. Filtration in porous media of fluids and gases containing associated with them (dissolved, particulate) solid substances is accompanied by the diffusion of these substances and mass transfer between the liquid (gas) and solid stages. The most common types of mass transfer are sorption and desorption, ion exchange, dissolution and crystallization, mudding, sulfation and suffusion, waxing. We consider the system of equations modeling the process of non-equilibrium sorption. We formulate a difference approximation of the differential problem by an implicit scheme. The solution to the difference problem is constructed by the sweep method. Basing on the numerical results, we can conclude the following: as the relaxation time decreases, the solution to the non-equilibrium problem tends to the solution of the equilibrium problem as the time increases.
{"title":"Numerical modeling of the non-equilibrium sorption process","authors":"I. A. Kaliev, S. Mukhambetzhanov, G. S. Sabitova","doi":"10.13108/2016-8-2-39","DOIUrl":"https://doi.org/10.13108/2016-8-2-39","url":null,"abstract":". Filtration in porous media of fluids and gases containing associated with them (dissolved, particulate) solid substances is accompanied by the diffusion of these substances and mass transfer between the liquid (gas) and solid stages. The most common types of mass transfer are sorption and desorption, ion exchange, dissolution and crystallization, mudding, sulfation and suffusion, waxing. We consider the system of equations modeling the process of non-equilibrium sorption. We formulate a difference approximation of the differential problem by an implicit scheme. The solution to the difference problem is constructed by the sweep method. Basing on the numerical results, we can conclude the following: as the relaxation time decreases, the solution to the non-equilibrium problem tends to the solution of the equilibrium problem as the time increases.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75522598","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 proposes an analogue of Vishik-Lyusternik-Vasileva-Imanalieva boundary functions method for constructing a uniform asymptotic expansion of solutions to bisingular perturbed problems. By means of this method we construct the uniform asymptotic expansion for the solution to the Dirichlet problem for bisingular perturbed second order elliptic equation with two independent variables in a circle. By the maximum principle we justify formal asymptotic expansion of the solution, that is, an estimate for the error term is established.
{"title":"Asymptotic expansions of solutions to Dirichlet problem for elliptic equation with singularities","authors":"D. Tursunov, U. Erkebaev","doi":"10.13108/2016-8-1-97","DOIUrl":"https://doi.org/10.13108/2016-8-1-97","url":null,"abstract":"The paper proposes an analogue of Vishik-Lyusternik-Vasileva-Imanalieva boundary functions method for constructing a uniform asymptotic expansion of solutions to bisingular perturbed problems. By means of this method we construct the uniform asymptotic expansion for the solution to the Dirichlet problem for bisingular perturbed second order elliptic equation with two independent variables in a circle. By the maximum principle we justify formal asymptotic expansion of the solution, that is, an estimate for the error term is established.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88009837","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 devoted to Cauchy problem for equation uxx Qpxqu P puq 0, where Qpxq is a π-periodic function. It is known that for a wide class of the nonlinearities P puq the “most part” of solutions of Cauchy problem for this equation are singular, i.e., they tend to infinity at some finite point of the real axis. Earlier in the case P puq u3 this fact allowed us to propose an approach for a complete description of solutions to this equation bounded on R. One of the ingredients in this approach is the studying of the set U L introduced as the set of the points pu , u1 q in the initial data plane, for which the solutions to the Cauchy problem up0q u , uxp0q u 1 are not singular in the segment r0;Ls. In the present work we prove a series of statements on the set U L and on their base, we classify all possible type of the geometry of such sets. The presented results of the numerical calculations are in a good agreement with theoretical statements.
本文研究了方程uxx Qpxqu P puq0的柯西问题，其中Qpxq是π周期函数。已知对于一类广泛的非线性P - puq，该方程的柯西问题的“大部分”解是奇异的，即它们在实轴的某有限点趋于无穷。在前面的例子P puqu3中，这一事实使我们能够提出一种方法来完整描述以r为界的方程的解。该方法的一个组成部分是研究集合U L作为初始数据平面上的点pu, u1 q的集合，其中柯西问题up0q U, uxp0q q 1的解在段r0;Ls中不是奇异的。本文证明了集合ll上的一系列命题，并在它们的基上对集合的所有可能的几何类型进行了分类。本文给出的数值计算结果与理论结论吻合较好。
{"title":"On solutions of Cauchy problem for equation $u_{xx}+Q(x)u-P(u)=0$ without singularities in a given interval","authors":"G. Alfimov, P. P. Kizin","doi":"10.13108/2016-8-4-24","DOIUrl":"https://doi.org/10.13108/2016-8-4-24","url":null,"abstract":"The paper is devoted to Cauchy problem for equation uxx Qpxqu P puq 0, where Qpxq is a π-periodic function. It is known that for a wide class of the nonlinearities P puq the “most part” of solutions of Cauchy problem for this equation are singular, i.e., they tend to infinity at some finite point of the real axis. Earlier in the case P puq u3 this fact allowed us to propose an approach for a complete description of solutions to this equation bounded on R. One of the ingredients in this approach is the studying of the set U L introduced as the set of the points pu , u1 q in the initial data plane, for which the solutions to the Cauchy problem up0q u , uxp0q u 1 are not singular in the segment r0;Ls. In the present work we prove a series of statements on the set U L and on their base, we classify all possible type of the geometry of such sets. The presented results of the numerical calculations are in a good agreement with theoretical statements.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74191017","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}
{"title":"Symmetry reduction and invariant solutions for nonlinear fractional diffusion equation with a source term","authors":"Stanislav Yur'evich Lukashchuk","doi":"10.13108/2016-8-4-111","DOIUrl":"https://doi.org/10.13108/2016-8-4-111","url":null,"abstract":"","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84068522","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}
. We consider a hydrodynamic type system, waterbag model, that admits a dispersionless Lax representation with a logarithmic Lax function. Using the Lax representation, we construct a recursion operator of the system. We note that the constructed recursion operator is not compatible with the natural Hamiltonian representation of the system.
{"title":"Recursion operator for a system with non-rational Lax representation","authors":"K. Zheltukhin","doi":"10.13108/2016-8-2-112","DOIUrl":"https://doi.org/10.13108/2016-8-2-112","url":null,"abstract":". We consider a hydrodynamic type system, waterbag model, that admits a dispersionless Lax representation with a logarithmic Lax function. Using the Lax representation, we construct a recursion operator of the system. We note that the constructed recursion operator is not compatible with the natural Hamiltonian representation of the system.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78463150","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 we consider gradient type iterative methods for solving the Stokes problem in bounded regions, where the pressure serves as the control; they are obtained by reducing the problem to that of a variational type. In the differential form the proposed methods are very close to the algorithms in the Uzawa family. We construct consistent finite-difference algorithms and we present their approbation on the sequence of grids for solving two-dimensional problem with a known analytic solution.
{"title":"Gradient methods for solving Stokes problem","authors":"I. I. Golichev, Timur Sharipov, N. I. Luchnikova","doi":"10.13108/2016-8-2-22","DOIUrl":"https://doi.org/10.13108/2016-8-2-22","url":null,"abstract":"In the present paper we consider gradient type iterative methods for solving the Stokes problem in bounded regions, where the pressure serves as the control; they are obtained by reducing the problem to that of a variational type. In the differential form the proposed methods are very close to the algorithms in the Uzawa family. We construct consistent finite-difference algorithms and we present their approbation on the sequence of grids for solving two-dimensional problem with a known analytic solution.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88722139","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 we consider the operator L in L2[0,+∞) generated by the differential expression L(y) = y(4) − 2(p(x)y′)′ + q(x)y and boundary conditions y(0) = y′′(0) = 0 in the “degenerate” case, when the roots of associated characteristic equation has different growth rate at the infinity. Assuming a power growth for functions p and q, under some additional conditions of smoothness and regularity kind, we obtain an asymptotic equation for the spectrum allowing us to write out several first terms in the asymptotic expansion for the eigenvalues of the operator L.
{"title":"Asymptotics for the eigenvalues of a fourth order differential operator in a “degenerate” case","authors":"Kh. K. Ishkin, Khairulla Khabibullovich Murtazin","doi":"10.13108/2016-8-3-79","DOIUrl":"https://doi.org/10.13108/2016-8-3-79","url":null,"abstract":"In the paper we consider the operator L in L2[0,+∞) generated by the differential expression L(y) = y(4) − 2(p(x)y′)′ + q(x)y and boundary conditions y(0) = y′′(0) = 0 in the “degenerate” case, when the roots of associated characteristic equation has different growth rate at the infinity. Assuming a power growth for functions p and q, under some additional conditions of smoothness and regularity kind, we obtain an asymptotic equation for the spectrum allowing us to write out several first terms in the asymptotic expansion for the eigenvalues of the operator L.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82511363","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 exponential series with complex exponents, whose real and imaginary parts are integer. We prove that each function analytical in the vicinity of the closure of a bounded convex domain in the complex plain can be expanded into the above mentioned series and this series converges absolutely inside this domain and uniformly on compact subsets. The result is based on constructing a regular subset with a prescribed angular density of the sequence of all complex numbers, whose real and imaginary parts are integer.
{"title":"Representation of analytic functions","authors":"A. I. Abdulnagimov, A. Krivosheev","doi":"10.13108/2016-8-4-3","DOIUrl":"https://doi.org/10.13108/2016-8-4-3","url":null,"abstract":". In this paper we consider exponential series with complex exponents, whose real and imaginary parts are integer. We prove that each function analytical in the vicinity of the closure of a bounded convex domain in the complex plain can be expanded into the above mentioned series and this series converges absolutely inside this domain and uniformly on compact subsets. The result is based on constructing a regular subset with a prescribed angular density of the sequence of all complex numbers, whose real and imaginary parts are integer.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78928491","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}
{"title":"On solutions of second order elliptic equations in cylindrical domains","authors":"A. V. Neklyudov","doi":"10.13108/2016-8-4-131","DOIUrl":"https://doi.org/10.13108/2016-8-4-131","url":null,"abstract":"","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79290141","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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