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Asymptotic solutions of the eigenvalue problem for an Euler operator in a neighborhood of a regular singular point are considered. We find a condition under which the asymptotic expansion is free of logarithms. Eigenvalues expressed in terms of elementary functions in the form of a finite sum of quasi-polynomials are obtained for third-order Euler operators and also for commuting Euler operators of sixth and ninth orders. The problem on common eigenfunctions for commuting Euler operators is examined. In the case of operators of rank 2 and 3, it can be reduced to second- and third-order Bessel equations by differential substitutions.

For the second-order nonlinear ordinary differential equation $$ {u}_{xx}^{\hbox{'}\hbox{'}}={u}^5-{tu}^3-x, $$ we prove the existence and uniqueness of a strictly increasing solution, which satisfies an initial condition and a limit condition at infinity and whose graph lies between the zero equation and the continuous graph of the root of the nondifferential equation u5 − tu3 − x = 0. For this solution, we find an asymptotics, which is uniform on the ray t ∈ (−∞,−Mt) as x → +∞; separately, we construct asymptotics on the ray s > Ms and on the segment 0 ≤ s ≤ Ms, where s = |t|−5/2x is the variable compressed with respect to x. Using the method of matching of asymptotic expansions, we construct a composite asymptotic expansion of the solution to the Cauchy problem whose initial conditions are found from the theorem on the existence of solutions to the original problem. Finally, we construct a uniform asymptotic expansion under the restriction t ≤ 0 as x2 + t2 →∞.

In this paper, we consider the sine-Gordon equation with a high-frequency parametric pumping and a weak dissipative force. We examine the class of π-kink-type solutions that are soliton solutions of the nonperturbed sine-Gordon equation. In contrast to stable 2π-kinks, these solutions are unstable. We prove that the time of decaying of π-kinks due to small perturbations is proportional to the cube of the inverse period of fast oscillations of the parametric pumping. We derive a two-time asymptotic expansion of a solution of the boundary-value problem and analyze evolution of a wave packet whose leading term has the form of a π-kink. Numerical simulations of solutions confirm the good qualitative agreement with asymptotic expansions.

We consider a one-parameter family of number series involving the generalized harmonic series and study asymptotic properties of the remainders. Using $$ R\left(N,p\right)\equiv \sum \limits_{n=N}^{\infty }1/{n}^p $$ as an example, we describe the typical obtained results: we obtain the integral representation, find the complete asymptotic expansion with respect to the parameter 2N − 1 as N →∞, and prove that R(N, p) is enveloped by its asymptotic series. The possibilities of the proposed approach are demonstrated by the problem of exact two-sided estimates for the central binomial coefficient.

We consider the Schrodinger operator on a graph consisting of two infinite edges, a loop, and a glued (at the start and end points of the loop) graph obtained by e−1 times contraction of some fixed graph. The Kirchhoff conditions are imposed at interior vertices and the Dirichlet or Neumann conditions are imposed at boundary vertices of the graph. We show that the resolvent of the Schrodinger operator is holomorphic with respect to the small parameter e and write out the first three terms of the asymptotic expansion of the resolvent.

We study the spectral problem for a two-dimensional Hartree type operator with smooth selfaction potential. We find asymptotic eigenvalues and eigenfunctions and construct an asymptotic expansion for quantum averages near the lower boundaries of spectral clusters.

We consider a continuous-time branching random walk on ℤ d with birth and death of particles at a periodic set of points (sources of branching). Spectral properties of the evolution operator of the mean number of particles are studied. We derive a representation of the mean value of particle number in a form of asymptotic series.

We examine asymptotic expansions of the third Painleve transcendents for αδ ≠ = 0 and γ = 0 in the neighborhood of infinity in a sector of aperture

This paper continues a series of publications on the shortwave diffraction of the plane wave on prolate bodies of revolution with axial symmetry in the Neumann problem. The approach, which is based on the Leontovich–Fock parabolic equation method for the two parameter asymptotic expansion of the solution, is briefly described. Two correction terms are found for the Fock’s main integral term of the solution expansion in the boundary layer. This solution can be continuously transformed into the ray solution in the illuminated zone and decays exponentially in the shadow zone. If the observation point is in the shadow zone near the scatterer, then the wave field can be obtained with the help of residue theory for the integrals of the reflected field, because the incident field does not reach the shadow zone. The obtained residues are necessary for the unique construction of the creeping waves in the boundary layer of the scatterer in the shadow zone.

We consider a linear parabolic boundary-value problem in a thin 3D starlike joint that consists of a finite number of thin curvilinear cylinders connected through a domain (node) with diameter O(e). We develop a procedure for the construction of the complete asymptotic expansion of the solution as e → 0, i.e., in the case where the starlike joint is transformed into a graph. By using the method of matching of the asymptotic series, we deduce the limit problem (e = 0) on a graph with the corresponding Kirchhoff-type conjugation conditions at the vertex. We also prove asymptotic estimates, which enable us to trace the influence of the geometric shape of the node and the physical processes running in the node on the global asymptotic behavior of the solution.

An asymptotic expansion for the solution to a nonhomogeneous difference equation of general type is obtained. The influence of the roots of the characteristic equation is taken into account. The asymptotic behavior of the remainder is established, depending on the asymptotics of the nonhomogeneous term of the equation.

The paper studies asymptotic behavior of posterior distribution of a real parameter centered by a $$ \sqrt{n} $$ -consistent estimate. The uniform analog of the Bernstein–von Mises theorem is proved. This result is extended to asymptotic expansion of the posterior distribution in powers of n−1/2. This expansion is generalized as the expansion of expectations of functions with polynomial majorant with respect to posterior distribution.

The paper is a continuation of previous papers of the authors dealing with the exploration of shortwave diffraction by smooth and strictly convex bodies of revolution (the axisymmetric case). In these problems, the boundary layer method contains two large parameters: one is the Fock parameter M and the second is Λ that characterizes the oblongness of the scatterer. This naturally gives the possibility of using the two-scaled asymptotic expansion, where both M and Λ are regarded as independent. The approximate formulas for the wave field in this situation depend on the mutual strength between the large parameters and may vary. In the paper, we carry out numerical experiments with our formulas, in the case where the Fock analytical solution is in good coincidence with the exact solution of a model problem, in order to examine the influence of the parameter Λ on the wave field. It follows from our numerical experiments that the influence of the oblongness of the scatterer on the wave field is really insignificant if the method of Leontovich–Fock parabolic equation does not meet mathematical difficulties.

We consider a nonuniform Neumann boundary value problem for the Poisson equation in a thin 3D aneurysm type domain consisting of thin curvilinear cylinders joined through an aneurysm of diameter ϐ(e). We develop a rigorous procedure for constructing a complete asymptotic expansion of the solution as e → 0. We prove energy and uniform pointwise estimates, which allows us to observe the impact of the aneurysm. Bibliography: 21 titles. Illustrations: 5 figures.

A certain convolution equation is studied on a large finite interval. This equation arose in acoustics for description of a wave conductor surface with a bed of ice. The main feature of this equation is that the symbol of the corresponding operator has zeros of power order in the dual variable, so that the inverse operator is a long-range one. A complete power-order asymptotic expansion is constructed for the kernel of the inverse operator as the length of the interval tends to infinity.

We construct an expansion of a solution to the Goursat problem for the sine-Gordon equation with respect to a small parameter that is involved in the data on characteristics. We prove that the expansion obtained is an asymptotic series.

We consider the transmission conditions at vertices of the graph modeling a periodic rectangular lattice of thin quantum waveguides described by the spectral Dirichlet problem for the Laplace operator. The type of transmission conditions is determined by the structure of the space B o of bounded solutions to the boundary layer problem in a cross-shaped waveguide with a circular core of radius R. We describe all variants of the structure of the space B st of nondecaying solutions and present methods for constructing hardly probable and very probable examples. Based on the method of matched asymptotic expansion, we construct all possible transmission conditions. We discuss numerical methods for computing critical radii, construction of the space B st , and classification of “trapped”/“almost standing” waves.

The paper generalizes results by E. Janvresse, T. de la Rue, and Y. Velenik and results by the second author on the fluctuations in the ergodic sums for the Pascal adic transformation in the case of an arbitrary ergodic invariant measure and arbitrary cylinder function.

We consider an elliptic operator in a multidimensional domain with frequent alternation of the Dirichlet condition and the Robin boundary condition in the case where the homogenized operator contains only the original Robin boundary condition. We prove the uniform resolvent convergence of the perturbed operator to the homogenized operator and obtain order sharp estimates for the rate of convergence. We construct a complete asymptotic expansion for the resolvent in the case where the resolvent acts on sufficiently smooth functions and the alternation of boundary conditions is strictly periodic and is given on a multidimensional hyperplane. Bibliography: 23 titles.

We construct an asymptotic expansion for the fundamental system of solutions of a linear singularly perturbed system of m-order differential equations with degenerate principal matrix at the higher-order derivatives. The case where the corresponding characteristic polynomial has a multiple spectrum is investigated. It is proved that, in this case, the asymptotic expansions are constructed in the fractional powers of the small parameter. The recurrence formulas for the coefficients of these expansions are obtained.

In a thin isotropic homogeneous three-dimensional plate of thickness h, we consider the limit passage of elastic fields as h → +0. It is assumed that the connected component Γ N of the boundary of the median cross section ω is a broken line with links of length ah, where a > 0 is a fixed parameter. We consider the Lame equations with the Neumann condition (a free surface) on the plate bases, the Dirichlet condition (a rigidly clamped surface) on a smooth part, and some linear contact conditions on a ribbed part of the lateral surface. For the solution to the boundary value problem we obtain an asymptotic expansion with different boundary layers. In the two-dimensional model, new boundary conditions, different from the contact ones, arise on the limit smooth contour Γ0 , thereby for the spatial elasticity problem, we confirm the Babuska paradox caused by linear boundary conditions on the plate edge, but not unilateral constraints of Signorini type. Bibliography: 42 titles. Illustrations: 5 figures.

We consider a linear system of partial differential equations with the Stokes operator in the principal part and coefficients that rapidly oscillate in time. In the case where the limiting (averaged) stationary part of the system degenerates, we use the boundary layer method for constructing a complete formal asymptotic expansion of a time-periodic solution.

We construct and substantiate an asymptotic expansion for the solution of an inhomogeneous Neumann boundary-value problem for the Poisson equation whose right-hand side depends on longitudinal and transverse variables in a thin cascade domain. We find the asymptotic energy and establish uniform pointwise estimates for the difference between the solution of the original problem and the solution of the corresponding limit problem.

An asymptotic analysis of surface water waves is made. The gradient of the ocean depth is assumed to be small. An ansatz for the asymptotic expansion in this small parameter is suggested. The goal is to construct the complete asymptotic expansion for a solution localized near a moving line on the liquid surface. The approach is based on the space-time ray method. The eiconal and amplitudes are found in the form of formal power series. Chains of recurrent equations for their terms are derived. Their solvability is proved. The first term is obtained in explicit form. Bibliography: 11 titles.

UDC 539.375 We perform computation of a prefracture zone in the vicinity of a corner point of an interface from which an interface crack emanates by the Wiener–Hopf method under plane strain. The zone is modeled by a segment of discontinuity of normal displacement. We investigate the dependence of the length of the prefracture zone and its opening displacement at the corner point on load, the angle of interface fracture, and elastic parameters of the media. On the basis of the strain criterion, the conditions of start of the crack are investigated. According to the modern notions of fracture of brittle bodies, a prefracture zone, which is a region of weakened interparticle bonds, forms ahead of the front of a crack. Upon the attainment of a certain limit distance between individual layers of the material in the prefracture zone, adhesion forces between them cannot resist the tensile stress, and bonds rupture, which leads to the propagation of the crack. The determination of the conditions of start of interface cracks in piecewise homogeneous bodies is of great interest for the fracture mechanics of composites, welded and glued joints, building materials, and structures. However, most works on this theme are devoted to cracks located on a plane interface of media [10, 14, 21]. At the same time, the more general problem of an interface crack that emanates from a corner point of a broken interface of media is insufficiently investigated. A corner point of an interface of two different media is a stress concentrator with a power singularity [6], the order of which is determined by the root of the corresponding characteristic equation that is the smallest in the interval (−1, 0) . The propagation of an interface crack from the corner point changes substantially the character of the stress-strain state (SSS) in the vicinity of the vertex of the corner; in particular, in certain intervals of interfacial angles, the characteristic equation may have complex conjugate roots, which cause physically incorrect space oscillations of stresses and displacements [2, 17]. To each root from the interval (−1, 0) there corresponds a singular term in the expansion of stresses into asymptotic series in the vicinity of the corner point. In the investigation of the fine structure of the tip of a crack, one most often restricts oneself to the largest term of the asymptotic expansion. Stress concentration at the corner point leads to the formation of a prefracture zone near it. At the beginning of the stage, when its size is much smaller than the length of the crack, the configuration of the prefracture zone depends substantially on the character of the SSS near the tip of the crack, which is determined by the asymptotic solution of an analogous problem without a prefracture zone. In [4, 5, 11, 13], a calculation of model prefracture zones at the end of a mode I crack reaching a nonsmooth interface of two different media was performed. In [3], a calculation of the initial prefracture zone in the vicinity of the tip of an interface crack at a corner point of an interface of two elastic materials was performed with regard for only one summand with a power singularity. At the same time, ranges of interfacial angles, to which the oscillating singular summands in the expansion of stresses correspond, remained unstudied.

We study the asymptotic behavior of positive solutions to nonlinear elliptic equations of Emden–Fowler type with absorption term. For operators with variable coefficients we obtain conditions on coefficients under which the solutions have the same asymptotics as solutions to the model equation Δu = −x| p |u| σ−1 u. For positive solutions we obtain lower order terms of the asymptotic expansion at infinity. Bibliography: 10 titles.

In a plane domain with several small holes of diameter O(e), we consider the von Karman equations describing the flexure of a thin isotropic plate. We construct asymptotic expansions for solutions to the nonlinear and the corresponding linearized problems. The coefficients of expansions turn out to be holomorphic and rational functions of |ln e|−1. The asymptotic results for the linear problem (the Kirchhoff plate) are interpreted within the framework of the theory of selfadjoint extensions of differential operators by using the tool of weighted spaces with separated asymptotics. We also present a model of a nonlinear singularly perturbed problem that provides high accuracy asymptotic formulas. This problem includes the generalized Sobolev conditions at the points to which the holes shrink. Bibliography: 34 titles. Illustrations: 1 figure.

We propose a method of double asymptotic expansion of an approximate analytic solution of the problem on thermal irradiation of annular radiator fins with a trapezoidal cross-section, which is reduced to a nonlinear differential equation of the second order with variable coefficients. The method is based on using the Poincare small parameter method and method of phase integrals.

External asymptotic expansions of the solutions of the problem of nonstationary thermal conductivity of laminated anisotropic inhomogeneous shells under different boundary conditions on faces are constructed. We analyze the obtained two-dimensional resolving equations and investigate the asymptotic properties of the solutions of the problem of thermal conductivity. A physical justification of some features of the asymptotic expansion of temperature is presented.

A complete asymptotic expansion is constructed for solutions of the Cauchy problem for nth order linear ordinary differential equations with rapidly oscillating coefficients, some of which may be proportional to ω n/2, where ω is oscillation frequency. A similar problem is solved for a class of systems of n linear first-order ordinary differential equations with coefficients of the same type. Attention is also given to some classes of first-order nonlinear equations with rapidly oscillating terms proportional to powers ω d . For such equations with d ∈ (1/2, 1], conditions are found that allow for the construction (and strict justification) of the leading asymptotic term and, in some cases, a complete asymptotic expansion of the solution of the Cauchy problem.

We consider initial boundary value problems for parabolic differential equations with rapidly oscillating coefficients in thin perforated domains of rapidly varying thickness. Under certain symmetry conditions on the domain and coefficients, we construct an asymptotic expansion of a solution to the problem with homogeneous third kind conditions on the exterior boundary and the boundary of cavities. In the case of inhomogeneous Neumann conditions, we construct an asymptotic solution without symmetry assumptions and prove an asymptotic estimate in the corresponding Sobolev space. Bibliography: 27 titles. Illustrations: 1 figure.

This paper considers systems of ordinary differential equations of the 1st order whose terms oscillate with a high frequency ω, among which there exist those proportional to \( \sqrt \omega \), the large high-frequency summands. For the problem on periodic solutions, the author justifies the averaging method and establishes a posteriori bounds for the error of partial sums of the complete asymptotic expansion for the solution.

The behavior of the remainder term of the asymptotic expansion for solutions of a quasi-linear parabolic Cauchy-Dirichlet problem in a sequence of domains with fine-granulated boundary is studied. By using a modification of the asymptotic expansion and new pointwise estimates of solutions of the model problem, the uniform convergence of the remainder term to zero is proved.

The author obtains an asymptotic expansion of solutions of Problem (1.1), (1.3) using the Vishik-Lyusternik method.

The semimartingale stochastic approximation procedure, precisely, the Robbins-Monro type SDE, is introduced, which naturally includes both generalized stochastic approximation algorithms with martingale noises and recursive parameter estimation procedures for statistical models associated with semimartingales. General results concerning the asymptotic behavior of the solution are presented. In particular, the conditions ensuring the convergence, the rate of convergence, and the asymptotic expansion are established. The results concerning the Polyak weighted averaging procedure are also presented.

We consider statistics of the form $$Q_n = \sum\limits_{j = 1}^N {a_{jj} } (X_j^2 - \mu _2 ) + \sum\limits_{1 \leqslant j \ne k \leqslant N} {a_{jk} } X_j X_k $$ , where the Xj are i.i.d. random variables with finite sixth moment. We obtain the rate of convergence in the central limit theorem for the one-term Edgeworth expansion. Furthermore, applications to Toeplitz matrices, quadratic form of ARMA-processes, goodness-of-fit, as well as spacing statistics are included. Bibliography: 16 titles.

A high-frequency asymptotic expansion of the boundary-layer type is constructed for flexural waves localized in the vicinity of the free edge of a Kirchhoff-Love elastic plate. Unlike previous papers on the subject, the boundary of the plate is not assumed to be rectilinear. Expressions for the leading-order terms of the expansion are obtained, which are then used in the problem of describing the eigenmodes of an arbitrary bounded plate with smooth boundary. Bibliography: 16 titles.

An asymptotic expansion of the solution to the Neumann problem for a second-order equation in a thin domain with peak-like edge is constructed and justified. Owing to the sharpness of the edge, the procedure of dimension reduction leads to a degenerate limit equation on the longitudinal cross-section of the domain and a solution has irregular behavior near the boundary. Bibliography: 20 titles.

This article deals with the asymptotic behavior as e → 0 of the solution {uɛ, pɛ} of the plane Stokes problem in a perforated domain. The limit problem is constructed and estimates for the speed of convergence are obtained. It is shown that the speed of convergence is of order O(e3/2).

The Steklov problem considered in the paper describes free two-dimensional oscillations of an ideal, incompressible, heavy fluid in a half-plane covered by a rigid dock with two symmetric gaps. Equivalent reduction of the problem to two spectral problems for integral operators allows us to find limits for all eigenfrequencies as the distance between the gaps tends to zero or infinity. For the fundamental eigenfrequency and the corresponding eigenfunction, two terms are found in the asymptotic expansion as the distance tends to infinity. It is proved that all eigenvalues are simple for any distance. Bibliography: 15 titles.

The concordance method of asymptotic expansions applied for constructing uniform asymptotic expansions of singularly-perturbed partial differential equations and systems is presented.

In the paper, we consider problems in perforated domains with rapidly oscillating boundary. We assume that the oscillating boundary has a special structure. The width of the “hills” has the same size as the cell of periodicity inside the domain and the order of the height of the “hills” is bigger. Under these assumptions we construct the homogenized problem, prove the homogenization theorem, construct the correctors, and obtain estimates of the deviation of the solution to the initial problem from the leading terms of the asymptotic series.