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We investigated the asymptotic dynamics of a nonlinear system modeling binary mixture of solids with delay term. Using the recent quasi-stability methods introduced by Chueshov and Lasiecka, we prove the existence, smoothness and finite dimensionality of a global attractor. We also prove the existence of exponential attractors. Moreover, we study the upper semicontinuity of global attractors with respect to small perturbations of the delay terms.

We discuss the dynamic mean-variance (MV) problem for pairs trading with the assumptions that one of the security prices satisfies a stochastic volatility model (SVM) and the corresponding price spread follows an Ornstein–Uhlenbeck (OU) process. We provide a semi-closed-form of the optimal strategy based on the solution of a PDE, which is difficult to solve explicitly. Thus, we assume that one of the security prices satisfies the Scott model, a fast-mean-reverting volatility model, and give a closed-form approximation for the optimal strategy. Empirical studies, by using historical data from Chinese security markets, show that the Scott model produces a more stable strategy by better capturing mean-reverting volatility.

Several studies have been conducted on scaling limits for Markov-modulated infinite-server queues. To the best of our knowledge, most of these studies adopt an approach to prove the convergence of the moment-generating function (or characteristic function) of the random variable that represents a scaled version of the number of busy servers and show the weak law of large numbers and the central limit theorem (CLT). In these studies, an essential assumption is the finiteness of the phase process and, in most of them, the CLT for the number of busy servers conditional on the phase (or the joint states) has not been considered. This paper proposes a new method called the moment approach to address these two limitations in an infinite-server batch service queue, which is called the M/MX/∞ queue. We derive the conditional weak law of large numbers and a recursive formula that suggests the conditional CLT. We derive series expansion of the conditional raw moments, which are used to confirm the conditional CLT by a symbolic algorithm.

In this article we obtain a 1-dimensional asymptotic model for a junction of thin hyperelastic rods as the thickness goes to zero. We show, under appropriate hypotheses on the loads, that the deformations that minimize the total energy weakly converge in a Sobolev space towards the minimum of a 1 D-dimensional energy for elastic strings by using techniques from Γ-convergence.

We study the process of two-phase flow in thin porous media domains of Brinkman-type. This is generally described by a model of coupled, mixed-type differential equations of fluids' saturation and pressure. To reduce the model complexity, different approaches that utilize the thin geometry of the domain have been suggested. We focus on a reduced model that is formulated as a single nonlocal evolution equation of saturation. It is derived by applying standard asymptotic analysis to the dimensionless coupled model, however, a rigid mathematical derivation is still lacking. In this paper, we prove that the reduced model is the analytical limit of the coupled two-phase flow model as the geometrical parameter of domain's width--length ratio tends to zero. Precisely, we prove the convergence of weak solutions for the coupled model to a weak solution for the reduced model as the geometrical parameter approaches zero.

One-stage stochastic linear complementarity problem (SLCP) is a special case of a multi-stage stochastic linear complementarity problem, which has important applications in economic engineering and operations management. In this paper, we establish asymptotic analysis results of a sample-average approximation (SAA) estimator for the SLCP. The asymptotic normality analysis results for the stochastic-constrained optimization problem are extended to the SLCP model and then the conditions, which ensure the convergence in distribution of the sample-average approximation estimator for the SLCP to multivariate normal with zero mean vector and a covariance matrix, are obtained. The results obtained are finally applied for estimating the confidence region of a solution for the SLCP.

To perform an asymptotic analysis of spectra of singularly perturbed periodic waveguides, it is required to estimate remainders of asymptotic expansions of eigenvalues of a model problem on the periodicity cell uniformly with respect to the Floquet parameter. We propose two approaches to this problem. The first is based on the max?min principle and is sufficiently easily realized, but has a restricted application area. The second is more universal, but technically complex since it is required to prove the unique solvability of the problem on the cell for some value of the spectral parameter and the Floquet parameter in a nonempty closed segment, which is verified by constructing an almost inverse operator of the operator of an inhomogeneous model problem in variational setting. We consider boundary value problems on the simplest periodicity cell: a rectangle with a row of fine holes.

One of the current directions in the development of the modern theory of oscillations is the elaboration of effective methods for analyzing the stability of solutions of dynamical systems. The aim of the work is to develop a new asymptotic method for studying the nonlinear monotonic stability of the amplitude of plane oscillations in a dynamic system of equations with one fast phase. The method is based on the use of the method of variation of an arbitrary constant, the averaging method, and the classical method of mathematical research of the function of one independent variable. It is assumed that the resulting approximate analytical function is defined and twice continuously differentiable on the entire considered interval of change of the independent variable. It describes the nonlinear and monotonic evolution of the oscillation amplitude on the entire considered interval of change of the independent variable. In the paper, this method is applied to the problem of nonlinear monotonic aerodynamic damping of the amplitude of oscillations of the angle of attack during the descent of a symmetric spacecraft in the atmosphere of Mars. The method presented in this paper made it possible to find all characteristic cases of nonlinear monotonic stability and instability of the oscillation amplitude of the angle of attack. In addition, one should speak of a symmetrical quantity of different cases of stability and instability, located on different sides of the zero value of the first average derivative of the angle of attack.

In this paper, we consider the Poisson equation on a “long” domain which is the Cartesian product of a one-dimensional long interval with a (d − 1)-dimensional domain. The right-hand side is assumed to have a rank-1 tensor structure. We will present and compare methods to construct approximations of the solution which have tensor structure and the computational effort is governed by only solving elliptic problems on lower-dimensional domains. A zero-th order tensor approximation is derived by using tools from asymptotic analysis (method 1). The resulting approximation is an elementary tensor and, hence has a fixed error which turns out to be very close to the best possible approximation of zero-th order. This approximation can be used as a starting guess for the derivation of higher-order tensor approximations by a greedy-type method (method 2). Numerical experiments show that this method is converging towards the exact solution. Method 3 is based on the derivation of a tensor approximation via exponential sums applied to discretized differential operators and their inverses. It can be proved that this method converges exponentially with respect to the tensor rank. We present numerical experiments which compare the performance and sensitivity of these three methods.

In this paper, we deal with the asymptotic modeling of the behavior of a reinforced rectangular plate with a thin layer of high thermal conductivity. We focus on a thermo-elastic model described by a set of nonlinear time dependent partial differential equations, accounting for nonlinear mechanical and nonlinear thermal coupling. Our aim is to model this junction and reproduce the effect of the thin body by means of approximate boundary conditions, obtained by an asymptotic analysis with respect to the thickness of this latter. The study is carried out for two kinds of layers: Stiff and soft. Different limit behaviors occur according to the rigidity or the softness of this latter.

This work is concerned with the problem of identifying the shape, size and location of a small embedded tumor from measured temperature on the skin surface. The temperature distribution in the biological tissue is governed by the Pennes model equation. The proposed approach is based on the Kohn–Vogelius formulation and the topological sensitivity analysis method. The ill-posed geometric inverse problem is reformulated as a topology optimization. The temperature field perturbation, caused by the presence of a small anomaly, is analyzed and estimated. A topological asymptotic formula, describing the variation of the considered Kohn–Vogelius type functional with respect to the presence of a small anomaly is derived.

This paper considers a tandem queueing network with a Poisson arrival process of incoming calls, two servers, and two infinite orbits by the method of asymptotic analysis. The servers provide services for incoming calls for exponentially distributed random times. Blocked customers at each server join the orbit of that server and retry to enter the server again after an exponentially distributed time. Under the condition of low retrial rates, we prove that the joint stationary distribution of scaled numbers of calls in the orbits weakly converges to a two-variable Normal distribution.

This work addresses the mathematical analysis – by means of asymptotic analysis – of a penalisation strategy for the full discretisation of elastic wave propagation problems in quasi-incompressible media that has been recently developed by the authors. We provide a convergence analysis of the solution to the continuous version of the penalised problem towards its formal limit when the penalisation parameter tends to infinity. Moreover, as a fundamental intermediate step we provide an asymptotic analysis of the convergence of solutions to quasi-incompressible problems towards solutions to purely incompressible problems when the incompressibility parameter tends to infinity. Finally, we further detail the regularity assumptions required to guarantee that the mentioned convergence holds.

We investigate the boundary layers of a singularly perturbed reaction-diffusion equation in a 3D channel domain. The equation is supplemented with a Robin boundary condition especially when the smooth function on the boundary, appearing in the Robin boundary condition, depends on the perturbation parameter. By constructing an explicit function, called corrector, which describes behavior of the perturbed solution near the boundary, we obtain an asymptotic expansion of the perturbed solution as the sum of the corresponding limit solution and the corrector, and show the convergence in L 2 of the perturbed solution to the limit solution as the perturbation parameter tends to zero.

We give sufficient conditions for the existence of weak solutions to quasilinear elliptic Dirichlet problem driven by the A-Laplace operator in a bounded domain Ω. The techniques, based on a variant of the symmetric mountain pass theorem, exploit variational methods. We also provide information about the asymptotic behavior of the solutions as a suitable parameter goes to 0 + . In this case, we point out the existence of a blow-up phenomenon. The analysis developed in this paper extends and complements various qualitative and asymptotic properties for some cases described by homogeneous differential operators.

International audience; The Petrowsky type equation $y_{tt}^\eps+\eps y_{xxxx}^\eps - y_{xx}^\eps=0$, $\eps>0$ encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order $\sqrt{\eps}$ occurring at the extremities, these boundary controls get singular as $\eps$ goes to $0$. Using the matched asymptotic method, we describe the boundary layer of the solution $y^\eps$ and derive a rigorous second order asymptotic expansion of the control of minimal weighted $L^2-$norm, with respect to the parameter $\eps$. The weight in the norm is chosen to guarantee the smoothness of the control. In particular, we recover and enrich earlier results due to J-.L.Lions in the eighties showing that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation. The asymptotic analysis also provides a robust discrete approximation of the control for any $\eps$ small enough. Numerical experiments support our study.

We introduce the random exponential recursive tree in which at each point of discrete time every node recruits a child (new leaf) with probability p, or fails to do so with probability 1 − p. We study the distribution of the size of these trees and the average level composition, often called the profile. We also study the size and profile of an exponential version of the plane-oriented recursive tree (PORT), wherein every insertion positions in the “gaps” between the edges recruits a child (new leaf) with probability p, or fails to do so with probability 1 − p. We use martingales in conjunction with distributional equations to establish strong laws for the size of both exponential flavors; in both cases, the limit laws are characterized by their moments. Via generating functions, we get an exact expression for the average expectation of the number of nodes at each level. Asymptotic analysis reveals that the most populous level is $\frac {p}{p+1} n$ in exponential recursive trees, and is $\frac {p}{2p+1} n$ in exponential PORTs.

For a small frequency, scattering of waves propagating along a Kirchhoff plate in the shape of locally perturbed strip with traction-free edges is studied. An asymptotic analysis shows that both, bending and twisting, waves do not in main detect distortion of the originally straight edges, that is the transmission coefficient differs a little from 1 while other scattering coefficients become small. In other words, an effect similar to the Weinstein anomalies in an acoustic waveguide is observed. Asymptotic procedures are based on a detailed investigation of the spectrum of an auxiliary operator pencil and the corresponding stationary problem. Justification of the derived asymptotic expansions is performed by means of the technique of weighted spaces with detached asymptotics.

The paper deals with a system consisting of $$n$$ identical elements and one repairing device. While one element is working, others stay in reserve. The distribution of working and repairing times of elements are supposed to be exponential. The asymptotic distribution of the system lifetime under the conditions of its high reliability is investigated.

A numerical-asymptotic approach is used to solve some coefficient inverse problems of tracer diffusion in the atmosphere. An asymptotic solution of the direct problem for an effective prognostic equation in the near-field zone of the source is obtained via a rigorous asymptotic analysis of a multidimensional singularly perturbed reaction–diffusion–advection problem. This solution is used as a priori information to construct a numerical algorithm for solving the inverse problem of recovering the parameters of an anthropogenic pollution source. The algorithm is implemented using sounding data on the Earth’s atmospheric composition obtained from the Russian Resurs-P satellite with highest available spatial resolution. For the first time, atmospheric pollutant emissions (nitrogen dioxide) from an isolated industrial source have been estimated by applying high-precision space monitoring and mathematical methods.

In this paper, we are concerned with a diffusive host-vector epidemic model with a nonlocal spatiotemporal interaction. When the delay kernel takes some special form, by employing linear chain techniques and geometric singular perturbation theory, we establish the existence of travelling front solutions connecting the two spatially uniform steady states for sufficiently small delays. Furthermore, by employing standard asymptotic theory, we also obtain the asymptotic behavior of traveling wave fronts of this model.

An asymptotic analysis of the eigenvalues and eigenfunctions in the Orr–Sommerfeld problem is carried out in the case when the velocity of the main plane-parallel shear flow in a layer of a Newtonian viscous fluid is low in a certain measure. The eigenvalues and corresponding eigenfunctions in the layer at rest are used as a zero approximation. For their perturbations, explicit analytical expressions are obtained in the linear approximation. It is shown that, FOR low velocities of the main shear flow, the perturbations of eigenvalues corresponding to monotonic decay near the rest in a viscous layer are such that, regardless of the velocity profile, the decay decrement remains the same, but an oscillatory component appears that is smaller in order by one than this decrement.

We consider singularly perturbed convection-diffusion equations on one-dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling condition at inner vertices are derived that guarantee conservation of mass as well as dissipation of a mathematical energy which allows us to prove stability and well-posedness. For single intervals and appropriately specified initial conditions, it is well-known that the solutions of the convection-diffusion problem converge to that of the transport problem with order $O(\sqrt{\epsilon})$ in the $L^\infty(L^2)$-norm with diffusion $\epsilon \to 0$. In this paper, we prove a corresponding result for problems on one-dimensional networks. The main difficulty in the analysis is that the number and type of coupling conditions changes in the singular limit which gives rise to additional boundary layers at the interior vertices of the network. Since the values of the solution at these network junctions are not known a-priori, the asymptotic analysis requires a delicate choice of boundary layer functions that allows to handle these interior layers.

First, we consider the problem of hedging in complete binomial models. Using the discrete-time F\"ollmer-Schweizer decomposition, we demonstrate the equivalence of the backward induction and sequential regression approaches. Second, in incomplete trinomial models, we examine the extension of the sequential regression approach for approximation of contingent claims. Then, on a finite probability space, we investigate stability of the discrete-time F\"ollmer-Schweizer decomposition with respect to perturbations of the stock price dynamics and, finally, perform its asymptotic analysis under simultaneous perturbations of the drift and volatility of the underlying discounted stock price process, where we prove stability and obtain explicit formulas for the leading order correction terms., Comment: 17 pages, 3 figures. This paper is a part of an REU project conduced in Summer 2019 at the University of Connecticut. Accepted in Involve

We propose a model for optimizing the last-mile delivery of n packages from a distribution center to their final recipients, using a strategy that combines the use of ride-sharing platforms (e.g., Uber or Lyft) with traditional in-house van delivery systems. The main objective is to compute the optimal reward offered to private drivers for each of the n packages such that the total expected cost of delivering all packages is minimized. Our technical approach is based on the formulation of a discrete sequential packing problem, in which bundles of packages are picked up from the warehouse at random times during the interval [Formula: see text]. Our theoretical results include both exact and asymptotic (as [Formula: see text]) expressions for the expected number of packages that are picked up by time T. They are closely related to the classical Rényi’s parking/packing problem. Our proposed framework is scalable with the number of packages.

In the paper, a single-server retrial queueing system with MMPP arrivals and an exponential law of the service time is studied. Unserviced calls go to an orbit and stay there during random time distributed exponentially, they access to the server according to a random multiple access protocol. In the system, a Poisson process of negative calls arrives, which delete servicing positive calls. The method of the asymptotic analysis under the heavy load condition for the system studying is proposed. It is proved that the asymptotic characteristic function of a number of calls on the orbit has the gamma distribution with the obtained parameters. The value of the system capacity is obtained, so, the condition of the system stationary mode is found. The results of a numerical comparison of the asymptotic distribution and the distribution obtained by simulation are presented. Conclusions about the method applicability area are made.

In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn-Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3-24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn-Hilliard equation for the tumor cell fraction, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, well-posedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding well-posedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases., Comment: Key words: fractional operators, Cahn-Hilliard systems, well-posedness, regularity of solutions, tumor growth models, asymptotic analysis

It is a well-known fact that the onset of Rayleigh–Benard convection occurs via a long-wavelength instability when the horizontal boundaries are thermally insulated. The aim of this paper is to quantify the exact dimensions of a cylinder of rectangular cross-section wherein stable three-dimensional Rayleigh–Benard convection sets in via a long-wavelength instability from the motionless state at the same value of the critical Rayleigh number as the corresponding horizontally unbounded problem when the bounding horizontal walls have infinite thermal conductance. Hence, we consider three-dimensional Rayleigh–Benard convection in a cell of infinite extent in the x-direction, confined between two vertical walls located at $$y= \pm H$$ and horizontal boundaries located at $$z=0$$ and $$z=d$$ . Our analysis predicts the existence of the sought stable state for experimental velocity boundary conditions at the vertical walls provided the aspect ratio $$\delta = H/d$$ takes a certain value. In the limit $$H \rightarrow \infty $$ , we retrieve the stability characteristics of the horizontally unbounded problem. As expected, the analysis predicts two counter-rotating rolls aligned along the y-direction of period $$2 \pi /\delta $$ equal to the period of the roll in the y-direction of the corresponding unbounded problem. A long-scale asymptotic analysis leads to the derivation of an evolution partial differential equation (PDE) that is fourth order in space and contains a single bifurcation parameter. The PDE, valid for a specific value of $$\delta $$ , is analyzed analytically and numerically as function of the bifurcation parameter and for a variety of velocity boundary conditions at the vertical walls to seek the stable steady-state solutions. The same analysis is also extended to the case of convection in a fluid-saturated porous medium.

We provide a comprehensive study on the asymptotic solutions of an interface problem corresponding to an elliptic partial differential equation with Dirichlet boundary condition and transmission condition, subject to the small geometric perturbation and/or the high contrast ratio of the conductivity. All asymptotic terms can be solved in the unperturbed reference domains, which significantly reduces computations in practice, especially for random perturbations. Our setting is quite general and allows two types of elliptic problems: the perturbation of the domain boundary where the Dirchlet condition is imposed and the perturbation of the interface where the transmission condition is imposed. As the perturbation size and the ratio of the conductivities tends to zero, the two-parameter asymptotic expansions on the reference domain are derived to any order after the single parameter expansions are solved beforehand. The results suggest the emergence of the Neumann or Robin boundary condition, depending on the relation of the two asymptotic parameters. Our method is the classic asymptotic analysis techniques but in a new unified approach to both problems.

peer-reviewed The full text of this article will not be available in ULIR until the embargo expires on the 01/08/2021 A recent asymptotic model for the operation of a vanadium redox flow battery (VRFB) is extended to include the dissociation of sulphuric acid - a bulk chemical reaction that occurs in the battery’s porous flow-through electrodes, but which is often omitted from VRFB models. Using asymptotic methods and time-dependent two-dimensional numerical simulations, we show that the charge-dischargecurve for the model with the dissociation reaction is almost identical to that for the model without, even though the concentrations of the ionic species in the recirculating tanks, although not the state of charge, are considerably different in the two models. The ability of the asymptotic model to extract both the qualitative and quantitative behaviour of the considerably more time-consuming numerical simulations correctly indicates that it should be possible to add further physical phenomena to the model without incurring significant computational expense.

We consider a multidimensional singularly perturbed stationary diffusion model with a cubic nonlinearity. For models of this type, a modified asymptotic method of boundary functions, which extends the classical asymptotic analysis methods to the case of multidimensional problems, and the asymptotic method of differential inequalities, which is based on the comparison principle, are used to study the existence of asymptotically Lyapunov stable solutions with internal layers as stationary solutions of the corresponding parabolic problems. Sufficient conditions are established for the existence of such solutions in the form of some conditions on the coefficients of the equation, an asymptotic approximation to the solution of an arbitrary accuracy order with coefficients is constructed in closed form, and the formal constructions are justified. This result can be used for creating efficient numerical algorithms for direct and coefficient inverse problems for stationary equations of the reaction–diffusion–advection type as well as for constructing test examples. Heat and mass transfer problems occurring in chemical industry are pointed out as possible applications of our results.

The ruin probability in the classical Brownian risk model can be explicitly calculated for both finite and infinite time horizon. This is not the case for the simultaneous ruin probability in the two-dimensional Brownian risk model. Relying on asymptotic theory, we derive in this contribution approximations for both simultaneous ruin probability and simultaneous ruin time for the two-dimensional Brownian risk model when the initial capital increases to infinity.

The goal of this article is to study the asymptotic analysis of an incompressible Herschel-Bulkley fluid in a thin domain with Tresca boundary conditions. The yield stress and the constant viscosity are assumed to vary with respect to the thin layer parameter e. Firstly, the problem statement and variational formulation are formulated. We then obtained the existence and the uniqueness result of a weak solution and the estimates for the velocity field and the pressure independently of the parameter e. Finally, we give a specific Reynolds equation associated with variational inequalities and prove the uniqueness.

A distributed optimal control problem for a phase field system which physical context is that of tumor growth is discussed. The system we are going to take into account consists of a Cahn-Hilliard equation for the phase variable (relative concentration of the tumor), coupled with a reaction-diffusion equation for the nutrient. The cost functional is of standard tracking-type and the control variable models the intensity with which it is possible to dispense a medication. The model we deal with presents two small and positive parameters which are introduced in previous contributions as relaxation terms. Here, starting from the already investigated optimal control problem for the relaxed model, we aim at confirming the existence of optimal control and characterizing the first-order optimality condition, via asymptotic schemes, when one of the two occurring parameters goes to zero.

The retrial queueing system of M=M=1 type with Poisson flow of arrivals, impatient cus- tomers, collisions and unreliable service device is considered in the paper. The novelty of our contribution is the inclusion of breakdowns and repairs of the service into our previous study to make the problem more realistic and hence more complicated. Retrial time of customers in the orbit, service time, impa- tience time of customers in the orbit, server lifetime (depending on whether it is idle or busy) and server recovery time are supposed to be exponentially distributed. An asymptotic analysis method is used to find the stationary distribution of the number of customers in the orbit. The heavy load of the system and long time patience of customers in the orbit are proposed as asymptotic conditions. Theorem about the Gaussian form of the asymptotic probability distribution of the number of customers in the orbit is formulated and proved. Numerical examples are given to show the accuracy and the area of feasibility of the proposed method

Discrete integrable nonlinear differential difference equations (NDDEs) have various mathematical structures and properties, such as Lax pair, infinitely many conservation laws, Hamiltonian structure, and different kinds of symmetries, including Lie point symmetry, generalized Lie bäcklund symmetry, and master symmetry. Symmetry is one of the very effective methods used to study the exact solutions and integrability of NDDEs. The Toda lattice equation is a famous example of NDDEs, which may be used to simulate the motions of particles in lattices. In this paper, we investigated the generalized Toda lattice equation related to 3×3 matrix linear spectral problem. This discrete equation is related to continuous linear and nonlinear partial differential equations under the continuous limit. Based on the known 3×3 Lax pair of this equation, the discrete generalized (m,3N−m)-fold Darboux transformation was constructed for the first time and extended from the 2×2 Lax pair to the 3×3 Lax pair to give its rational solutions. Furthermore, the limit states of such rational solutions are discussed via the asymptotic analysis technique. Finally, the exponential–rational mixed solutions of the generalized Toda lattice equation are obtained in the form of determinants.

An infinite-server queueing model with state-dependent arrival process and exponential distribution of service time is analyzed. It is assumed that the difference between the value of the arrival rate and total service rate becomes positive starting from a certain value of the number of customers in the system. In this paper, time until reaching this value by the number of customers in the system is called the pseudo steady-state period (PSSP). Distribution of duration of PSSP, its raw moments and its simple approximation under a certain scaling of the number of customers in the system are analyzed. Novelty of the considered problem consists of an arbitrary dependence of the rate of customer arrival on the current number of customers in the system and analysis of time until reaching from below a certain level by the number of customers in the system. The relevant existing papers focus on the analysis of time interval since exceeding a certain level until the number of customers goes down to this level (congestion period). Our main contribution consists of the derivation of a simple approximation of the considered time distribution by the exponential distribution. Numerical examples are presented, which confirm good quality of the proposed approximation.

In this paper, a multi-server retrial queue with two orbits is considered. There are two arrival processes of positive customers (with two types of customers) and one process of negative customers. Every positive customer requires some amount of resource whose total capacity is limited in the system. The service time does not depend on the customer’s resource requirement and is exponentially distributed with parameters depending on the customer’s type. If there is not enough amount of resource for the arriving customer, the customer goes to one of the two orbits, according to his type. The duration of the customer delay in the orbit is exponentially distributed. A negative customer removes all the customers that are served during his arrival and leaves the system. The objects of the study are the number of customers in each orbit and the number of customers of each type being served in the stationary regime. The method of asymptotic analysis under the long delay of the customers in the orbits is applied for the study. Numerical analysis of the obtained results is performed to show the influence of the system parameters on its performance measures.

We study expectations of powers and correlation functions for characteristic polynomials of $N \times N$ non-Hermitian random matrices. For the $1$-point and $2$-point correlation function, we obtain several characterizations in terms of Painlev\'e transcendents, both at finite-$N$ and asymptotically as $N \to \infty$. In the asymptotic analysis, two regimes of interest are distinguished: boundary asymptotics where parameters of the correlation function can touch the boundary of the limiting eigenvalue support and bulk asymptotics where they are strictly inside the support. For the complex Ginibre ensemble this involves Painlev\'e IV at the boundary as $N \to \infty$. Our approach, together with the results in \cite{HW17} suggests that this should arise in a much broader class of planar models. For the bulk asymptotics, one of our results can be interpreted as the merging of two `planar Fisher-Hartwig singularities' where Painlev\'e V arises in the asymptotics. We also discuss the correspondence of our results with a normal matrix model with $d$-fold rotational symmetries known as the \textit{lemniscate ensemble}, recently studied in \cite{BGM, BGG18}. Our approach is flexible enough to apply to non-Gaussian models such as the truncated unitary ensemble or induced Ginibre ensemble; we show that in the former case Painlev\'e VI arises at finite-$N$. Scaling near the boundary leads to Painlev\'e V, in contrast to the Ginibre ensemble., Comment: Typos corrected, 39 pages, 4 figures, 1 table

This paper investigates a relativistic Toda lattice system with an arbitrary parameter that is a very remarkable generalization of the usual Toda lattice system, which may describe the motions of particles in lattices. Firstly, we study some integrable properties for this system such as Hamiltonian structures, Liouville integrability and conservation laws. Secondly, we construct a discrete generalized (m,2N−m)-fold Darboux transformation based on its known Lax pair. Thirdly, we obtain some exact solutions including soliton, rational and semi-rational solutions with arbitrary controllable parameters and hybrid solutions by using the resulting Darboux transformation. Finally, in order to understand the properties of such solutions, we investigate the limit states of the diverse exact solutions by using graphic and asymptotic analysis. In particular, we discuss the asymptotic states of rational solutions and exponential-and-rational hybrid solutions graphically for the first time, which might be useful for understanding the motions of particles in lattices. Numerical simulations are used to discuss the dynamics of some soliton solutions. The results and properties provided in this paper may enrich the understanding of nonlinear lattice dynamics.

The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of the numerical solving process. One of the approaches to solving inverse problems of the class under consideration is the use of methods of asymptotic analysis. These methods, under certain conditions, make it possible to construct the so-called reduced formulation of the inverse problem. Usually, a differential equation in this formulation has a lower dimension/order with respect to the differential equation, which is included in the full statement of the inverse problem. In this paper, we consider an example that leads to a reduced formulation of the problem, the solving of which is no less a time-consuming procedure in comparison with the numerical solving of the problem in the full statement. In particular, to obtain an approximate numerical solution, one has to use the methods of the numerical diagnostics of the solution’s blow-up. Thus, it is demonstrated that the possibility of constructing a reduced formulation of the inverse problem does not guarantee its more efficient solving. Moreover, the possibility of constructing a reduced formulation of the problem does not guarantee the existence of an approximate solution that is qualitatively comparable to the true one. In previous works of the authors, it was shown that an acceptable approximate solution can be obtained only for sufficiently small values of the singular parameter included in the full statement of the problem. However, the question of how to proceed if the singular parameter is not small enough remains open. The work also gives an answer to this question.

This paper analyses a class of parabolic linear cooperative systems in a cylindrical domain with degenerate spatio-temporal potentials. In other words, potentials vanish in some non-empty connected subdomains which are disjoint and increase in size temporally. Then, the vanishing subdomains for the potentials are not cylindrical. Following a similar idea to the semiclassical analysis behaviour, but done here for parabolic problems, under these geometrical assumptions, the asymptotic behaviour of the system is ascertained when a parameter, in front of these potentials, goes to infinity. In particular, the strong convergence of the solutions of the system is obtained using energy methods and the theory associated with the $$\Gamma $$ Γ -convergence. Also, the exponential decay of the solutions to zero in the exterior of the subdomains where the potentials vanish is achieved.

In this paper, we obtain the Nth-order rational solutions for the defocusing nonlocal nonlinear Schrodinger equation by the Darboux transformation and some limit technique. Then, via an improved asymptotic analysis method relying on the balance between different algebraic terms, we derive the explicit expressions of all asymptotic solitons of the rational solutions with the order 1=2 are stronger than those in the exponential and exponential-and-rational solutions., 28 pages, 7 figures

Steady-state Navier–Stokes equations in a thin tube structure with the Bernoulli pressure inflow–outflow boundary conditions and no-slip boundary conditions at the lateral boundary are considered. Applying the Leray–Schauder fixed point theorem, we prove the existence and uniqueness of a weak solution. An asymptotic approximation of a weak solution is constructed and justified by an error estimate.

The paper considers the features of numerical reconstruction of the advection coefficient when solving the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type. Information on the position of a reaction front is used as data of the inverse problem. An important question arises: is it possible to obtain a mathematical connection between the unknown coefficient and the data of the inverse problem? The methods of asymptotic analysis of the direct problem help to solve this question. But the reduced statement of the inverse problem obtained by the methods of asymptotic analysis contains a nonlinear integral equation for the unknown coefficient. The features of its solution are discussed. Numerical experiments demonstrate the possibility of solving problems of such class using the proposed methods.