Asymptotic Integration Research Articles

Asymptotic integrability of nonlinear wave equations.

We introduce the notion of asymptotic integrability into the theory of nonlinear wave equations. It means that the Hamiltonian structure of equations describing propagation of high-frequency wave packets is preserved by hydrodynamic evolution of the large-scale background wave so that these equations have an additional integral of motion. This condition is expressed mathematically as a system of equations for the carrier wave number as a function of the background variables. We show that a solution of this system for a given dispersion relation of linear waves is related to the quasiclassical limit of the Lax pair for the completely integrable equation having the corresponding dispersionless and linear dispersive behavior. We illustrate the theory with several examples.

Analysis of the three‐dimensional elasticity theory problem for a radially inhomogeneous cylinder

AbstractIn this paper an axisymmetric problem of elasticity theory for a radially inhomogeneous cylinder of small thickness is studied. It is assumed that homogeneous mixed boundary conditions are specified on lateral surfaces of the cylinder, and boundary conditions that leave the cylinder in equilibrium are specified on the ends of the cylinder. It is considered that elastic moduli are arbitrary continuous functions of a variable along the radius of the cylinder. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thinness of the cylinder walls. To determine its solution, the method of asymptotic integration is used, based on three iteration processes. All solutions of the equilibrium equation are constructed, satisfying the specified homogeneous boundary conditions on lateral surfaces. It is obtained that the solution of the problem consists of a penetrating solution and a solution of the nature of the boundary layer. An analysis of stress‐strain states corresponding to different types of solutions is carried out. Based on the constructed asymptotic solutions, a connection is found between the solutions obtained by the equation of elasticity theory and by the applied theory of shells. It is shown that the penetrating solution determines the internal stress‐strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the principal vector of stress acting in a cross section perpendicular to the axis of the cylinder.The first terms of asymptotic expansions of the penetrating solution in a small parameter coincide with solutions in the applied theory of shells. New classes of solutions are defined as the character of a boundary layer, which are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in the applied theories of shells. The first terms of the asymptotic expansion of the solution, having the nature of a boundary layer, are equivalent to the Saint‐Venant edge effect in the theory of heterogeneous plates. Asymptotic formulas for displacement and stresses are constructed, allowing one to calculate the three‐dimensional stress‐strain state of a radially heterogeneous cylinder of small thickness with any predetermined accuracy. Based on the obtained asymptotic expansions, one can estimate the applicability areas of applied theories and construct a refined applied theory for radially heterogeneous cylindrical shells.

PERIODIC ATEB-FUNCTIONS AND THE VAN DER POL METHOD FOR CONSTRUCTING SOLUTIONS OF TWO-DIMENSIONAL NONLINEAR OSCILLATIONS MODELS OF ELASTIC BODIES

In the process of operation, the simplest elements (hereinafter elastic bodies) of machines and mechanisms under the influence of external and internal factors carry out complex oscillations ‒ a combination of longitudinal, bending and torsion combinations in various combinations. In general, mathematical models of the process of such complex phenomena in elastic bodies, even for one-dimensional calculation models, are boundary value problems for systems of partial differential equations. A two-dimensional mathematical model of oscillatory processes in a nonlinear elastic body is considered. A method of constructing an analytical solution of the corresponding boundary-value problems for nonlinear partial differential equations is proposed, which is based on the use of Ateba functions, the Van der Pol method, ideas of asymptotic integration, and the principle of single-frequency oscillations. For "undisturbed" analogues of the model equations, single-frequency solutions were obtained in an explicit form, and for "perturbed" ‒ analytical dependences of the basic parameters of the oscillation process on a small perturbation. The dependence of the main frequency of oscillations on the amplitude and non-linearity parameter of elastic properties in the case of single-frequency oscillations of "unperturbed motion" is established. An asymptotic approximation of the solution of the autonomous "perturbed" problem is constructed. Graphs of changes in amplitude and frequency of oscillations depending on the values of the system parameters are given.

The asymptotic solution of the elasticity theory problem for a radially inhomogeneous cylinder

The elasticity theory problem for a radially inhomogeneous cylinder of small thickness, whose elastic moduli are arbitrary continuous functions depending on the radius of the cylinder, is considered. It is assumed that the side surface of the cylinder is stress-free, and the boundary conditions that keep the cylinder in equilibrium are given at its seats. Asymptotic solutions are constructed by the asymptotic integration method. It is shown that the asymptotic solution consists of the sum of the penetrating solution, simple boundary effect and boundary layer solutions. The character of the stress-strain state corresponding to the penetrating solution, simple boundary effect and boundary layer solutions is determined. Asymptotic formulas are obtained for displacements and stresses, which allow to calculate the stress-strain state of a cylinder.The problem of torsion of a radially inhomogeneous cylinder is studied, with its lateral surface free from stress and the boundary conditions keeping it in equilibrium at its seats. By applying the asymptotic integration method, it is determined that the asymptotic solution for the torsion problem consists of the sum of the penetrating solution and boundary layer solutions.Numerical analysis is performed and the effect of material inhomogeneity on the stress-strain state of the cylinder is evaluated.

Quasibound and quasinormal modes of a thick brane in Rastall gravity

In this work, we study the gravitational quasinormal modes of the thick brane in Rastall gravity. Using the asymptotic iteration and direct integration methods, we solve the quasinormal frequencies of the Rastall thick brane. We also obtained the waveforms of these quasinormal modes through numerical evolution. The results indicate that although the Rastall thick brane lacks a bound zero mode, when the Rastall parameter λ ≳ 0, a long-lived quasinormal mode appears. This long-lived quasinormal mode may restore the four-dimensional effective Newtonian potential on the brane on a large scale. This may provide a new perspective for the localization of gravity on thick branes, that a thick brane does not necessarily require the gravity to be localized, perhaps quasi-localized is sufficient.

ON THE PROBLEM OF THE THEORY OF ELASTICITY FOR THE RADIALLY INHOMOGENEOUS TRANSVERSAL ISOTROPIC CYLINDER WITH A FIXED LATERAL SURFACE

Using the method of asymptotic integration of the equations of the theory of elasticity, the axisymmetric problem of the theory of elasticity for a radially inhomogeneous transversally isotropic cylinder of small thickness is studied. Suppose that the elastic moduli are arbitrary continuous functions of the radius of the cylinder. It is assumed that the side part of the cylinder is fixed, and stresses are set at the ends of the cylinder, leaving the cylinder in equilibrium. Solutions have been determined having the nature of a boundary layer and are localized at the ends of the cylinder. The first terms of its asymptotic expansion coincide with the Saint-Venant edge effect in the theory of plates. The nature of the stress-strain state has been studied. It is shown that some boundary layer solutions may not be extinguished by propagating far from the ends.

Construction of homogeneous solutions of the torsion problem for a radially inhomogeneous transversely isotropic cylinder

The torsion problem for a radially inhomogeneous transversely isotropic cylinder of small thickness was investigated by the method of asymptotic integration of elasticity theory equations. It is assumed that the side part of the cylinder is stress-free, and boundary conditions are set at the ends of the cylinder, leaving the cylinder in equilibrium. The elastic moduli are thought to be arbitrary continuous functions of the variable along the cylinder radius. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thin-walledness of the cylinder. Homogeneous solutions are built, i.e. any solutions of the equilibrium equation satisfying the condition of no stresses on the side surfaces. It is shown that the solution of the torsion problem consists of a penetrating solution and a boundary layer character solution similar to Saint-Venant's edge effect in the theory of inhomogeneous plates. The penetrating solution determines the internal stress-strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the torsional moments of stresses acting in the cross-section perpendicular to the cylinder axis. Solutions having the boundary layer character are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in applied shell theories. Asymptotic formulas for displacement and stresses are built, which make it possible to calculate the three-dimensional stress-strain state of a radially inhomogeneous transversely isotropic cylinder of small thickness. Based on the obtained asymptotic expansions, it is possible to assess the applicability of applied theories and build a refined applied theory for radially inhomogeneous cylindrical shells

Asymptotic long-wave model for a high-contrast two-layered elastic plate

The paper is concerned with the derivation of asymptotically consistent equations governing the long-wave flexural response of a two-layered rectangular plate with high-contrast elastic properties. In the general case, the plate is under dynamic and variable surface, volume, and edge forces. Performing the asymptotic integration of the three-dimensional (3D) elasticity equations in the transverse direction and satisfying boundary conditions on the faces and interface, we derived the sequence of two-dimensional (2D) differential equations with respect to required functions in the first two approximations. The eight independent restraints for the generalized displacements and stress resultants are considered to formulate the 16 independent variants of boundary conditions. One of the main results of the paper is the Timoshenko–Reissner type equation capturing the effect of the softer layer and taking into account the in-plane deformation induced by the edge forces. Comparative calculations of natural frequencies were carried out based on our and alternative models.

Interpolating asymptotic integration methods for second-order differential equations

The problem of asymptotic behaviour at infinity of solutions to second-order differential equation can be reduced via the Liouville transform to that of an equation with almost constant coefficients. In the present paper, we compare various methods of asymptotic integration in application to the reduced equation $u"-(\lambda^2+\varphi(t))u=0$ and interpolate the corresponding results in the case $\operatorname{Re}\lambda>0$, provided that a complex-valued function $\varphi(t)$ is in a certain sense small for large values of the argument.

Interpolating methods of asymptotic integration for second order differential equations

Для дифференциальных уравнений второго порядка на полуоси вопрос о поведении их решений на бесконечности может быть сведен посредством преобразования Лиувилля к аналогичной задаче для уравнения с почти постоянными коэффициентами. В настоящей работе проводится сравнительный анализ различных методов и интерполяция результатов асимптотического интегрирования применительно непосредственно к редуцированному дифференциальному уравнению $u"-(\lambda^2+\varphi(t))u=0$ в случае, когда $\operatorname{Re}\lambda>0$, а комплекснозначная функция $\varphi(t)$ в том или ином смысле мала при больших значениях аргумента. Библиография: 15 наименований.

Asymptotic model of long-wave vibrations of an ultrathin strip-beam taking into account surface effects

The paper is concerned with the derivation of asymptotically consistent equations governing long-wave vibrations of an ultrathin elastic strip-beam taking into account surface effects within the framework of the Gurtin-Murdoch theory of the surface elasticity. Two dimensional equations of motion of an elastic isotropic medium are used as the initial ones. In the general case, the strip-beam is under the action of variable unsteady surface forces. The presence of residual shear stresses is assumed on the face surfaces. The ratio of the strip thickness to the characteristic length of bending deformation is considered as a small parameter. Within the framework of the Gurtin-Murdoch theory, two cases are considered, providing for the presence of large a) residual stresses on the faces, b) effects of surface inertia. Using the method of asymptotic integration over the transverse direction, relations for displacements and stresses in an ultra-thin strip-beam are obtained, and equivalent one dimensional the Timoshenko type equations taking into account surface effects are derived. As an example, free vibrations of a simply supported beam accounting for the surface effects are considered.

Solution of a Singularly Perturbed Mixed Problem on the Half-Line for a Parabolic Equation with a Strong Turning Point of the Limit Operator

We study singularly perturbed problems in the presence of spectral singularities of the limit operator using S.A. Lomov’s regularization method. In particular, a regularized asymptotic solution is constructed for a singularly perturbed inhomogeneous mixed problem on the half-line for a parabolic equation with a strong turning point of the limit operator. Based on the idea of asymptotic integration of problems with unstable spectrum, it is shown how regularizing functions and additional regularizing operators should be introduced, the formalism of the regularization method for this type of singularity is described in detail, this algorithm is justified, and an asymptotic solution of any order in a small parameter is constructed.

New Qualitative Outcomes for Ordinary Differential Systems of Second Order

This paper deals with a nonlinear ordinary differential system of second order. In the paper, qualitative properties of solutions of the system called asymptotic stability (AS), uniform stability (US), boundedness, ultimately boundedness (UB) and integrability of solutions, are investigated by using the second method of Lyapunov. We give four new qualitative results and an example as a numerical application of the results. The results of this article extend and improve some earlier ones in the literature.

Regularized Asymptotics of the Solution of a Singularly Perturbed Mixed Problem on the Semiaxis for the Schrödinger Equation with the Potential Q = X2

In this paper, we study the solution of a singularly perturbed inhomogeneous mixed problem on the half-axis for the Schrödinger equation in the presence of a “strong” turning point for the limit operator on time interval that do not contain focal points. Based on the ideas of the regularization method for asymptotic integration of problems with an unstable spectrum, it is shown how regularizing functions should be constructed for this type of singularity. The paper describes in detail the formalism of the regularization method, justifies the algorithm, constructs an asymptotic solution of any order in a small parameter, and proves a theorem on the asymptotic convergence of the resulting series.

Dynamic Axisymmetric Tension of a Thin Round Ideally Rigid-Plastic Layer

We consider the stress-strain state that occurs during dynamic tension of a homogeneous round layer of an incompressible ideally rigid-plastic material that obeys the Mises–Genka criterion. The upper and lower bases are stress-free, and the radial velocity is set on the lateral boundary. The possibility of thickening or thinning of the layer is taken into account, which simulates neck formation and further development of the neck. Two characteristic tension modes are revealed. First one is associated with a rather high rate of removal of the side boundary of the layer from the center, the second one is associated with acceleration. In the second case, we have carried out an analysis using the method of asymptotic integration, which makes it possible to approximately find the parameters of the stress-strain state.

Dynamic Regimes of Biaxial Stretching of a Thin Ideally Rigid-Plastic Rectangular Plate

The stress-strain state arising during dynamic tension of a homogeneous rectangular plate of an incompressible ideally rigid-plastic material, which obeys the Mises–Hencky criterion, is considered. The upper and lower bases are stress-free, longitudinal velocities are set at the ends. The possibility of deformation of the upper and lower sides of the plate is taken into account, which simulates neck formation and further development of the neck. A small geometric parameter is introduced – the ratio of the average thickness of the plate to its length along one of the directions. At different time intervals, the order of smallness of the dimensionless functions characterizing the dynamic stretching mode may be different with respect to the geometric parameter, which determines one or another stretching mode. Two such characteristic modes have been identified, one is associated with a sufficiently high rate of removal of the ends of the plate from each other, the second with acceleration. In the second case, an analysis was carried out using the method of asymptotic integration, which allows us to approximately find the parameters of the stress-strain state.

On the fundamental qualitative properties of integro-delay differential equations

This paper discusses qualitative properties of solutions of certain unperturbed and perturbed systems of nonlinear integro-delay differential equations (IDDEs), namely asymptotic stability, uniform stability, integrability and boundedness. Here, four new theorems are proved on these properties of solutions by using Lyapunov–Krasovskiǐ functional (LKF) technique. As illustrations and applications of our results, we also provide two examples, solve them numerically, and plot the trajectories of their solutions. The results of this paper include weaker sufficient conditions than the ones found in the literature, e.g., some superfluous conditions are removed here, and the results have also new contributions to the qualitative theory of integro-differential equations (IDEs) and IDDEs.

Methodology of investigation the dynamics of longitudinally moving systems under the action of impulse perturbations

A methodology for researching oscillations of one-dimensional elastic systems under the influence of impulse disturbances has been developed. The mathematical model of the considered process is boundary value problems for hyperbolic equations of the second order with an irregular right-hand side and a mixed derivative with respect to linear and time variables. As for the irregularity of the right part, it takes into account the impulse effect on the research object and has the most important theoretical and practical periodic character; and with the help of the mixed derivative in the Euler variables, the longitudinal movement of the studied systems is partially taken into account. Using the basic ideas of asymptotic integration of equations with partial derivatives, the basic propositions of the wave theory of motion adapted to the considered class of problems, the principle of single-frequency oscillations in nonlinear systems with concentrated masses and distributed parameters, the method of regularization of impulse disturbances, equations in the standard form were obtained that describe the determining parameters of nonlinear oscillations for resonant and non-resonant cases. It is shown: that the longitudinal movement of the systems has a significant effect on the natural frequency of oscillations, and therefore on the period of impulse disturbances during which the resonance process takes place; the amplitude of the passage of resonance depends significantly on the speed of movement. The results obtained in the work can serve as a basis for choosing the operating modes of some classes of systems characterized by longitudinal movement in order to avoid resonant processes in them.

A Regularized Asymptotic Solution of the Cauchy Problem for the Nonhomogeneous Schroedinger Equation in the Quasiclassical Approximation in the Presence of a "strong" Turning Point of the Limit Operator

The article is devoted to the development of the regularization method by S.A. Lomov on singularly perturbed problems in the presence of spectral singularities of the limit operator. In particular, a regularized asymptotic solution is constructed for the singularly perturbed inhomogeneous Cauchy problem that arises in the quasiclassical approximation in the Schroedinger equation in the coordinate representation. The potential energy profile chosen in the paper leads to a singularity in the spectrum of the limit operator in the form of a <> turning point. Based on the ideas of asymptotic integration of problems with unstable spectrum, S.A. Lomov and A.G. Eliseev, it is indicated how and from what considerations regularizing functions and additional regularizing operators should be introduced, the formalism of the regularization method for the indicated type of singularity is described in detail, this algorithm is substantiated, and an asymptotic solution of any order with respect to a small parameter is constructed.

On the Fundamental Analyses of Solutions to Nonlinear Integro-Differential Equations of the Second Order

In this article, a scalar nonlinear integro-differential equation of second order and a non-linear system of integro-differential equations with infinite delays are considered. Qualitative properties of solutions called the global asymptotic stability, integrability and boundedness of solutions of the second-order scalar nonlinear integro-differential equation and the nonlinear system of nonlinear integro-differential equations with infinite delays are discussed. In the article, new explicit qualitative conditions are presented for solutions of both the second-order scalar nonlinear integro-differential equations with infinite delay and the nonlinear system of integro-differential equations with infinite delay. The proofs of the main results of the article are based on two new Lyapunov–Krasovskiĭ functionals. In particular cases, the results of the article are illustrated with three numerical examples, and connections to known tests are discussed. The main novelty and originality of this article are that the considered integro-differential equation and system of integro-differential equations with infinite delays are new mathematical models, the main six qualitative results given are also new.