In the work we construct two modifications of the classical Faà-di-Bruno formula. We consider the applications of these formulae in the integrability theory for nonlinear partial differential equations. We discuss the problem on integration by parts in the Gelfand-Olver-Sanders formal variational calculus.
{"title":"On applications of Faà-di-Bruno formula","authors":"A. Shabat, Magomed Khochalaevich Efendiev","doi":"10.13108/2017-9-3-131","DOIUrl":"https://doi.org/10.13108/2017-9-3-131","url":null,"abstract":"In the work we construct two modifications of the classical Faà-di-Bruno formula. We consider the applications of these formulae in the integrability theory for nonlinear partial differential equations. We discuss the problem on integration by parts in the Gelfand-Olver-Sanders formal variational calculus.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"22 1","pages":"131-136"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90851667","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":"G{aa}rding inequality for higher order elliptic operators with a non-power degeneration and its applications","authors":"S. A. Iskhokov, M. G. Gadoev, I. Yakushev","doi":"10.13108/2016-8-1-51","DOIUrl":"https://doi.org/10.13108/2016-8-1-51","url":null,"abstract":"","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"99 1","pages":"51-67"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85904340","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":"Quantum aspects of the integrability of the third Painlevé equation and a non-stationary time Schrödinger equation with the Morse potential","authors":"B. Suleimanov","doi":"10.13108/2016-8-3-136","DOIUrl":"https://doi.org/10.13108/2016-8-3-136","url":null,"abstract":"","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"24 1","pages":"136-154"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82984614","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 nonlinear optimal control problems for semi-linear elliptic equations with discontinuous data and solutions (states) with controls in the boundary conditions of conjugation of heterogeneous media and in the right hand side of the state equation. We prove the differentiability and Lipshitz continuity for the grid analogue of the cost functional for extremum problems.
{"title":"On Frech`{e}t differentiability of cost functional in optimal control of coefficients of elliptic equations","authors":"A. Manapova, F. Lubyshev","doi":"10.13108/2016-8-1-79","DOIUrl":"https://doi.org/10.13108/2016-8-1-79","url":null,"abstract":"In the work we consider nonlinear optimal control problems for semi-linear elliptic equations with discontinuous data and solutions (states) with controls in the boundary conditions of conjugation of heterogeneous media and in the right hand side of the state equation. We prove the differentiability and Lipshitz continuity for the grid analogue of the cost functional for extremum problems.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"1 1","pages":"79-96"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90178342","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 a new general method allowing us to study the problem on constructing hyperbolicity and stability regions for nonlinear dynamical systems. The method is based on a modification of the method by M. Rozo for studying the stability of linear systems with periodic coefficients depending on a small parameter and on the asymptotic formulae in the perturbation theory of linear operators. We obtain approximate formulae describing the boundary of hyperbolicity and stability regions. As an example, we provide the scheme for constructing the stability regions for Mathieu equation.
{"title":"The asymptotic formulae in the problem on constructing hyperbolicity and stability regions of dynamical systems","authors":"L. S. Ibragimova, I. Mustafina, M. Yumagulov","doi":"10.13108/2016-8-3-58","DOIUrl":"https://doi.org/10.13108/2016-8-3-58","url":null,"abstract":"The paper proposes a new general method allowing us to study the problem on constructing hyperbolicity and stability regions for nonlinear dynamical systems. The method is based on a modification of the method by M. Rozo for studying the stability of linear systems with periodic coefficients depending on a small parameter and on the asymptotic formulae in the perturbation theory of linear operators. We obtain approximate formulae describing the boundary of hyperbolicity and stability regions. As an example, we provide the scheme for constructing the stability regions for Mathieu equation.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"1 1","pages":"58-78"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83036813","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}
As it is known, Darboux transformations play a key role in applications in ISP. In the paper we expose the theory of such transformations for the Schrödinger equations with compactly supported potential, which are not necessarily smooth. We study a new class of transformations connected with the zeroes of the reflection coefficient located at conjugated points in the complex plane.
{"title":"Darboux transformations in the inverse scattering problem","authors":"Mukhtar Shamil'evich Badakhov, A. Shabat","doi":"10.13108/2016-8-4-42","DOIUrl":"https://doi.org/10.13108/2016-8-4-42","url":null,"abstract":"As it is known, Darboux transformations play a key role in applications in ISP. In the paper we expose the theory of such transformations for the Schrödinger equations with compactly supported potential, which are not necessarily smooth. We study a new class of transformations connected with the zeroes of the reflection coefficient located at conjugated points in the complex plane.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"47 1","pages":"42-51"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88184319","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 discuss some growth rates of compositions of entire and meromorphic functions on the base of generalized relative order and generalized relative lower order of meromorphic functions with respect to entire functions.
{"title":"Some results on generalized relative orders of meromorphic functions","authors":"D. Kumar, Biswas Tanmay, Das Pranab","doi":"10.13108/2016-8-2-95","DOIUrl":"https://doi.org/10.13108/2016-8-2-95","url":null,"abstract":"In this paper we discuss some growth rates of compositions of entire and meromorphic functions on the base of generalized relative order and generalized relative lower order of meromorphic functions with respect to entire functions.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"1 1","pages":"95-103"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89433750","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 the expansion of a meromorphic function in partial fractions","authors":"Maergoiz Lev Sergeevich","doi":"10.13108/2016-8-2-104","DOIUrl":"https://doi.org/10.13108/2016-8-2-104","url":null,"abstract":"","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"41 1","pages":"104-111"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76580879","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 obtain the upper bound for the decay rate of the solution to the Dirichlet initial boundary value problem for a linear parabolic second order equation with a double degeneracy 𝜇 ( 𝑥 ) 𝑢 𝑡 = ( 𝜌 ( 𝑥 ) 𝑎 𝑖𝑗 ( 𝑡, 𝑥 ) 𝑢 𝑥 𝑖 ) 𝑥 𝑗 in an unbounded domain. For a wide class of revolution domains we prove a lower bound. We adduce the examples showing that the upper and lower bounds are in some sense sharp. We prove the unique solvability of the problem in an unbounded domain by Galerkin’s approximations method.
{"title":"On decay of solution to linear parabolic equation with double degeneracy","authors":"V. F. Vil'danova","doi":"10.13108/2016-8-1-35","DOIUrl":"https://doi.org/10.13108/2016-8-1-35","url":null,"abstract":". We obtain the upper bound for the decay rate of the solution to the Dirichlet initial boundary value problem for a linear parabolic second order equation with a double degeneracy 𝜇 ( 𝑥 ) 𝑢 𝑡 = ( 𝜌 ( 𝑥 ) 𝑎 𝑖𝑗 ( 𝑡, 𝑥 ) 𝑢 𝑥 𝑖 ) 𝑥 𝑗 in an unbounded domain. For a wide class of revolution domains we prove a lower bound. We adduce the examples showing that the upper and lower bounds are in some sense sharp. We prove the unique solvability of the problem in an unbounded domain by Galerkin’s approximations method.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"50 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78270630","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 employ the integral Laplace transform to invert the generalized RiemannLiouville operator in a closed form. We establish that the inverse generalized RiemannLiouville operator is a differential or integral-differential operator. We establish a relation between Riemann-Liouville operator and Temlyakov-Bavrin operator. We provide new examples of generalized Riemann-Liouville operator.
{"title":"Inverting of generalized Riemann - Liouville operator by means of integral Laplace transform","authors":"I. I. Bavrin, O. Yaremko","doi":"10.13108/2016-8-3-41","DOIUrl":"https://doi.org/10.13108/2016-8-3-41","url":null,"abstract":"We employ the integral Laplace transform to invert the generalized RiemannLiouville operator in a closed form. We establish that the inverse generalized RiemannLiouville operator is a differential or integral-differential operator. We establish a relation between Riemann-Liouville operator and Temlyakov-Bavrin operator. We provide new examples of generalized Riemann-Liouville operator.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"19 1","pages":"41-48"},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83778492","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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