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In this paper, using the exponential conformal map for the Hankel contour we show a new Schlafli-type integral representation for the Wright function. We apply the steepest descent method and the Lagrange expansion to find the asymptotic expansions of Wright function for the large parameter. We study two cases for the stationary points and discuss the associated asymptotic expansions. The results extend the asymptotic expansions of the Bessel functions of the first and second kinds.

We provide complete asymptotic expansions for some sequences of Bernstein-type and Meyer–Konig and Zeller-type operators preserving the monomials e 0 and e j , j > 1 . In particular, this answers a conjecture related to a Voronovskaja-type theorem.

We do a thorough asymptotic analysis of nonoscillatory solutions of the q-difference equation D q ( r ( t ) D q y ( t ) ) + p ( t ) y ( q t ) = 0 considered on the lattice { q k : k ∈ N 0 } , q > 1 . We classify the solutions according to various aspects that take into account their asymptotic behavior. We show relations among the asymptotic classes. For every positive solution we establish asymptotic formulae. Several discrepancies are revealed, when comparing the results with their existing differential equations or difference equations counterparts; however, it should be noted that many of our observations in the q-case have not their continuous or discrete analogies yet. Important roles in our considerations are played by the theory of q-regular variation and various transformations. The results are illustrated by examples.

We establish a homogenization result for a fully nonlinear degenerate parabolic system with critical growth arising from the heat and moisture flow through a partially saturated porous media. Existence of a global weak solution of the mesoscale problem is proven by means of a semidiscretization in time, a priori estimates and passing to the limit from discrete approximations. After that, porous material exhibiting periodic spatial oscillations is considered and the two-scale convergence (as the oscillation period vanishes) to a corresponding homogenized problem is rigorously proven.

We study the asymptotic behavior of the solution to some time harmonic wave problems when the wave number is taken as a small asymptotic parameter. Our basic strategy is to introduce suitable Lagrangian multipliers into the governing equations, and transforming them into saddle point problems. These saddle point problems are uniformly invertible with respect to the wave number k ∈ [ 0 , k 0 ] , with k 0 being an arbitrary but fixed positive number. The asymptotic expansion is then derived by the standard regular perturbation technique.

We introduce a new isometric geometric property, namely asymptotic midpoint uniform convexity and investigate its connection to similar asymptotic properties.

A parabolic system with small viscosity is considered in a two dimensional channel. With the help of discussions on suitable boundary conditions of the corresponding hyperbolic equations on both sides of the channel, the existence and stability of multiple boundary layers are proved by using the matched asymptotic analysis and energy estimates. The results therefore consequently justify the zero viscosity limit when e tends to 0.

This paper deals with monic orthogonal polynomials generated by a Geronimus canonical spectral transformation of the Laguerre classical measure: \[ \frac{1}{x-c}x^{\alpha }e^{-x}dx+N\delta (x-c), \] for $x\in[0,\infty)$, $\alpha>-1$, a free parameter $N\in \mathbb{R}_{+}$ and a shift $c, Comment: 17 pages, no figures

In this paper, we study the asymptotic properties of minimizers for a class of constraint minimization problems derived from the Maxwell-Schrodinger-Poisson system − Δ u − ( | u | 2 ⁎ | x | − 1 ) u − α | u | 2 p u − μ p u = 0 , x ∈ R 3 on the L 2 -spheres A λ = { u ∈ H 1 ( R 3 ) : ∫ R 3 | u | 2 d x = λ } , where α , p > 0 . Let λ ⁎ = ‖ Q 2 3 ‖ 2 2 , and Q 2 3 is the unique (up to translations) positive radial solution of − 3 p 2 Δ u + 2 − p 2 u − | u | 2 p u = 0 in R 3 with p = 2 3 . We prove that if λ α − 3 2 λ ⁎ , then minimizers are relatively compact in A λ as p ↗ 2 3 . On the contrary, if λ > α − 3 2 λ ⁎ , by directly using asymptotic analysis, we prove that all minimizers must blow up and give the detailed asymptotic behavior of minimizers.

We consider a semi-linear parabolic problem in a thick junction Ω e , which is the union of a domain Ω 0 and a lot of joined thin trees situated e-periodically along some manifold on the boundary of Ω 0 . The trees have finite number of branching levels. The following nonlinear Robin boundary condition ∂ ν v e + e α i μ ( t , x 2 , v e ) = e β g e is given on the boundaries of the branches from the i-th branching layer; { α i } and β are real parameters. The asymptotic analysis of this problem is made as e → 0 , i.e., when the number of the thin trees infinitely increases and their thickness vanishes. In particular, the corresponding homogenized problem is found and the existence and uniqueness of its solution in an anisotropic Sobolev space of multi-sheeted functions is proved. We construct the asymptotic approximation for the solution v e and prove the corresponding asymptotic estimate in the space C ( [ 0 , T ] ; L 2 ( Ω e ) ) ∩ L 2 ( 0 , T ; H 1 ( Ω e ) ) , which shows the influence of the parameters { α i } and β on the asymptotic behavior of the solution.

This paper deals with an anomalous diffusion system which describes the spread of epidemics among a population. The analysis includes some results of the asymptotic behavior of global bounded solutions for this system with homogeneous Neumann boundary conditions.

This paper presents the exponential stability of a one-dimensional wave equation with viscoelastic damping. Using the asymptotic analysis technique, we prove that the spectrum of the system operator consists of two parts: the point and continuous spectrum. The continuous spectrum is a set of N points which are the limits of the eigenvalues of the system, and the point spectrum is a set of three classes of eigenvalues: one is a subset of N isolated simple points, the second is approaching to a vertical line which parallels to the imagine axis, and the third class is distributed around the continuous spectrum. Moreover, the Riesz basis property of the generalized eigenfunctions of the system is verified. Consequently, the spectrum-determined growth condition holds true and the exponential stability of the system is then established.

The p -Laplace equations have some applications in continuum mechanics. On the basis of this background detail, we study oscillation and asymptotic behavior of solutions to two classes of higher-order delay damped differential equations with p -Laplacian like operators. Some new criteria are presented that improve the related contributions to the subject. Several examples are provided to illustrate the relevance of new theorems.

Let μ be a probability measure on the set R of real numbers and μ ˆ ( t ) : = ∫ R e − i t λ d μ ( λ ) ( t ∈ R ) be the Fourier transform of μ ( i is the imaginary unit). Then, under suitable conditions, asymptotic formulae for | μ ˆ ( t / x ) | 2 x in 1 / x as x → ∞ are derived. These results are applied to the so-called quantum Zeno effect to establish asymptotic formulae for its occurrence probability in the inverse of the number N of measurements made in a time interval as N → ∞ .

We examine the asymptotic behavior as n → + ∞ of the coefficients G n appearing in an asymptotic expansion for the Gamma function.

This paper deals with a class of integro-differential equations arising in evolutionary biology to model the dynamics of specialist and generalist species related by mutualistic interactions. The effects of mutation events, proliferative phenomena and competition are taken into account. Specialist population is assumed to be structured by a continuous phenotypical trait related to the ability of individuals to ingest specific resources and a parameter ϵ is introduced to model the average size of mutations. A well-posedness result is here proposed and the asymptotic behavior of the density of specialist individuals in the space of the phenotypical traits is studied in the limit ϵ → 0 . In particular, under a suitable time rescaling, we prove the weak convergence of such a density to a sum of Diracʼs masses. A characterization of the set of concentration points is provided.

In this paper, we study the structural properties of a class of vector-valued hyperbolic equations with appropriate boundary conditions, including the spectrum determined growth condition. We prove that the equations associate with a C 0 semigroup. By the structural analysis, we obtained a sufficient and necessary condition for being at least an eigenvalue on the imaginary axis. In particular, using the asymptotic analysis technique we prove that the spectrum of the operator determined by the equations is distributed in a strip parallel to the imaginary axis and is union of finitely many separable sets. Furthermore, we prove that the root vectors of the operator are complete and there is a sequence of root vectors that forms a Riesz basis with parentheses for the Hilbert state space. As applications of our results, we give some concrete examples in controlled complex network of Euler–Bernoulli beams.

In this paper, we derive recursions of some RNA secondary structures with certain properties under two new representations. Furthermore, by making use of methods of asymptotic analysis and generating functions we present asymptotic enumeration of these RNA secondary structures.

This paper is concerned about a reaction–diffusion equation on n-dimensional isotropically growing domain, which describes the insect dispersal. The model for growing domains is first derived, and the comparison principle is then presented. The asymptotic behavior of the solution to the reaction–diffusion problem is given by constructing upper and lower solutions. Our results show that the growth of domain takes a positive effect on the asymptotic stability of positive steady state solution while it takes a negative effect on the asymptotic stability of the trivial solution. Numerical simulations are also performed to illustrate the analytical results.

Under investigation in this paper is the Hirota–Maccari equation, which is a generalized ( 2 + 1 ) -dimensional model in fluid dynamics, plasma physics and optical fiber communication. With the aid of the Hirota bilinear method and symbolic computation, the corresponding N-soliton solutions are given and illustrated. The characteristic face method and asymptotic analysis are applied to discuss the solitonic propagation and collision, including the bidirectional solitons, elastic interactions and inelastic interactions. Finally, a kind of special phenomenon with the parameters varying is investigated, which might provide people with useful information on the dynamics of the relevant fields.

Deterministic descriptions of three biochemical reaction channels formerly considered by Bratsun et al. (2005) [19] are studied. These descriptions are based on the mass action law and for the simple protein production with delayed degradation differ from that proposed by Bratsun et al. An explicit solution to this model is calculated. For the model of reaction with negative feedback and delayed production, global stability of a unique positive steady state is proved. According to the models of these two reaction channels considered in the present paper there cannot appear delayed induced oscillations. For the model of reaction with negative feedback, dimerisation and delayed protein production, local stability for a unique positive steady state is shown for some range of parameters. It is also proved that for some range of parameters the destabilisation due to the increasing delay can occur and delayed induced oscillations may appear.

In this paper we study the asymptotic limiting behavior of the solutions to the initial boundary value problem for linearized one-dimensional compressible Navier–Stokes equations. We consider the characteristic boundary conditions, that is we assume that an eigenvalue of the associated inviscid Euler system vanishes uniformly on the boundary. The aim of this paper is to understand the evolution of the boundary layer, to construct the asymptotic ansatz which is uniformly valid up to the boundary, and to obtain rigorously the uniform convergence to the solution of the Euler equations without the weakness assumption on the boundary layer.

The authors derive and justify two models for the bending-stretching of a viscoelastic rod by using the asymptotic expansion method. © 2010 Elsevier Inc.

The qualitative theory of differential equations is applied to the K ( 2 , 2 ) equation with osmosis dispersion. Smooth, peaked and cusped solitary wave solutions of the osmosis K ( 2 , 2 ) equation under inhomogeneous boundary condition are obtained. The parametric conditions of existence of the smooth, peaked and cusped solitary wave solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for smooth, peaked and cusped solitary wave solutions of the osmosis K ( 2 , 2 ) equation.

It is well known that, due to Boutroux, the first Painleve equation admits solutions characterized by divergent asymptotic expansions near infinity in specified sectors of the complex plane. In this paper, we show that such solutions exist for higher order analogues of the first Painleve equation (the first Painleve hierarchy) as well.

We continue investigation of certain classes of sequences of positive real numbers which are important in asymptotic analysis of divergent processes. Among others, we define the notion of quotient speed, give representation results for some kinds of sequences and improve several our earlier results concerning selection principles and games.

The initial value problem of the Chern–Simons–Dirac equations in one space dimension is studied. We prove the existence of global solution and investigate asymptotic behaviors.

In this paper, we obtain a bidimensional shallow water model with polynomial dependence on depth. With this aim, we introduce a small non-dimensional parameter ε and we study three-dimensional Euler equations in a domain depending on ε (in such a way that, when ε becomes small, the domain has small depth). Then, we use asymptotic analysis to study what happens when ε approaches to zero. Asymptotic analysis allows us to obtain a new bidimensional shallow water model that not only computes the average velocity (as the classical model does) but also provides the horizontal velocity at different depths. This represents a significant improvement over the classical model. We must also remark that we obtain the model without making assumptions about velocity or pressure behavior (only the usual ansatz in asymptotic analysis). Finally, we present some numerical results showing that the new model is able to approximate the non-constant in depth solutions to Euler equations, whereas the classical model can only obtain the average velocity.

We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere S d in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials | x − y | − s with d − 2 ⩽ s d . For a given axis-supported external field, the support and the density of the corresponding extremal measure on S d is determined. The special case s = d − 2 yields interesting phenomena, which we investigate in detail. A weak ∗ asymptotic analysis is provided as s → ( d − 2 ) + .

In this paper, a new approach is provided to study the asymptotic behavior of functions. A Tauberian theorem is improved and applied to describe the asymptotic behavior of abstract functional differential equations of the form u ˙ ( t ) = A u ( t ) + [ B u ] ( t ) + f ( t ) , t ∈ R . An example is given to illustrate our results.

We study the quasi-neutral limit in one-dimensional steady-state Euler–Poisson equations with junction layers. Typically, the junction layer phenomenon occurs in a ballistic diode of a semiconductor device where the doping profile is a discontinuous function. We derive the junction layer equations and prove the existence of their solutions which decay exponentially. Finally, we justify the quasi-neutral limit with junction layers by giving uniform error estimates.

Interferons are active biomolecules, which help fight viral infections by spreading from infected to uninfected cells and activate effector molecules, which confer resistance from the virus on cells. We propose a new model of dynamics of viral infection, including endocytosis, cell death, production of interferon and development of resistance. The novel element is a specific biologically justified mechanism of interferon action, which results in dynamics different from other infection models. The model reflects conditions prevailing in liquid cultures (ideal mixing), and the absence of cells or virus influx from outside. The basic model is a nonlinear system of five ordinary differential equations. For this variant, it is possible to characterise global behaviour, using a conservation law. Analytic results are supplemented by computational studies. The second variant of the model includes age-of-infection structure of infected cells, which is described by a transport-type partial differential equation for infected cells. The conclusions are: (i) If virus mortality is included, the virus becomes eventually extinct and subpopulations of uninfected and resistant cells are established. (ii) If virus mortality is not included, the dynamics may lead to extinction of uninfected cells. (iii) Switching off the interferon defense results in a decrease of the sum total of uninfected and resistant cells. (iv) Infection-age structure of infected cells may result in stabilisation or destabilisation of the system, depending on detailed assumptions. Our work seems to constitute the first comprehensive mathematical analysis of the cell-virus-interferon system based on biologically plausible hypotheses.

Based on the new representation of RNA secondary structures, we obtain the basic relations about secondary structures with a prescribed size m for hairpin loops and minimum stack length l. Furthermore, we make an asymptotic analysis on RNA secondary structures with certain additional constrains.

We obtain linear continuous operators providing a solution to the Stieltjes moment problem in the framework of Gelfand–Shilov spaces of rapidly decreasing smooth functions. The construction rests on an interpolation procedure due to R. Estrada for general rapidly decreasing smooth functions, and adapted by S.-Y. Chung, D. Kim and Y. Yeom to the case of Gelfand–Shilov spaces. It requires a linear continuous version of the so-called Borel–Ritt–Gevrey theorem in asymptotic theory.

New explicit sufficient conditions for the asymptotic stability of the zero solution of higher order difference equations are obtained. These criteria can be applied to autonomous and nonautonomous equations. The celebrated Clark asymptotic stability criterion is improved. Also, applications to models from mathematical biology and macroeconomics are given.

We consider three singularly perturbed convection-diffusion problems defined in three-dimensional domains: (i) a parabolic problem $-\epsilon(u_{xx}+u_{yy})+u_t +v_1u_x+v_2u_y=0$ in an octant, (ii) an elliptic problem $-\epsilon(u_{xx}+u_{yy}+u_{zz}) +v_1u_x+v_2u_y+v_3u_z=0$ in an octant and (iii) the same elliptic problem in a half space. We consider for all of these problems discontinuous boundary conditions at certain regions of the boundaries of the domains. For each problem, an asymptotic approximation of the solution is obtained from an integral representation when the singular parameter $\epsilon\to 0^+$. The solution is approximated by a product of two error functions, and this approximation characterizes the effect of the discontinuities on the small $\epsilon-$ behaviour of the solution and its derivatives in the boundary layers or the internal layers.

The complete asymptotic expansion of power means in terms of Bell polynomials is obtained. Some results recently obtained by M. Bjelica are generalized.

A one-dimensional singularly perturbed problem of mixed type is considered. The domain under consideration is partitioned into two subdomains. In the first subdomain a parabolic reaction–diffusion problem is given and in the second one an elliptic convection–diffusion–reaction problem. The solution is decomposed into regular and singular components. The problem is discretized using an inverse-monotone finite volume method on condensed Shishkin meshes. We establish an almost second-order global pointwise convergence in the space variable, that is uniform with respect to the perturbation parameter.

We consider a production planning problem for a jobshop with unreliable machines producing a number of products. There are upper and lower bounds on intermediate parts and an upper bound on finished parts. The machine capacities are modelled as finite state Markov chains. The objective is to choose the rate of production so as to minimize the total discounted cost of inventory and production. Finding an optimal control policy for this problem is difficult. Instead, we derive an asymptotic approximation by letting the rates of change of the machine states approach infinity. The asymptotic analysis leads to a limiting problem in which the stochastic machine capacities are replaced by their equilibrium mean capacities. The value function for the original problem is shown to converge to the value function of the limiting problem. The convergence rate of the value function together with the error estimate for the constructed asymptotic optimal production policies are established.

The paper studies certain (probability) measures of binomial type defined in a recursive way on the unit interval. These measures are related to the sum-of-digit function and similar quantities. In particular, we undertake an asymptotic analysis of the moments of the corresponding distributions. This is done by a combination of a method based on the Mellin transform and the depoissonisation technique.

In this paper, a general reaction–diffusion food-limited population model with time-delay is proposed. Accordingly, the existence and uniqueness of the periodic solutions for the boundary value problem and the asymptotic periodicity of the initial-boundary value problem are considered. Finally, the effect of the time-delay on the asymptotic behavior of the solutions is given.

This paper deals with boundary value problems whose nonlinear part involves periodic functions and such that the linear part has a one-dimensional solution space. We shall study the existence and multiplicity of solutions using various methods of Nonlinear Analysis such as the Lyapunov–Schmidt reduction and methods of critical point theory. The proofs are based on some general results on the oscillation and asymptotic behavior of certain parametric integrals.

The convergence for the radial minimizers of a second-order energy functional, when the parameter tends to 0 is studied. And the location of the zeros of the radial minimizers of this functional is presented. Based on this result, the uniqueness of the radial minimizer is discussed.

For linear functional difference equations, we obtain some results on the asymptotic behavior of solutions, which correspond to a Perron-type theorem for linear ordinary difference equations. We also apply our results to Volterra difference equations with infinite delay.

Recently Kirk introduced the notion of asymptotic contractions on a metric space and using ultrapower techniques he obtained an asymptotic version of the Boyd–Wong fixed point theorem. In this paper we extend this result and moreover, we give a constructive proof of it. Furthermore, we obtain a complete characterization of asymptotic contractions on a compact metric space. As a by-product we establish a separation theorem for upper semicontinuous functions satisfying some limit condition.

Within the framework of the Koiter's linear elastic shell theory, we study the limit model of a Lipschitz curved arch whose mid-surface is periodically waved. The magnitude and the period of the wavings are of the same order. To achieve the asymptotic analysis, we consider a mixed formulation, for which we perform a two-scale homogenization technique. We prove the convergence of the displacements, the rotation of the normal, and the membrane strain. From the limit formulation, we derive an effective model for curved critically wrinkled arches. It introduces two membrane strain functions—instead of one in the classical case—and exhibits a corrector membrane term to the coupling between the rotation of the normal and the membrane strain.

We consider a singularly perturbed elliptic convection–diffusion problem on the unit square. A new asymptotic expansion of its solution is constructed, giving precise conditions under which the solution can be decomposed in a particularly opportune way into a sum of smooth and layer functions.

A flexible structure consisting of a Euler–Bernoulli beam with a tip mass is considered. To stabilize this system we use a boundary control laws: −uxxx(1, t) + mutt(1, t) = −αut(1, t) + βuxxxt(1, t) and uxx(1, t) = −γuxt(1, t). A sensitivity asymptotic analysis of the system's eigenvalues and eigenfunctions is set up. We prove that all of the generalized eigenfunctions of (2.9) form a Riesz basis of H . By a new method, we prove that the operator A generates a C0 contraction semigroup T(t), t ≥ 0. Furthermore T(t), t ≥ 0, is uniformly exponentially stable and the optimal exponential decay rate can be obtained from the spectrum of the system.