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This work deals with the solution and asymptotic analysis for a porous-elastic system with internal damping of the fractional derivative type. We consider an augmented model. The energy function is presented and establishes the dissipativity property of the system. We use the semigroup theory. The existence and uniqueness of the solution are obtained by applying the well-known Lumer-Phillips Theorem. We present two results for the asymptotic behavior: Strong stability of the C 0 -semigroup associated with the system using Arendt-Batty and Lyubich-Vũ's general criterion and polynomial stability applying Borichev and Tomilov's Theorem. [ABSTRACT FROM AUTHOR]

Asymptotic analysis for an elastic layer under light fluid loading was developed. The ratio of fluid and solid densities was chosen as the main small parameter determining a novel scaling. The leading- and next-order approximations were derived from the full dispersion relation corresponding to long-wave, low-frequency, antisymmetric motions. The asymptotic plate models, including the equations of motion and the impenetrability condition, motivated by the aforementioned shortened dispersion equations, were derived for a plane-strain setup. The key findings included, in particular, the necessity of taking into account transverse plate inertia at the leading order, which is not the case for heavy fluid loading. In addition, the transverse shear deformation, rotation inertia, and a number of other corrections appeared at the next order, contrary to the previous asymptotic developments for fluid-loaded plates not assuming a light fluid loading scenario. [ABSTRACT FROM AUTHOR]

The problem of description of quasi-static behavior is studied for a planar thermoelastic body incorporating an inhomogeneity, which geometrically is a strip with a small cross-section. This problem contains a small positive parameter δ describing the thickness of the inhomogeneity, i.e., the size of the cross-section. Relying on the variational formulation of the problem, we investigate the behavior of solutions as δ tends to zero. As the result, by the version of the method of formal asymptotic expansions, we derive a closed limit model in which the inhomogeneity is thin (of zero width). After this, using the Galerkin method and the classical techniques of derivation of energy estimates, we prove existence, uniqueness, and stability of a weak solution to this model. [ABSTRACT FROM AUTHOR]

Being able to characterise objects at low frequencies, but in situations where the modelling error in the eddy current approximation of the Maxwell system becomes large, is important for improving current metal detection technologies. Importantly, the modelling error becomes large as the frequency increases, but the accuracy of the eddy current model also depends on the object topology and on its materials, with the error being much larger for certain geometries compared to others of the same size and materials. Additionally, the eddy current model breaks down at much smaller frequencies for highly magnetic conducting materials compared to non-permeable objects (with similar conductivities, sizes and shapes) and, hence, characterising small magnetic objects made of permeable materials using the eddy current at typical frequencies of operation for a metal detector is not always possible. To address this, we derive a new asymptotic expansion for permeable highly conducting objects that is valid for small objects and holds not only for frequencies where the eddy current model is valid but also for situations where the eddy current modelling error becomes large and applying the eddy approximation would be invalid. The leading-order term we derive leads to new forms of object characterisations in terms of polarizability tensor object descriptions where the coefficients can be obtained from solving vectorial transmission problems. We expect these new characterisations to be important when considering objects at greater stand-off distance from the coils, which is important for safety critical applications, such as the identification of landmines, unexploded ordnance and concealed weapons. We also expect our results to be important when characterising artefacts of archaeological and forensic significance at greater depths than the eddy current model allows and to have further applications parking sensors and improving the detection of hidden, out-of-sight, metallic objects. [ABSTRACT FROM AUTHOR]

The creation of hairpin or lambda vortices, typical for the early stages of the laminar-turbulent transition process in various boundary layer flows, in some sense may be associated with blow-up solutions of the Fisher--Kolmogorov--Petrovsky--Piskunov equation. In contrast to the usual applications of this nonlinear evolution equation of the reaction-diffusion type, the solution quantity in the present context needs to stay neither bounded nor positive. We focus on the solution behavior beyond a finite-time point blow-up event, which consists of two moving singularities (representing the cores of the vortex legs) propagating in opposite directions, and their initial motion is determined with the method of matched asymptotic expansions. After resolving subtleties concerning the transition between logarithmic and algebraic expansion terms regarding asymptotic layers, we find that the internal singularity structure resembles a combination of second- and first-order poles in the form of a singular traveling wave with a time-dependent speed imprinted through the characteristics of the preceding blow-up event. [ABSTRACT FROM AUTHOR]

We consider the propagation of acoustic waves in a waveguide containing a penetrable dissipative inclusion. We prove that as soon as the dissipation, characterized by some coefficient η, is non zero, the scattering solutions are uniquely defined. Additionally, we give an asymptotic expansion of the corresponding scattering matrix when η → 0+ (small dissipation) and when η → +∞ (large dissipation). Surprisingly, at the limit η → +∞, we show that no energy is absorbed by the inclusion. This is due to the so-called skin-effect phenomenon and can be explained by the fact that the field no longer penetrates into the highly dissipative inclusion. These results guarantee that in monomode regime, the amplitude of the reflection coefficient has a global minimum with respect to η. The situation where this minimum is zero, that is when the device acts as a perfect absorber, is particularly interesting for certain applications. However it does not happen in general. In this work, we show how to perturb the geometry of the waveguide to create 2D perfect absorbers in monomode regime. Asymptotic expansions are justified by error estimates and theoretical results are supported by numerical illustrations. [ABSTRACT FROM AUTHOR]

We perform an asymptotic analysis with respect to the parameter ε > 0 of the solution of the scalar advection–diffusion equation y t ε + M (x , t) y x ε − ε y x x ε = 0 , (x , t) ∈ (0 , 1) × (0 , T) , supplemented with Dirichlet boundary conditions. For small values of ε, the solution y ε exhibits a boundary layer of size O (ε) in the neighborhood of x = 1 (assuming M > 0) and an internal layer of size O (ε 1 / 2) in the neighborhood of the characteristic starting from the point (0 , 0). Assuming that these layers interact each other after a finite time T > 0 and using the method of matched asymptotic expansions, we construct an explicit approximation P ε satisfying ‖ y ε − P ε ‖ L ∞ (0 , T ; L 2 (0 , 1)) = O (ε 1 / 2). We emphasize the additional difficulties with respect to the case M constant considered recently by the authors. [ABSTRACT FROM AUTHOR]

We present new alternative complete asymptotic expansions for the time harmonic low‐frequency magnetic field perturbation caused by the presence of a conducting permeable object as its size tends to zero for the eddy current approximation of the Maxwell system. Our new alternative formulations enable a natural extension of the well‐known rank 2 magnetic polarizability tensor (MPT) object characterization to higher order tensor descriptions by introducing generalized MPTs (GMPTs) using multi‐indices. In particular, we identify the magnetostatic contribution, provide new results on the symmetries of GMPTs, derive explicit formulae for the real and imaginary parts of GMPT coefficients and also describe the spectral behavior of GMPT coefficients. We also introduce the concept of harmonic GMPTs (HGMPTs) that have fewer coefficients than other GMPT descriptions of the same order. We describe the scaling, translation and rotational properties of HGMPTs and describe an approach for obtaining those HGMPT coefficients that are invariant under the action of a symmetry group. Such an approach is one candidate for selecting features in object classification for hidden object identification using HGMPTs. [ABSTRACT FROM AUTHOR]

We find asymptotic expansions as for integrals of the form , where sufficiently smooth functions and satisfy natural assumptions on their behavior at infinity and all critical points of in the set are nondegenerate. These asymptotic expansions play a crucial role in analyzing stochastic models for nonlinear waves systems. We generalize a result of Kuksin that a similar asymptotic expansion occurs in a particular case where is a nondegenerate quadratic form of signature with even . [ABSTRACT FROM AUTHOR]

For a specific set of parameters, we analyze the stability of a one-spike equilibrium solution to the one-dimensional Gierer-Meinhardt reaction-diffusion model with delay in the components of the reaction-kinetics terms. Assuming slow activator diffusivity, we consider instabilities due to Hopf bifurcation in both the spike position and the spike profile for increasing values of the time-delay parameter T. Using method of matched asymptotic expansions it is shown that the model can be reduced to a system of ordinary differential equations representing the position of the slowly evolving spike solution. The reduced evolution equations for the one-spike solution undergoes a Hopf bifurcation in the spike position in two cases: when the negative feedback of the activator equation is delayed, and when delay is in both the negative feedback of the activator equation and the non-linear production term of the inhibitor equation. Instabilities in the spike profile are also considered, and it is shown that the spike solution is unstable as T is increased beyond a critical Hopf bifurcation value TH, and this occurs for the same cases as in the spike position analysis. In all cases, the instability in the profile is triggered before the positional instability. If however the degradation of activator is delayed, we find stable positional oscillations can occur in this system. [ABSTRACT FROM AUTHOR]

In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions. Two different perturbed versions of this "background" situation are investigated, when (i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition on a "small" subset ω ε of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on a "small" subset ω ε of the Dirichlet boundary. The relevant quantity that measures the "smallness" of the subset ω ε differs in the two cases: while it is the harmonic capacity of ω ε in the former case, we introduce a notion of "Neumann capacity" to handle the latter. In the first part of this work we derive representation formulas that catch the structure of the first non trivial term in the asymptotic expansion of the voltage potential, for a general ω ε , under the sole assumption that it is "small" in the appropriate sense. In the second part, we explicitly calculate the first non trivial term in the asymptotic expansion of the voltage potential, in the particular geometric situation where the subset ω ε is a vanishing surfacic ball. [ABSTRACT FROM AUTHOR]

The Fourier transform, F, on RN (N > 1) transforms the Cauchy problem for a strongly damped beam equation with structural damping utt - Δut + α(Δ²)u - Δu = 0; α > 0, to an ordinary differential equation in time. With u(t; x) being the weak solution of the problem given by the Fourier transform, the goal of the paper is to determine the asymptotic expansion of the squared L²-norm of u(t; x) as t - ∞. With suitable, additional assumptions on the initial data u(0; x) and ut(0; x), we establish the behavior of the squared L²-norm of u(t; x) as t - ∞. [ABSTRACT FROM AUTHOR]

For spectral actions consisting of the average number of particles and arising from open systems made of general free q -particles (including Bose, Fermi and classical ones corresponding to q = ±1 and 0, respectively) in thermal equilibrium, we compute the asymptotic expansion with respect to the natural cut-off. We treat both relevant situations relative to massless and non relativistic massive particles, where the natural cut-off is 1/ β = k B T and 1 / β , respectively. We show that the massless situation enjoys less regularity properties than the massive one. We also treat in some detail the relativistic massive case for which the natural cut-off is again 1/ β. We then consider the passage to the continuum describing infinitely extended open systems in thermal equilibrium, by also discussing the appearance of condensation phenomena occurring for Bose-like q -particles, q ∈ (0, 1]. We then compare the arising results for the finite volume situation (discrete spectrum) with the corresponding infinite volume one (continuous spectrum). [ABSTRACT FROM AUTHOR]

Localized wave oscillations in an open system that do not decay or grow in time, despite their frequency lying within a continuous spectrum of radiation modes carrying energy to or from infinity, are known as bound states in the continuum (BIC). Small perturbations from the typically delicate conditions for BIC almost always result in the waves weakly coupling with the radiation modes, leading to leaky states called quasi-BIC that have a large quality factor. We study the asymptotic nature of this weak coupling in the case of acoustic waves interacting with a rigid substrate featuring a partially partitioned slit--a setup that supports quasi-BIC that exponentially approach BIC as the slit is made increasingly narrow. In that limit, we use the method of matched asymptotic expansions in conjunction with reciprocal relations to study those quasi-BIC and their resonant excitation. In particular, we derive a leading approximation for the exponentially small imaginary part of each wavenumber eigenvalue (inversely proportional to quality factor), which is beyond all orders of the expansion for the wavenumber eigenvalue itself. Furthermore, we derive a leading approximation for the exponentially large amplitudes of the states in the case where they are resonantly excited by a plane wave at oblique incidence. These resonances occur in exponentially narrow wavenumber intervals and are physically manifested in cylindrical-dipolar waves emanating from the slit aperture and exponentially large field enhancements inside the slit. The asymptotic approximations are validated against numerical calculations. [ABSTRACT FROM AUTHOR]

We study scattering by a high aspect ratio particle using boundary integral equation methods. This problem has important applications in nanophotonics problems, including sensing and plasmonic imaging. To illustrate the effect of parity and the need for adapted methods in the presence of high aspect ratio particles, we consider the scattering in two dimensions by a sound-hard, high aspect ratio ellipse. This fundamental problem highlights the main challenge and provides valuable insights to tackle plasmonic problems and general high aspect ratio particles. For this problem, we find that the boundary integral operator is nearly singular due to the collapsing geometry from an ellipse to a line segment. We show that this nearly singular behavior leads to qualitatively different asymptotic behaviors for solutions with different parities. Without explicitly taking this nearly singular behavior and this parity into account, computed solutions incur a large error. To address these challenges, we introduce a new method called quadrature by parity asymptotic expansions (QPAX) that effectively and efficiently addresses these issues. We first develop QPAX to solve the Dirichlet problem for Laplace's equation in a high aspect ratio ellipse. Then, we extend QPAX for scattering by a sound-hard, high aspect ratio ellipse. We demonstrate the effectiveness of QPAX through several numerical examples. [ABSTRACT FROM AUTHOR]

The one‐dimensional radial vibration model of piezoelectric disks has been widely used to determine the relevant material coefficients from admittance measurements. However, the one‐dimensional model assumes infinitely thin disks, and therefore cannot predict their axial displacements. We extend the one‐dimensional model by performing an asymptotic analysis of the axisymmetric radial vibration of thin disks. The asymptotic expansions include the asymptotic axial displacement and the second‐order corrections to the admittance and the radial displacement in the one‐dimensional model. We verify the asymptotic expansions and the one‐dimensional model with the Chebyshev tau method. In the one‐dimensional model, the frequencies of the maximum admittance fn in the first and second radial modes are accurate to 1% for Pz27 disks with thickness‐to‐diameter ratios of 0.15 and 0.065, respectively. For a general piezoelectric disk in the forced vibration, the error of fn in the one‐dimensional model can be estimated from the second‐order correction of the asymptotic resonance frequency in the free vibration. [ABSTRACT FROM AUTHOR]

In this paper, we consider a microscopic semilinear elliptic equation posed in periodically perforated domains and associated with the Fourier-type condition on internal micro-surfaces. The first contribution of this work is the construction of a reliable linearization scheme that allows us, by a suitable choice of scaling arguments and stabilization constants, to prove the weak solvability of the microscopic model. Asymptotic behaviors of the microscopic solution with respect to the microscale parameter are thoroughly investigated in the second theme, based upon several cases of scaling. In particular, the variable scaling illuminates the trivial and non-trivial limits at the macroscale, confirmed by certain rates of convergence. Relying on classical results for homogenization of multiscale elliptic problems, we design a modified two-scale asymptotic expansion to derive the corresponding macroscopic equation, when the scaling choices are compatible. Moreover, we prove the high-order corrector estimates for the homogenization limit in the energy space , using a large amount of energy-like estimates. A numerical example is provided to corroborate the asymptotic analysis. [ABSTRACT FROM AUTHOR]

We study the time-domain acoustic scattering problem by a cluster of small holes (i.e., sound-soft obstacles). Based on the retarded boundary integral equation method, we derive the asymptotic expansion of the scattered field as the size of the holes goes to zero. Under certain geometrical constraints on the size and the minimum distance between the holes, we show that the scattered field is approximated by a linear combination of point-sources where the weights are given by the capacitance of each hole and the causal signals (of these point-sources) can be computed by solving a retarded in time linear algebraic system. A rigorous justification of the asymptotic expansion and the unique solvability of the linear algebraic system is shown under natural conditions on the cluster of holes. As an application of the asymptotic expansion, we derive, in the limit case when the holes are densely distributed and occupy a bounded domain, the equivalent effective acoustic medium (an equivalent mass density characterized by the capacitance of the holes) that generates, approximately, the same scattered field as the cluster of holes. Conversely, given a locally variable, smooth, and positive mass density, satisfying a certain subharmonicity condition, we can design a perforated material with holes, having appropriate capacitances, that generates approximately the same acoustic field as the acoustic medium modeled by the given mass density (and constant speed of propagation). Finally, we numerically verify the asymptotic expansions by comparing the asymptotic approximations with the numerical solutions of the scattered fields via the finite element method. [ABSTRACT FROM AUTHOR]

In previous work the authors found the asymptotic expansion of the L 2 -norm of the solution u (t , x) of the strongly damped wave equation u t t − Δ u t − Δ u = 0 and also of the L 2 -norm of the difference between u (t , x) and its asymptotic approximation ν (t , x). This was done in space dimension N ⩾ 3. In the present work results are extended to the exceptional cases N = 1 and N = 2. This extension is achieved by deriving new lemmas on the asymptotic expansion of some parameter dependent integrals. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this study is sensitivity analysis of the L2-norm and H1-seminorm of the solution of a diffusive–convective–reactive problem to topological changes of the underlying material. Design/methodology/approach: The topological derivative method is used to devise a simple and efficient topology design algorithm based on a level-set domain representation method. Findings: Remarkably simple analytical expressions for the sensitivities are derived, which are useful for practical applications including heat exchange topology design and membrane eigenvalue maximization. Originality/value: The topological asymptotic expansion associated with a diffusive–convective–reactive equation is rigorously derived, which is not available in the literature yet. [ABSTRACT FROM AUTHOR]

In this paper, we devise an asymptotic model for calculating electromagnetic diffraction and absorption in planar multilayered structures with a shallow surface-relief grating. Far from the grating, we assume that the solution can be written as a power series in terms of the grating thickness δ, the coefficients of this expansion being smooth up to the grating. However, the expansion approximates the solution only sufficiently far from the grating (far field approximation). Near the grating, we assume that there exists another expansion in powers of δ (near field approximation). Moreover, there is an overlapping zone where both expansion are valid. The proposed model is based on matching the two expansions on this overlapping domain. Then, by truncating terms of order δ2 or higher, we obtain explicitly the equations satisfied by the lowest order terms in the power series. Under appropriate assumptions, we prove second order convergence of the error with respect to δ. Finally, an alternative form, more convenient for implementation, is derived and discretized with finite elements to perform some numerical tests. [ABSTRACT FROM AUTHOR]

This paper deals with the derivation and analysis of reduced order elliptic PDE models on fractured domains. We use a Fourier analysis to obtain coupling conditions between subdomains when the fracture is represented as a hypersurface embedded in the surrounded rock matrix. We compare our results to prominent examples from the literature for diffusive models. In a second step, we present error estimates for the reduced order models in terms of the fracture width. For the proofs, we rely on a combination of Fourier analysis, asymptotic expansions, and functional analysis. Finally, we study the behavior of the error of the reduced order solutions on numerical test cases when the fracture width tends to zero. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider the propagation of surface water waves in a straight planar channel perturbed at the bottom by several thin curved tunnels and wells. We propose a method to construct non reflecting underwater topographies of this type at an arbitrary prescribed wave number. To proceed, we compute asymptotic expansions of the diffraction solutions with respect to the small parameter of the geometry taking into account the existence of boundary layer phenomena. We establish error estimates to validate the expansions using advances techniques of weighted spaces with detached asymptotics. In the process, we show the absence of trapped surface waves for perturbations small enough. This analysis furnishes asymptotic formulas for the scattering matrix and we use them to determine underwater topographies which are non-reflecting. Theoretical and numerical examples are given. [ABSTRACT FROM AUTHOR]

As ε goes to zero, the unique solution of the scalar advection‐diffusion equation ytε−εyxxε+Myxε=0, (x,t)∈(0,1)×(0,T) with Dirichlet boundary conditions exhibits a boundary layer of size O(ε) and an internal layer of size O(ε). If the time T is large enough, these thin layers, where the solution yε displays rapid variations, intersect and interact with each other. Using the method of matched asymptotic expansions, we show how we can construct an explicit approximation P˜ε of the solution yε satisfying ‖yε−P˜ε‖L∞(0,T;L2(0,1))=O(ε3/2) and ‖yε−P˜ε‖L2(0,T;H1(0,1))=O(ε) for all ε small enough. [ABSTRACT FROM AUTHOR]

Excitation of surface-plasmon resonances of closely spaced nanometallic structures is a key technique used in nanoplasmonics to control light on subwavelength scales and generate highly confined electric-field hotspots. In this paper, we develop asymptotic approximations in the near-contact limit for the entire set of surface-plasmon modes associated with the prototypical sphere dimer geometry. Starting from the quasi-static plasmonic eigenvalue problem, we employ the method of matched asymptotic expansions between a gap region, where the boundaries are approximately paraboloidal, pole regions within the spheres and close to the gap, and a particle-scale region where the spheres appear to touch at leading order. For those modes that are strongly localised to the gap, relating the gap and pole regions gives a set of effective eigenvalue problems formulated over a half space representing one of the poles. We solve these problems using integral transforms, finding asymptotic approximations, singular in the dimensionless gap width, for the eigenvalues and eigenfunctions. In the special case of modes that are both axisymmetric and odd about the plane bisecting the gap, where matching with the outer region introduces a logarithmic dependence upon the dimensionless gap width, our analysis follows Schnitzer [Singular perturbations approach to localized surface-plasmon resonance: nearly touching metal nanospheres. Phys. Rev. B92(23), 235428 (2015)]. We also analyse the so-called anomalous family of even modes, characterised by field distributions excluded from the gap. We demonstrate excellent agreement between our asymptotic formulae and exact calculations. [ABSTRACT FROM AUTHOR]

A two-dimensional time dependent model of an interaction between a thin elastic plate and a Newtonian viscous fluid described by the non-steady Stokes equations is considered. It depends on a small parameter ε that is the ratio of the thicknesses of the plate and the fluid layer. The Young's modulus of the plate and its density may be great or small parameters equal to some powers (positive or negative) of ε while the density and the viscosity of the fluid are supposed to be of order one. An asymptotic expansion is constructed and justified for various magnitudes of the rigidity and density of the plate. The limit problems are studied in all these cases. They are Stokes equations with some special coupled or uncoupled boundary conditions modeling the interaction with the plate. The estimates of the difference between the exact solution and a truncated asymptotic expansion are established. These estimates justify the asymptotic approximations. [ABSTRACT FROM AUTHOR]

We derive the effective equations describing the behavior of the fluid in strap with rugged boundary. Starting from the Navier--Stokes system in domain with periodically perturbed boundary, we compute the asymptotic expansion of the solution. Using the expansion we obtain a 2D version of the Darcy--Weisbach law. [ABSTRACT FROM AUTHOR]

Abstract The Fourier transform, F , on R N (N ≥ 3) transforms the Cauchy problem for the strongly damped wave equation u t t − Δ u t − Δ u = 0 to an ordinary differential equation in time. We let u (t , x) be the solution of the problem given by the Fourier transform, and ν (t , ξ) be the asymptotic profile of F (u) (t , ξ) = u ˆ (t , ξ) found by Ikehata in the paper Asymptotic profiles for wave equations with strong damping (2014). In this paper we study the asymptotic expansions of the squared L 2 -norms of u (t , x) , u ˆ (t , ξ) − ν (t , ξ) , and ν (t , ξ) as t → ∞. With suitable initial data u (0 , x) and u t (0 , x) , we establish the rate of decay of the squared L 2 -norms of u (t , x) and ν (t , ξ) as t → ∞. By noting the cancellation of leading terms of their respective expansions, we conclude that the rate of convergence between u ˆ (t , ξ) and ν (t , ξ) in the L 2 -norm occurs quickly relative to their individual behaviors. This observation is similar to the diffusion phenomenon, which has been well studied. [ABSTRACT FROM AUTHOR]

In this paper, we study polynomials orthogonal with respect to a Pollaczek–Jacobi type weight wpJ(x,t)=e−txxα(1−x)β,t≥0,α>0,β>0,x∈[0,1].The uniform asymptotic expansions for the monic orthogonal polynomials on the interval (0,1) and outside this interval are obtained. Moreover, near x=0, the uniform asymptotic expansion involves Airy function as ς=2n2t→∞,n→∞, and Bessel function of order α as ς=2n2t→0,n→∞; in the neighborhood of x=1, the uniform asymptotic expansion is associated with Bessel function of order β as n→∞. The recurrence coefficients and leading coefficient of the orthogonal polynomials are expressed in terms of a particular Painlevé III transcendent. We also obtain the limit of the kernel in the bulk of the spectrum. The double scaled logarithmic derivative of the Hankel determinant satisfies a σ‐form Painlevé III equation. The asymptotic analysis is based on the Deift and Zhou's steepest descent method. [ABSTRACT FROM AUTHOR]

In this paper we introduce a model describing a layered structure composed by two thermoelastic adherents and a thin adhesive subject to a degradation process. By an asymptotic expansion method, we derive a model of imperfect interface coupling damage and temperature evolution. Moreover, assuming that the behaviour of the adhesive is ruled by two different regimes, one in traction and one in compression, we derive a second limit model where unilateral contact conditions on the interface are also included. [ABSTRACT FROM AUTHOR]

This paper focuses on Generalized Impedance Boundary Conditions (GIBC) with second order derivatives in the context of linear elasticity and general curved interfaces. A condition of the Wentzell type modeling thin layer coatings on some elastic structures is obtained through an asymptotic analysis of order one of the transmission problem at the thin layer interfaces with respect to the thickness parameter. We prove the well-posedness of the approximate problem and the theoretical quadratic accuracy of the boundary conditions. Then we perform a shape sensitivity analysis of the GIBC model in order to study a shape optimization/optimal design problem. We prove the existence and characterize the first shape derivative of this model. A comparison with the asymptotic expansion of the first shape derivative associated to the original thin layer transmission problem shows that we can interchange the asymptotic and shape derivative analysis. Finally we apply these results to the compliance minimization problem. We compute the shape derivative of the compliance in this context and present some numerical simulations. [ABSTRACT FROM AUTHOR]

We perform the asymptotic analysis of the scalar advection-diffusion equation y t ε − ε y x x ε + M y x ε = 0 , (x , t) ∈ (0 , 1) × (0 , T) , M > 0 , with respect to the diffusion coefficient ε. We use the matched asymptotic expansion method which allows to describe the boundary layers of the solution. We then use the asymptotics to discuss the controllability property of the solution for T ⩾ 1 / M. [ABSTRACT FROM AUTHOR]

We introduce phononic box crystals, namely, arrays of adjoined perforated boxes, as a three-dimensional prototype for an unusual class of subwavelength metamaterials based on directly coupling resonating elements. In this case, when the holes coupling the boxes are small, we create networks of Helmholtz resonators with nearest-neighbor interactions. We use matched asymptotic expansions, in the small hole limit, to derive simple, yet asymptotically accurate, discrete wave equations governing the pressure field. These network equations readily furnish analytical dispersion relations for box arrays, slabs, and crystals that agree favorably with finite-element simulations of the physical problem. Our results reveal that the entire acoustic branch is uniformly squeezed into a subwavelength regime; consequently, phononic box crystals exhibit nonlinear-dispersion effects (such as dynamic anisotropy) in a relatively wide band, as well as a high effective refractive index in the long-wavelength limit. We also study the sound field produced by sources placed within one of the boxes by comparing and contrasting monopole- with dipole-type forcing; for the former the pressure field is asymptotically enhanced while for the latter there is no asymptotic enhancement and the translation from the microscale to the discrete description entails evaluating singular limits, using a regularized and efficient scheme, of the Neumann Green's function for a cube. We conclude with an example of using our asymptotic framework to calculate localized modes trapped within a defected box array. [ABSTRACT FROM AUTHOR]

Abstract By using the Darboux transformation, we obtain two new types of exponential-and-rational mixed soliton solutions for the defocusing nonlocal nonlinear Schrödinger equation. We reveal that the first type of solution can display a large variety of interactions among two exponential solitons and two rational solitons, in which the standard elastic interaction properties are preserved and each soliton could be either the dark or antidark type. By developing the asymptotic analysis method, we also find that the second type of solution can exhibit the elastic interactions among four mixed asymptotic solitons. But in sharp contrast to the common solitons, the mixed asymptotic solitons have the t -dependent velocities and their phase shifts before and after interaction also grow with | t | in the logarithmical manner. In addition, we discuss the degenerate cases for such two types of mixed soliton solutions when the four-soliton interaction reduces to a three-soliton or two-soliton interaction. Highlights • We derive two new types of exponential-and-rational mixed soliton solutions. • The elastic interactions are discovered among exponential and rational solitons, or among mixed solitons. • The mixed asymptotic solitons have the t-dependent velocities and phase shift upon an interaction. • The asymptotic analysis method is developed by considering the balance between the algebraic and exponential terms. [ABSTRACT FROM AUTHOR]

In this companion paper to "Optimal Investment with Transaction Costs and Stochastic Volatility Part I: Infinite Horizon,"" we give an accuracy proof for the finite time optimal investment and consumption problem under fast mean-reverting stochastic volatility of a joint asymptotic expansion in a time scale parameter and the small transaction cost. [ABSTRACT FROM AUTHOR]

A periodic, axially symmetric time-dependent model of interaction between a viscous fluid and a thin cylindrical elastic tube is considered. The problem depends on a small parameter , representing the ratio of the thickness of the wall and the radius of the cylinder. We consider Young's modulus of the elastic medium and its density great or small parameters equal to some powers of . An asymptotic expansion is constructed for various magnitudes of the rigidity and of the density of the elastic tube. The generality of the considered model requires to take into account six different combinations of the three parameters depending on . We obtain in this way different limit problems which are Stokes equations with some special boundary conditions. The expansion is justified by the high-order estimates for the difference of the exact solution and truncated asymptotic approximation. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study a time-dependent flow of an incompressible micropolar fluid through a pipe with arbitrary cross-section. The effective behavior of the flow is found by means of rigorous asymptotic analysis with respect to the small parameter representing the pipe's thickness. The complete asymptotic expansion (up to an arbitrary order) of the solution is constructed with detailed analysis of the boundary layers provided. The convergence of the expansion is proved justifying the usage of the formally derived asymptotic model. [ABSTRACT FROM AUTHOR]

In this paper we tackle the asymptotic behavior of a family of orthogonal polynomials with respect to a nonstandard inner product involving the forward operator Δ. Concretely, we treat the generalized Charlier weights in the framework of Δ -Sobolev orthogonality. We obtain an asymptotic expansion for these orthogonal polynomials where the falling factorial polynomials play an important role. [ABSTRACT FROM AUTHOR]

The kernel-based regularization method has two core issues: kernel design and hyperparameter estimation. In this paper, we focus on the second issue and study the properties of several hyperparameter estimators including the empirical Bayes (EB) estimator, two Stein’s unbiased risk estimators (SURE) (one related to impulse response reconstruction and the other related to output prediction) and their corresponding Oracle counterparts, with an emphasis on the asymptotic properties of these hyperparameter estimators. To this goal, we first derive and then rewrite the first order optimality conditions of these hyperparameter estimators, leading to several insights on these hyperparameter estimators. Then we show that as the number of data goes to infinity, the two SUREs converge to the best hyperparameter minimizing the corresponding mean square error, respectively, while the more widely used EB estimator converges to another best hyperparameter minimizing the expectation of the EB estimation criterion. This indicates that the two SUREs are asymptotically optimal in the corresponding MSE senses but the EB estimator is not. Surprisingly, the convergence rate of two SUREs is slower than that of the EB estimator, and moreover, unlike the two SUREs, the EB estimator is independent of the convergence rate of Φ T Φ ∕ N to its limit, where Φ is the regression matrix and N is the number of data. A Monte Carlo simulation is provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

Four binary discrimination methods are studied in the context of high-dimension, low sample size data with an asymptotic geometric representation, when the dimension increases while the sample sizes of the classes are fixed. We show that the methods support vector machine, mean difference, distance-weighted discrimination, and maximal data piling have the same asymptotic behavior as the dimension increases. We study the consistent, inconsistent, and strongly inconsistent cases in terms of angles between the normal vectors of the separating hyperplanes of the methods and the optimal direction for classification. A simulation study is done to assess the theoretical results. [ABSTRACT FROM AUTHOR]

In a smooth bounded domain Ω of R2 we consider the spectral problem - Ωuε = λ (ε )ρε uε with boundary condition ∂uε/∂ν = 0. The factor ρε plays the role of a mass density, and it is equal to a constant of order ε - 1 in an ε -neighborhood of the boundary and to a constant of order ε-1 in the rest of Ω. We study the asymptotic behavior of the eigenvalues λ(ε) and the eigenfunctions uε as ε tends to zero. We obtain explicit formulas for the first and second terms of the corresponding asymptotic expansions by exploiting the solutions of certain auxiliary boundary value problems. [ABSTRACT FROM AUTHOR]

We consider a family of linear viscoelastic shells with thickness 2 ε (where ε is a small parameter), clamped along a portion of their lateral face, all having the same middle surface S. We formulate the three-dimensional mechanical problem in curvilinear coordinates and provide existence and uniqueness of (weak) solution of the corresponding three-dimensional variational problem. We are interested in studying the limit behavior of both the three-dimensional problems and their solutions when ε tends to zero. To do that, we use asymptotic analysis methods. First, we formulate the variational problem in a fixed domain independent of ε. Then we assume an asymptotic expansion of the scaled displacements field, u ( ε ) = ( u i ( ε ) ) , and we characterize the zeroth order term as the solution of a two-dimensional scaled limit problem. Moreover, we find that, depending on the order of the applied forces, the limit of the field u ( ε ) is the solution of one of the two sets of two-dimensional variational equations derived, which can be described as viscoelastic membrane shell and viscoelastic flexural shell problems. In both cases, we find a model which presents a long-term memory that takes into account the deformations at previous times. We finally comment on the existence and uniqueness of solution for the two-dimensional variational problems found and announce convergence results. [ABSTRACT FROM AUTHOR]

We study a 2D semi-linear equation in a domain with a small Dirichlet obstacle of size . Using the method of matched asymptotic expansions, we compute an asymptotic expansion of the solution as tends to zero. Its relevance is justified by proving a rigorous error estimate. Then we construct an approximate model, based on an equation set in the limit domain without the small obstacle, which provides a good approximation of the far field of the solution of the original problem. The interest of this approximate model lies in the fact that it leads to a variational formulation which is very simple to discretize. We present numerical experiments to illustrate the analysis. [ABSTRACT FROM AUTHOR]

We consider a family of linearly elastic shells with thickness $2\varepsilon$ (where $\varepsilon$ is a small parameter). The shells are clamped along a portion of their lateral face, all having the same middle surface $S$ , and may enter in contact with a rigid foundation along the bottom face. We are interested in studying the limit behavior of both the three-dimensional problems, given in curvilinear coordinates, and their solutions (displacements $\boldsymbol{u}^{\varepsilon}$ of covariant components $u_{i}^{\varepsilon}$ ) when $\varepsilon$ tends to zero. To do that, we use asymptotic analysis methods. On one hand, we find that if the applied body force density is $O(1)$ with respect to $\varepsilon$ and surface tractions density is $O(\varepsilon)$ , a suitable approximation of the variational formulation of the contact problem is a two-dimensional variational inequality which can be identified as the variational formulation of the obstacle problem for an elastic membrane. On the other hand, if the applied body force density is $O(\varepsilon^{2})$ and surface tractions density is $O(\varepsilon^{3})$ , the corresponding approximation is a different two-dimensional inequality which can be identified as the variational formulation of the obstacle problem for an elastic flexural shell. We finally discuss the existence and uniqueness of solution for the limit two-dimensional variational problems found. [ABSTRACT FROM AUTHOR]

This work proposes new transmission conditions at the interfaces between the layers of a three-dimensional composite structures. The proposed transmission conditions are obtained by applying the asymptotic expansion technique in the framework of Lagrange-Hamilton's principle. The proposed conditions take into account interfacial effects of higher order, thus representing an extension of the classical zero-thickness interface models. In particular, the (small) thickness of the interface together with its inertia, stiffness and anisotropy are accounted for. The effect of the transmission conditions on the band structure of Bloch–Floquet waves propagating in a one dimensional phononic crystal is discussed based on numerical results. [ABSTRACT FROM AUTHOR]

This paper studies the planar deformations of a beam composed of a linearly elastic material. Starting from the field equations for the plane-stress problem and adopting a series expansion for the displacement vector about the bottom surface, we deduce the beam equations with two unknowns in a consistent manner. The success relies on using the field equations together with the bottom traction conditions to establish the exact recursion relations, such that all quantities can be represented in terms of the two leading expansion coefficients of the displacements. Another feature is that the remainders of the series can be carried over to the beam equations. Then, based on the general solutions and the error terms of the beam equations, pointwise error estimates for displacement and stress fields are rigorously established. Three benchmark problems are considered, for which the two-dimensional exact solutions are available. It is shown that this new beam theory recovers the exact solutions for these problems. Two cases with boundary layer effects are also discussed in the appendix. [ABSTRACT FROM AUTHOR]

Recent modeling of coffee bean roasting suggests that in the early stages of roasting, within each coffee bean, there are two emergent regions: a dried outer region and a saturated interior region. The two regions are separated by a transition layer (or, drying front). In this paper, we consider the asymptotic analysis of a recent multiphase model in order to gain a better understanding of its salient features. The model consists of a PDE system governing the thermal, moisture, and gas pressure profiles throughout the interior of the bean. By obtaining asymptotic expansions for these quantities in relevant limits of the physical parameters, we are able to determine the qualitative behavior of the outer and interior regions, as well as the dynamics of the drying front. Although a number of simplifications and scalings are used, we take care not to discard aspects of the model which are fundamental to the roasting process. Indeed, we find that for all of the asymptotic limits considered, our approximate solutions faithfully reproduce the qualitative features evident from numerical simulations of the full model. From these asymptotic results, we have a better qualitative understanding of the drying front (which is hard to resolve precisely in numerical simulations) and, hence, of the various mechanisms at play including heating, evaporation, and pressure changes. This qualitative understanding of solutions to the multiphase model is essential when creating more involved models that incorporate chemical reactions and solid mechanics effects. [ABSTRACT FROM AUTHOR]

We give in this paper an uniqueness result of the lubricated Elrod Adams model in stationary and transitory cases and in one dimensional space. We give also an asymptotic behaviour in time. [ABSTRACT FROM AUTHOR]