. 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":"7 1","pages":"39-43"},"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":"48 1","pages":"97-107"},"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":"157 1","pages":"24-41"},"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}
In the paper we establish some tests for absolute Cesáro summability of the Fourier series for almost periodic functions in the Besicovitch space. We consider the case, when the Fourier exponents have a limiting point at zero and as a structure characteristics of the studied function, we use a high order averaging modulus.
{"title":"On absolute Cesáro summablity of Fourier series for almost-periodic functions with limiting points at zero","authors":"Y. Khasanov","doi":"10.13108/2016-8-4-144","DOIUrl":"https://doi.org/10.13108/2016-8-4-144","url":null,"abstract":"In the paper we establish some tests for absolute Cesáro summability of the Fourier series for almost periodic functions in the Besicovitch space. We consider the case, when the Fourier exponents have a limiting point at zero and as a structure characteristics of the studied function, we use a high order averaging modulus.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"69 2","pages":"144-151"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72417440","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 work we find the minimal value that can be taken by the type of an entire function of order 𝜌 ∈ (0 , 1) with zeroes of prescribed upper and lower densities and located in an angle of a fixed opening less than 𝜋 . The main theorem generalizes the previous result by the author (the zeroes lie on one ray) and by A.Yu. Popov (only the upper density of zeros was taken into consideration). We distinguish and study in detail the case when the an entire function has a measurable sequence of zeroes. We provide applications of the obtained results to the uniqueness theorems for entire functions and to the completeness of exponential systems in the space of analytic in a circle functions with the standard topology of uniform convergence on compact sets.
{"title":"Minimal value for the type of an entire function of order $rhoin(0,,1)$, whose zeros lie in an angle and have a prescribed density","authors":"V. Sherstyukov","doi":"10.13108/2016-8-1-108","DOIUrl":"https://doi.org/10.13108/2016-8-1-108","url":null,"abstract":". In the work we find the minimal value that can be taken by the type of an entire function of order 𝜌 ∈ (0 , 1) with zeroes of prescribed upper and lower densities and located in an angle of a fixed opening less than 𝜋 . The main theorem generalizes the previous result by the author (the zeroes lie on one ray) and by A.Yu. Popov (only the upper density of zeros was taken into consideration). We distinguish and study in detail the case when the an entire function has a measurable sequence of zeroes. We provide applications of the obtained results to the uniqueness theorems for entire functions and to the completeness of exponential systems in the space of analytic in a circle functions with the standard topology of uniform convergence on compact sets.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"52 1","pages":"108-120"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72543994","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}
By the method of monotone operators we establish theorems on global existence and uniqueness, as well as estimats and methods of finding the solutions for various classes of nonlinear convolution type integral equations in the real space of 2πperiodic functions Lp(−π, π).
{"title":"Periodic solutions of convolution type equations with monotone nonlinearity","authors":"S. Askhabov","doi":"10.13108/2016-8-1-20","DOIUrl":"https://doi.org/10.13108/2016-8-1-20","url":null,"abstract":"By the method of monotone operators we establish theorems on global existence and uniqueness, as well as estimats and methods of finding the solutions for various classes of nonlinear convolution type integral equations in the real space of 2πperiodic functions Lp(−π, π).","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"91 1","pages":"20-34"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73162029","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 an universal solution to the KdV equation. This solution also satisfies a fifth order ordinary differential equation. We pose the problem on studying the behavior of this solution as t → ∞. For large time, the asymptotic solution has different structure depending on the slow variable s = x2/t. We construct the asymptotic solution in the domains s < −3/4, −3/4 < s < 5/24 and in the vicinity of the point s = −3/4. It is shown that a slow modulation of solution’s parameters in the vicinity of the point s = −3/4 is described by a solution to Painlevé IV equation.
本文考虑了KdV方程的一个通解。这个解也满足一个五阶常微分方程。我们提出了研究该解在t→∞时的行为的问题。对于大时间，随着慢变量s = x2/t的变化，渐近解具有不同的结构。构造了在s <−3/4，−3/4 < s < 5/24和点s =−3/4附近的渐近解。结果表明，在点s =−3/4附近溶液参数的缓慢调制可以用painlevev方程的解来描述。
{"title":"On simultaneous solution of the KdV equation and a fifth-order differential equation","authors":"R. Garifullin","doi":"10.13108/2016-8-4-52","DOIUrl":"https://doi.org/10.13108/2016-8-4-52","url":null,"abstract":"In the paper we consider an universal solution to the KdV equation. This solution also satisfies a fifth order ordinary differential equation. We pose the problem on studying the behavior of this solution as t → ∞. For large time, the asymptotic solution has different structure depending on the slow variable s = x2/t. We construct the asymptotic solution in the domains s < −3/4, −3/4 < s < 5/24 and in the vicinity of the point s = −3/4. It is shown that a slow modulation of solution’s parameters in the vicinity of the point s = −3/4 is described by a solution to Painlevé IV equation.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"104 1","pages":"52-61"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89935774","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":"7 1","pages":"79-94"},"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}
We consider a degenerate fractional order differential equationDα t Lu(t) = Mu(t) in a Hausdorff sequentially complete locally convex space. Under the p-regularity of the operator pair (L,M), we find the phase space of the equation and the family of its resolving operators. We show that the identity image of the latter coincides with the phase space. We prove an unique solvability theorem and obtain the form of the solution to the Cauchy problem for the corresponding inhomogeneous equation. We give an example of application the obtained abstract results to studying the solvability of the initial boundary value problems for the partial differential equations involving entire functions on an unbounded operator in a Banach space, which is a specially constructed Frechét space. It allows us to consider, for instance, a periodic in a spatial variable x problem for the equation with a shift along x and with a fractional order derivative with respect to time t.
考虑Hausdorff序列完备局部凸空间中的退化分数阶微分方程d α t Lu(t) = Mu(t)。在算子对(L,M)的p正则性下，我们得到了方程的相空间及其解析算子族。我们证明了后者的恒等像与相空间重合。证明了相应的非齐次方程的柯西问题的唯一可解定理，得到了柯西问题的解的形式。我们给出了一个应用所得到的抽象结果在Banach空间(这是一个特殊构造的frech空间)上研究无界算子上包含整个函数的偏微分方程初边值问题的可解性的例子。它允许我们考虑，例如，一个关于空间变量x的周期问题对于一个方程，它沿着x移动并且对时间t有分数阶导数。
{"title":"Degenerate fractional differential equations in locally convex spaces with a $sigma$-regular pair of operators","authors":"M. Kostic, V. Fedorov","doi":"10.13108/2016-8-4-98","DOIUrl":"https://doi.org/10.13108/2016-8-4-98","url":null,"abstract":"We consider a degenerate fractional order differential equationDα t Lu(t) = Mu(t) in a Hausdorff sequentially complete locally convex space. Under the p-regularity of the operator pair (L,M), we find the phase space of the equation and the family of its resolving operators. We show that the identity image of the latter coincides with the phase space. We prove an unique solvability theorem and obtain the form of the solution to the Cauchy problem for the corresponding inhomogeneous equation. We give an example of application the obtained abstract results to studying the solvability of the initial boundary value problems for the partial differential equations involving entire functions on an unbounded operator in a Banach space, which is a specially constructed Frechét space. It allows us to consider, for instance, a periodic in a spatial variable x problem for the equation with a shift along x and with a fractional order derivative with respect to time t.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"2007 1","pages":"98-110"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89513257","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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