{"title":"Approximation of solutions to singular integro-differential equations by Hermite-Fejer polynomials","authors":"A. Fedotov","doi":"10.13108/2018-10-2-109","DOIUrl":"https://doi.org/10.13108/2018-10-2-109","url":null,"abstract":"","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"63 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86825824","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 generalize the Lomov’s regularization method for partial differential equations with integral operators, whose kernel contains a rapidly varying exponential factor. We study the case when the upper limit of the integral operator coincides with the differentiation variable. For such problems we develop an algorithm for constructing regularized asymptotics. In contrast to the work by Imanaliev M.I., where for analogous problems with slowly varying kernel only the passage to the limit studied as the small parameter tended to zero, here we construct an asymptotic solution of any order (with respect to the parameter). We note that the Lomov’s regularization method was used mainly for ordinary singularly perturbed integro-differential equations (see detailed bibliography at the end of the article). In one of the authors’ papers the case of a partial differential equation with slowly varying kernel was considered. The development of this method for partial differential equations with rapidly changing kernel was not made before. The type of the upper limit of an integral operator in such equations generates two fundamentally different situations. The most difficult situation is when the upper limit of the integration operator does not coincide with the differentiation variable. As studies have shown, in this case, the integral operator can have characteristic values, and for the construction of the asymptotics, more strict conditions on the initial data of the problem are required. It is clear that these difficulties also arise in the study of an integro-differential system with a rapidly changing kernels, therefore in this paper the case of the dependence of the upper limit of an integral operator on the variable 𝑥 is deliberately avoided. In addition, it is assumed that the same regularity is observed in a rapidly decreasing kernel exponent integral operator. Any deviations from these (seemingly insignificant) limitations greatly complicate the problem from the point of view of constructing its asymptotic solution. We expect that in our further works in this direction we will succeed to weak these restrictions.
{"title":"Regularized asymptotics of solutions to integro-differential partial differential equations with rapidly varying kernels","authors":"A. Bobodzhanov, V. F. Safonov","doi":"10.13108/2018-10-2-3","DOIUrl":"https://doi.org/10.13108/2018-10-2-3","url":null,"abstract":". We generalize the Lomov’s regularization method for partial differential equations with integral operators, whose kernel contains a rapidly varying exponential factor. We study the case when the upper limit of the integral operator coincides with the differentiation variable. For such problems we develop an algorithm for constructing regularized asymptotics. In contrast to the work by Imanaliev M.I., where for analogous problems with slowly varying kernel only the passage to the limit studied as the small parameter tended to zero, here we construct an asymptotic solution of any order (with respect to the parameter). We note that the Lomov’s regularization method was used mainly for ordinary singularly perturbed integro-differential equations (see detailed bibliography at the end of the article). In one of the authors’ papers the case of a partial differential equation with slowly varying kernel was considered. The development of this method for partial differential equations with rapidly changing kernel was not made before. The type of the upper limit of an integral operator in such equations generates two fundamentally different situations. The most difficult situation is when the upper limit of the integration operator does not coincide with the differentiation variable. As studies have shown, in this case, the integral operator can have characteristic values, and for the construction of the asymptotics, more strict conditions on the initial data of the problem are required. It is clear that these difficulties also arise in the study of an integro-differential system with a rapidly changing kernels, therefore in this paper the case of the dependence of the upper limit of an integral operator on the variable 𝑥 is deliberately avoided. In addition, it is assumed that the same regularity is observed in a rapidly decreasing kernel exponent integral operator. Any deviations from these (seemingly insignificant) limitations greatly complicate the problem from the point of view of constructing its asymptotic solution. We expect that in our further works in this direction we will succeed to weak these restrictions.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"39 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84866074","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 consider the coercive properties of a nonlinear biharmonic operator with a matrix operator in the space L2(R n)l and we prove its separability in this space. The considered nonlinear operators are not small perturbation of linear operators. The case of the linear biharmonic operator is considered separately.
{"title":"On coercive properties and separability of biharmonic operator with matrix potential","authors":"Olimzhon Khudoiberdievich Karimov","doi":"10.13108/2017-9-1-54","DOIUrl":"https://doi.org/10.13108/2017-9-1-54","url":null,"abstract":"In the work we consider the coercive properties of a nonlinear biharmonic operator with a matrix operator in the space L2(R n)l and we prove its separability in this space. The considered nonlinear operators are not small perturbation of linear operators. The case of the linear biharmonic operator is considered separately.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"38 1","pages":"54-61"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74526449","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 means of the averaging method, we analyze two model problems on capture into resonance that leads us to the adiabatic approximation in the leading term in the asymptotics. The main aim is an approximate (by using a small parameter) description of the domain of capture into resonance. This domain is in the phase plane and it is formed by the initial points for the resonance solutions with an unboundedly increasing energy. The capture domain depends on an additional parameter involved in the equation. We show that the adiabatic approximation fails as the capture domain becomes narrow. In this case we have to modify substantially the averaging method. As a result, a system of nonlinear differential equations arises for the leading term in the asymptotics and this system is not always integrable.
{"title":"Adiabatic approximation in a resonance capture problem","authors":"L. Kalyakin","doi":"10.13108/2017-9-3-61","DOIUrl":"https://doi.org/10.13108/2017-9-3-61","url":null,"abstract":". By means of the averaging method, we analyze two model problems on capture into resonance that leads us to the adiabatic approximation in the leading term in the asymptotics. The main aim is an approximate (by using a small parameter) description of the domain of capture into resonance. This domain is in the phase plane and it is formed by the initial points for the resonance solutions with an unboundedly increasing energy. The capture domain depends on an additional parameter involved in the equation. We show that the adiabatic approximation fails as the capture domain becomes narrow. In this case we have to modify substantially the averaging method. As a result, a system of nonlinear differential equations arises for the leading term in the asymptotics and this system is not always integrable.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"131 1","pages":"61-75"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74690602","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 first paper of this series we have introduced a certain parametrix and the associated potential. The parametrix corresponds to a uniformly elliptic second order differential operator with locally Hölder continuous coefficients in the half-space. Here we show that the potential is an approximate left inverse of the differential operator modulo hyperplane integrals, with the error estimated in terms of the local Hölder norms. As a corollary, we calculate approximately the potential whose density and differential operator originate from the straightening of a special Lipschitz domain. This corollary is aimed for the future derivation of approximate formulae for harmonic functions.
{"title":"Dicrete Hölder estimates for a certain kind of parametrix. II","authors":"A. I. Parfenov","doi":"10.13108/2017-9-2-62","DOIUrl":"https://doi.org/10.13108/2017-9-2-62","url":null,"abstract":"In the first paper of this series we have introduced a certain parametrix and the associated potential. The parametrix corresponds to a uniformly elliptic second order differential operator with locally Hölder continuous coefficients in the half-space. Here we show that the potential is an approximate left inverse of the differential operator modulo hyperplane integrals, with the error estimated in terms of the local Hölder norms. As a corollary, we calculate approximately the potential whose density and differential operator originate from the straightening of a special Lipschitz domain. This corollary is aimed for the future derivation of approximate formulae for harmonic functions.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"53 1","pages":"62-91"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91374445","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 study an interpolation problem in the class of entire functions of exponential type determined by some majorant in a convergence class (non-quasianalytic majorant). In a smaller class, when the majorant possessed a concavity property, similar problem was studied by B. Berndtsson with the nodes at some subsequence of natural numbers. He obtained a solvability criterion for this interpolation problem. At that, he applied first the H¨ormander method for solving a 𝜕 -problem. In works by A.I. Pavlov, J. Korevaar and M. Dixon, interpolation sequences in the Berndtsson sense were applied successfully in a series of problems in the complex analysis. At that, there was found a relation with approximative properties of the system of powers { 𝑧 𝑝 𝑛 } and with the well known Polya and Macintyre problems. In this paper we establish the criterion of the interpolation property in a more general sense for an arbitrary sequence of real numbers. In the proof of the main theorem we employ a modification of the Berndtsson method.
{"title":"Pavlov - Korevaar - Dixon interpolation problem with majorant in convergence class","authors":"R. Gaisin","doi":"10.13108/2017-9-4-22","DOIUrl":"https://doi.org/10.13108/2017-9-4-22","url":null,"abstract":". We study an interpolation problem in the class of entire functions of exponential type determined by some majorant in a convergence class (non-quasianalytic majorant). In a smaller class, when the majorant possessed a concavity property, similar problem was studied by B. Berndtsson with the nodes at some subsequence of natural numbers. He obtained a solvability criterion for this interpolation problem. At that, he applied first the H¨ormander method for solving a 𝜕 -problem. In works by A.I. Pavlov, J. Korevaar and M. Dixon, interpolation sequences in the Berndtsson sense were applied successfully in a series of problems in the complex analysis. At that, there was found a relation with approximative properties of the system of powers { 𝑧 𝑝 𝑛 } and with the well known Polya and Macintyre problems. In this paper we establish the criterion of the interpolation property in a more general sense for an arbitrary sequence of real numbers. In the proof of the main theorem we employ a modification of the Berndtsson method.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"7 1","pages":"22-34"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89708829","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":"Dirichlet boundary value problem in half-strip for fractional differential equation with Bessel operator and Riemann - Liouville partial derivative","authors":"F. G. Khushtova","doi":"10.13108/2017-9-4-114","DOIUrl":"https://doi.org/10.13108/2017-9-4-114","url":null,"abstract":"","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"1 1","pages":"114-126"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82802514","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 find the leading term of the asymptotics at infinity for some fundamental system of solutions to a class of linear differential equations of arbitrary order τy = λy, where λ is a fixed complex number. At that we consider a special class of ShinZettl type and τy is a quasi-differential expression generated by the matrix in this class. The conditions we assume for the primitives of the coefficients of the quasi-differential expression τy, that is, for the entries of the corresponding matrix, are not related with their smoothness but just ensures a certain power growth of these primitives at infinity. Thus, the coefficients of the expression τy can also oscillate. In particular, this includes a wide class of differential equations of arbitrary even or odd order with distribution coefficients of finite order. Employing the known definition of two quasi-differential expressions with nonsmooth coefficients, in the work we propose a method for obtaining asymptotic formulae for the fundamental system of solutions to the considered equation in the case when the left hand side of this equations is represented as a product of two quasi-differential expressions. The obtained results are applied for the spectral analysis of the corresponding singular differential operators. In particular, assuming that the quasi-differential expression τy is symmetric, by the known scheme we define the minimal closed symmetric operator generated by this expression in the space of Lebesgue square-integrable on [1,+∞) functions (in the Hilbert space L2[1,+∞)) and we calculate the deficiency indices for this operator.
{"title":"Asymptotics of solutions to a class of linear differential equations","authors":"N. N. Konechnaja, K. A. Mirzoev","doi":"10.13108/2017-9-3-76","DOIUrl":"https://doi.org/10.13108/2017-9-3-76","url":null,"abstract":"In the paper we find the leading term of the asymptotics at infinity for some fundamental system of solutions to a class of linear differential equations of arbitrary order τy = λy, where λ is a fixed complex number. At that we consider a special class of ShinZettl type and τy is a quasi-differential expression generated by the matrix in this class. The conditions we assume for the primitives of the coefficients of the quasi-differential expression τy, that is, for the entries of the corresponding matrix, are not related with their smoothness but just ensures a certain power growth of these primitives at infinity. Thus, the coefficients of the expression τy can also oscillate. In particular, this includes a wide class of differential equations of arbitrary even or odd order with distribution coefficients of finite order. Employing the known definition of two quasi-differential expressions with nonsmooth coefficients, in the work we propose a method for obtaining asymptotic formulae for the fundamental system of solutions to the considered equation in the case when the left hand side of this equations is represented as a product of two quasi-differential expressions. The obtained results are applied for the spectral analysis of the corresponding singular differential operators. In particular, assuming that the quasi-differential expression τy is symmetric, by the known scheme we define the minimal closed symmetric operator generated by this expression in the space of Lebesgue square-integrable on [1,+∞) functions (in the Hilbert space L2[1,+∞)) and we calculate the deficiency indices for this operator.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"108 1","pages":"76-86"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77584518","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 propose a new approach for studying differential operators with a discontinuous weight function. We study the spectral properties of a differential operator on a finite segment with separated boundary conditions and with “matching” condition at the discontinuity point of the weight function. We assume that the potential of the operator is a summable function on the segment, on which the operator is considered. For large value of the spectral parameter we obtain an asymptotics for the fundamental system of solutions of the corresponding differential equation. By means of this asymptotics we study the “matching” conditions of the considered differential operator. Then we study the boundary conditions of the considered operator. As a result, we obtain an equation for the eigenvalues of the operator, which an entire function. We study the indicator diagram of the equation for the eigenvalues; this diagram is a regular octagon. In various sectors of the indicator diagram we find the asymptotics for the eigenvalues of the studied differential operator.
{"title":"Study of differential operator with summable potential and discontinuous weight function","authors":"S. I. Mitrokhin","doi":"10.13108/2017-9-4-72","DOIUrl":"https://doi.org/10.13108/2017-9-4-72","url":null,"abstract":". In the work we propose a new approach for studying differential operators with a discontinuous weight function. We study the spectral properties of a differential operator on a finite segment with separated boundary conditions and with “matching” condition at the discontinuity point of the weight function. We assume that the potential of the operator is a summable function on the segment, on which the operator is considered. For large value of the spectral parameter we obtain an asymptotics for the fundamental system of solutions of the corresponding differential equation. By means of this asymptotics we study the “matching” conditions of the considered differential operator. Then we study the boundary conditions of the considered operator. As a result, we obtain an equation for the eigenvalues of the operator, which an entire function. We study the indicator diagram of the equation for the eigenvalues; this diagram is a regular octagon. In various sectors of the indicator diagram we find the asymptotics for the eigenvalues of the studied differential operator.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"15 1","pages":"72-84"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87294228","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 obtain two results on the behavior of Dirichlet series on a real axis.Thefirst of them concerns the lower bound for the sum of the Dirichlet series on the system of segments [ 𝛼, 𝛼 + 𝛿 ]. Here the parameters 𝛼 > 0, 𝛿 > 0 are such that 𝛼 ↑ + ∞ , 𝛿 ↓ 0. The needed asymptotic estimates is established by means of a method based on some inequalities for extremal functions in the appropriate non-quasi-analytic Carleman class. This approach turns out to be more effective than the known traditional ways for obtaining similar estimates. The second result specifies essentially the known theorem by M.A. Evgrafov on existence of a bounded on R Dirichlet series. According to Macintyre, the sum of this series tends to zero on R . We prove a spectific estimate for the decay rate of the function in an Macintyre-Evgrafov type example.
{"title":"Estimate for growth and decay of functions in Macintyre - Evgrafov kind theorems","authors":"A. Gaisin, G. Gaisina","doi":"10.13108/2017-9-3-26","DOIUrl":"https://doi.org/10.13108/2017-9-3-26","url":null,"abstract":". In the paper we obtain two results on the behavior of Dirichlet series on a real axis.Thefirst of them concerns the lower bound for the sum of the Dirichlet series on the system of segments [ 𝛼, 𝛼 + 𝛿 ]. Here the parameters 𝛼 > 0, 𝛿 > 0 are such that 𝛼 ↑ + ∞ , 𝛿 ↓ 0. The needed asymptotic estimates is established by means of a method based on some inequalities for extremal functions in the appropriate non-quasi-analytic Carleman class. This approach turns out to be more effective than the known traditional ways for obtaining similar estimates. The second result specifies essentially the known theorem by M.A. Evgrafov on existence of a bounded on R Dirichlet series. According to Macintyre, the sum of this series tends to zero on R . We prove a spectific estimate for the decay rate of the function in an Macintyre-Evgrafov type example.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"84 1 1","pages":"26-36"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79538940","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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