
{"title":"Construction of the planar vector fields with nonsimple critical point of prescribed topological structure","authors":"S. Volkov","doi":"10.22363/2413-3639-2022-68-4-575-595","DOIUrl":"https://doi.org/10.22363/2413-3639-2022-68-4-575-595","url":null,"abstract":"The problem of constructing n-linear (n 2) plane vector elds with isolated critical point and given separatrices of prescribed types is considered. Such constructions are based on the use of vector algebra, the qualitative theory of second-order dynamic systems and classical methods for investigating their critical points. This problem is essentially an inverse problem of the qualitative theory of ordinary di erential equations, and its solution can be used to synthesize mathematical models of controlled dynamical systems of various physical nature.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"94 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124446672","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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