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## POWER GEOMETRY OF A NON-LINEAR DIFFERENTIAL EQUATION

Moscow Mathematical Journal. 2018. Vol. 18. No. 2. P. 387-402.

****In the article the solutions of Emden–Fowler-type equa- tions of any order are studied using methods of power geometry. It is shown that these methods can be successfully applied in the study of asymptotic behaviour of the solutions. Also, we find conditions for the existence (nonexistence) of solutions of new types having non- power (power-logarithmic) asymptotics. Some numerical characteristics of such solutions are given.

V.S.Samovol, Mathematical notes 2014 Vol. 95 No. 5 P. 708-720

The paper deals with solutions to Emden-Fowler-type equations of any arbitrary order. The asymptotic properties of solutions to these equations are studied and a systematic survey of numerous uncoordinated results of analysis of continuable and noncontinuable solutions is given. ...

Added: January 25, 2015

V.S. Samovol, Asymptotic Analysis 2019 Vol. 115 P. 223-239

The Riccaty equation with coefficients having power asymptotic in a neighbourhood of infinity is studied. We examine the solutions to these equations and describe their asymptotic form ...

Added: November 24, 2019

V.S.Samovol, Mathematical notes 2014 Vol. 95 No. 6 P. 843-855

The paper contains the results of the study of the asymptotic proprties of solutions with integer-valued asymptotics as wel as of solutions arising from the rapid decrease of the coefficient of the equation. To analize the asymptotic behavior of solutions of the equations, methods of power geometry are used. ...

Added: January 25, 2015

Gaydukov R. K., Fonareva A. V., Russian Journal of Mathematical Physics 2019 Vol. 26 No. 3 P. 334-343

The problem of viscous compressible fluid flow in an axially symmetric pipe with small periodic irregularities on the wall is considered for large Reynolds numbers. An asymptotic solution with double-deck structure of the boundary layer and unperturbed core flow is obtained. Numerical investigations of the influence of the density of the core flow on the flow behavior in ...

Added: September 2, 2019

V.S. Samovol, Doklady Mathematics 2018 Vol. 97 No. 3 P. 250-253

A Riccati equation with coefficients expandable into convergent power series in a neighborhood of
infinity is considered. Continuable solutions to equations of this type are studied. Conditions for the expansion
of these solutions into convergent series in a neighborhood of infinity are obtained by methods of power
geometry. ...

Added: July 6, 2018

Chernyshev V. L., Hilberdink T., Nazaikinskii V. E., Russian Journal of Mathematical Physics 2020 Vol. 27 No. 4 P. 456-468

We find the logarithmic asymptotics as T tends to infinity of the number of partitions of positive numbers not exceeded T into sums of positive numbers t_j with j less or equal k, where the counting function of t_j grows as a power of j. The asymptotics is uniform with respect to k. ...

Added: October 31, 2020

Ludmila I. Kuzmina, Osipov Y., Vetoshkin N., International Journal for Computational Civil and Structural Engineering 2018 Vol. 14 No. 1 P. 92-101

The filtration problem in a porous medium is an important part of underground hydromechanics. Filtration of suspensions and colloids determines the processes of strengthening the soil and creating waterproof walls in the ground while building the foundations of buildings and underground structures. It is assumed that the formation of a deposit is dominated by the ...

Added: July 1, 2018

V.S.Samovol, Mathematical notes 2019 Vol. 105 No. 4 P. 592-603

The Riccati equation with coefficients expandable in convergent power series in a neigh- borhood of infinity are considered. Extendable solutions of such equations are studied. Methods of power geometry are used to obtain conditions for convergent series expansions of these solutions. An algorithm for deriving such series is given. ...

Added: April 10, 2019

Gaydukov R., European Journal of Mechanics - B/Fluids 2021 Vol. 89 P. 401-410

An asymptotic solution with double-deck boundary layer structure is constructed for the problem of an incompressible fluid flow along a semi-infinite plate with small localized or periodic (fast-oscillating) irregularities on the surface whose shape depends on time. Numerical simulation of flow in the near-plate region is presented for two types of the shape change: oscillations of the ...

Added: January 4, 2021

Protasov V., Mathematics of Computation 2017 Vol. 86 No. 305 P. 1499-1524

For an arbitrary set of nonnegative integers, we consider the Euler binary partition function which equals the total number of binary expansions of an integer with ``digits'' from . By applying the theory of subdivision schemes and refinement equations, the asymptotic behaviour of as is characterized. For all finite , we compute the lower and upper exponents of growth of , find when ...

Added: May 11, 2017

V.S.Samovol, Doklady Mathematics 2020 Vol. 101 No. 1 P. 49-52

Scalar real Riccati equations with coefficients expandable in convergent power series in a neighborhood
of infinity are considered. Extendable solutions to equations of this kind are studied. Methods of
power geometry are used to obtain conditions for asymptotic series expansions of these solutions. ...

Added: October 10, 2020

Ludmila I. Kuzmina, Osipov Y., International Journal for Computational Civil and Structural Engineering 2013 Vol. 9 No. 3 P. 31-39

A classic pursuit problem is considered in which the pursuer is always moving towards the target. The shape of the mechanical trajectory is established, and the time of motion is calculated. An integral for the length of the pursuit curve is constructed, its asymptotics is calculated and compared with the result of the numerical computation. ...

Added: November 14, 2013

Badanin A., Korotyaev E., Mathematische Nachrichten 2020 Vol. 293 P. 210-225

We consider fourth order ordinary differential operators on the half-line and on the
line, where the perturbation has compactly supported coefficients. The Fredholm
determinant for this operator is an analytic function in the whole complex plane without
zero.We describe the determinant at zero.We show that in the generic case it has
a pole of order 4 in the case ...

Added: February 5, 2021

Samovol V.S., Doklady Mathematics 2017 Vol. 96 No. 1 P. 373-376

The Riccati equation is considered. Both continuable and noncontinuable solutions of this equation are studied. Asymptotic representations of its solutions are obtained by power geometry methods ...

Added: December 18, 2017

V.L. Chernyshev, Minenkov D. S., Nazaikinskii V. E., Functional Analysis and Its Applications 2016 Vol. 50 No. 4 P. 291-307

We find the asymptotics of the element counting function for an additive arithmetic semigroup with exponential growth of the counting function of prime generators. ...

Added: October 18, 2015

Fonareva A. V., Gaydukov R. K., Russian Journal of Mathematical Physics 2021 Vol. 28 No. 2 P. 224-243

A subsonic flow of a viscous compressible fluid in a two-dimensional channel with small periodic or localized irregularities on the walls for large Reynolds numbers is considered. A formal asymptotic solution with double-deck structure of the boundary layer is constructed. A nontrivial time hierarchy is discovered in the decks. An analysis of the scales of irregularities at ...

Added: March 22, 2021

Л.И. Кузьмина, Осипов Ю. В., Вестник Московского государственного строительного университета 2014 № 7 С. 34-40

A classic pursuit problem is studied in which two material points - a Pursuer and a Pursued - move in a plane at constant speeds. The velocity vector of the Pursued does not change its direction and the velocity vector of the Pursuer turns and always aims at the Pursued. If the Pursuer moves at ...

Added: August 3, 2014

Grushin V. V., Математические заметки 2017 Т. 102 № 6 С. 816-827

Asymptotic formulas are obtained for a class of integrals that are Fourier transforms of rapidly oscillating functions. These formulas contain special functions and generalize the well-known method of stationary phase. ...

Added: January 13, 2018

Gordin V. A., М. : Физматлит, 2013

Описаны аналитические и численные методы исследования уравнений и систем в частных производных: гиперболических, параболических, эллиптических и смешанного типа, линейных и нелинейных. Список этих методов и приемов велик, и они должны дополнять друг друга: интегральные преобразования, вариационное исчисление, специальные функции, асимптотические методы, сплайны, рациональные аппроксимации.
Книга адресована читателю, который использует и аналитические, и численные, компьютерные ...

Added: March 18, 2013

L.I. Kuzmina, Osipov Y. V., Вестник Московского государственного строительного университета 2017 No. 11 P. 1278-1283

Subject: a groundwater filtration affects the strength and stability of underground and hydro-technical constructions. Research objectives: the study of one-dimensional problem of displacement of suspension by the flow of pure water in a porous medium. Materials and methods: when filtering a suspension some particles pass through the porous medium, and some of them are stuck ...

Added: January 6, 2018

Grushin V. V., Наноструктуры. Математическая физика и моделирование 2012 Т. 7 № 2 С. 17-44

В работе получено асимптотическое разложение собственных значений оператора Лапласа с нулевыми данными Дирихле в бесконечных трубках, т. е. в бесконечных изогнутых цилиндрах с внутренним кручением при равномерном сжатии поперечных сечений, по малому параметру, характеризующему поперечные размеры трубки. Аналогичное разложение получено для оператора Шредингера с учетом магнитного поля для конечных изогнутых трубок. Предложен метод сведения задачи ...

Added: December 24, 2012

Samovol V. S., Математические заметки 2014 Т. 95 № 6 С. 911-925

The asymptotic properties of nonoscillating solutions of Emden-Fowler-type equations of arbitrary order are considered. The paper contains the results of the study of the asymptotic properties of solutions with integer-valued asymptotics as well as of solutions arising from the rapid decrease of the coefficient of the equation. To analyze the asymptotic behavior of solutions of ...

Added: February 13, 2014

Gaydukov R. K., Indian Journal of Pure and Applied Mathematics 2024

The paper is deals with the heat transfer in the laminar flow of a viscous incompressible fluid along a plate with a small heated localized irregularity. It is well known that double- and triple-deck structures of the boundary layer arise in such flow problems. These structures make it possible, among other things, to describe the phenomenon of ...

Added: May 21, 2024

LIUDMILA I. KUZMINA, Osipov Y., International Journal for Computational Civil and Structural Engineering 2020 Vol. 16 No. 4 P. 50-56

Filtration of suspension in a porous medium is actual in the construction of tunnels and underground structures.
A model of deep bed filtration with size-exclusion mechanism of particle capture is considered. The inverse filtration
problem – finding the Langmuir coefficient from a given concentration of suspended particles at the porous medium outlet is solved using the asymptotic solution ...

Added: July 2, 2021