Showing papers on "Method of matched asymptotic expansions published in 2018"

Asymptotic Theory of Wave-Propagation

Abstract: A general method is presented for finding asymptotic solutions of problems in wave-propagation. The method is applicable to linear symmetric-hyperbolic partial differential equations and to the integro-differential equations for the electromagnetic field in a dispersive medium. These equations may involve a large parameter λ. In the electromagnetic case λ is a characteristic frequency of the medium. The parameter may also appear in initial data or in the source terms of the equations, in a variety of different ways. This gives rise to a variety of different types of asymptotic solutions. The expansion procedure is a “ray method”, i.e., all the functions that appear in the expansion satisfy ordinary differential equations along certain space-time curves called rays. In general, these rays do not lie on characteristic surfaces, but may, for example, fill out the interior of a characteristic hypercone. They are associated with an appropriately defined “group velocity”. In subsequent papers the ray method developed here will be applied to the analysis of transients, Cerenkov radiation, transition radiation, and other phenomena of wave-propagation.

Parameter-Robust Numerical Scheme for Time-Dependent Singularly Perturbed Reaction–Diffusion Problem with Large Delay

Abstract: This work presents the development of numerical scheme for second-order time-dependent singularly perturbed reaction-diffusion problem with large delay in the undifferentiated term. These t...

Comparative Study between Optimal Homotopy Asymptotic Method and Perturbation-Iteration Technique for Different Types of Nonlinear Equations

Abstract: In this paper, we compare optimal homotopy asymptotic method and perturbation-iteration method to solve random nonlinear differential equations. Both of these methods are known to be new and very powerful for solving differential equations. We give some numerical examples to prove these claims. These illustrations are also used to check the convergence of the proposed methods.

On the homogenization of the Helmholtz problem with thin perforated walls of finite length

Abstract: In this work, we present a new solution representation for the Helmholtz transmission problem in a bounded domain in ℝ2 with a thin and periodic layer of finite length. The layer may consists of a periodic pertubation of the material coefficients or it is a wall modelled by boundary conditions with an periodic array of small perforations. We consider the periodicity in the layer as the small variable δ and the thickness of the layer to be at the same order. Moreover we assume the thin layer to terminate at re-entrant corners leading to a singular behaviour in the asymptotic expansion of the solution representation. This singular behaviour becomes visible in the asymptotic expansion in powers of δ where the powers depend on the opening angle. We construct the asymptotic expansion order by order. It consists of a macroscopic representation away from the layer, a boundary layer corrector in the vicinity of the layer, and a near field corrector in the vicinity of the end-points. The boundary layer correctors and the near field correctors are obtained by the solution of canonical problems based, respectively, on the method of periodic surface homogenization and on the method of matched asymptotic expansions. This will lead to transmission conditions for the macroscopic part of the solution on an infinitely thin interface and corner conditions to fix the unbounded singular behaviour at its end-points. Finally, theoretical justifications of the second order expansion are given and illustrated by numerical experiments. The solution representation introduced in this article can be used to compute a highly accurate approximation of the solution with a computational effort independent of the small periodicity δ .

Asymptotic expansions and unique continuation at Dirichlet-Neumann boundary junctions for planar elliptic equations

Abstract: We consider elliptic equations in planar domains with mixed boundary conditions of Dirichlet-Neumann type. Sharp asymptotic expansions of the solutions and unique continuation properties from the Dirichlet-Neumann junction are proved.

Vacuum solutions around spherically symmetric and static objects in the Starobinsky model

Abstract: The vacuum solutions around a spherically symmetric and static object in the Starobinsky model are studied with a perturbative approach. The differential equations for the components of the metric and the Ricci scalar are obtained and solved by using the method of matched asymptotic expansions. The presence of higher order terms in this gravity model leads to the formation of a boundary layer near the surface of the star allowing the accommodation of the extra boundary conditions on the Ricci scalar. Accordingly, the metric can be different from the Schwarzschild solution near the star depending on the value of the Ricci scalar at the surface of the star while matching the Schwarzschild metric far from the star.

Asymptotic approximations for the plasmon resonances of nearly touching spheres

Abstract: 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 [O. Schnitzer, \textit{Physical Review B}, \textbf{92} 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.

Reactions, diffusion, and volume exclusion in a conserved system of interacting particles

Abstract: Complex biological and physical transport processes are often described through systems of interacting particles. The effect of excluded volume on these transport processes has been well studied; however, the interplay between volume exclusion and reactions between heterogenous particles is less well studied. In this paper we develop a framework for modeling reaction-diffusion processes which directly incorporates volume exclusion. We consider simple reactions (unimolecular and bimolecular) that conserve the total number of particles. From an off-lattice microscopic individual-based model we use the Fokker-Planck equation and the method of matched asymptotic expansions to derive a low-dimensional macroscopic system of nonlinear partial differential equations describing the evolution of the particles. A biologically motivated, hybrid model of chemotaxis with volume exclusion is explored, where reactions occur at rates dependent upon the chemotactic environment. Further, we show that for reactions that require particle contact the appropriate reaction term in the macroscopic model is of lower order in the asymptotic expansion than the nonlinear diffusion term. However, we find that the next reaction term in the expansion is needed to ensure good agreement with simulations of the microscopic model. Our macroscopic model allows for more direct parametrization to experimental data than existing models.

The computation of biomarkers in pharmacokinetics with the aid of singular perturbation methods

Abstract: In pharmacokinetics, exact solutions to one-compartment models with nonlinear elimination kinetics cannot be found analytically, if dosages are assumed to be administered repetitively through extravascular routes (Tang and Xiao in J Pharmacokinet Pharmacodyn 34(6):807–827, 2007). Hence, for the corresponding impulsed dynamical system, alternative methods need to be developed to find approximate solutions. The primary purpose of this paper is to use the method of matched asymptotic expansions (Holmes Introduction to Perturbation Methods, vol 20. Springer Science & Business Media, Berlin, 2012), a singular perturbation method (Holmes, Introduction to Perturbation Methods, vol 20. Springer Science & Business Media, Berlin, 2012; Keener Principles of Applied Mathematics, Addison-Wesley, Boston, 1988), to obtain approximate solutions. With this method, we are able to rigorously determine conditions under which there is a stable periodic solution of the model equations. Furthermore, typical important biomarkers that enable the design of practical, efficient and safe drug delivery protocols, such as the time the drug concentration reaches the peak and the peak concentrations, are theoretically estimated by the perturbation method we employ.

Asymptotic Solution of a Dispersive Hyperbolic Equation With Variable Coefficients

Abstract: : Initial-boundary value problems are considered for an energy conserving dispersive hyperbolic equation, the Klein-Gordon equation This equation contains the main feature of dispersion: The speed of propagation depends on the frequency The asymptotic expansion of solutions obtained by a technique which we call the ray method is compared with the asymptotic expansion of the exact solution In every case considered, the solutions agree Solutions are obtained for a series of initial-boundary value problems in one space dimension with variable coefficients A new feature which is called space-time diffraction is found This phenomenon has the following physical interpretation: A portion of the energy of a wave reaches a boundary surface and then gradually leaks off, leaving a diminishing residue on the boundary for all time (Author)

On impedance conditions for circular multiperforated acoustic liners

Abstract: The acoustic damping in gas turbines and aero-engines relies to a great extent on acoustic liners that consists of a cavity and a perforated face sheet. The prediction of the impedance of the liners by direct numerical simulation is nowadays not feasible due to the hundreds to thousands repetitions of tiny holes. We introduce a procedure to numerically obtain the Rayleigh conductivity for acoustic liners for viscous gases at rest, and with it define the acoustic impedance of the perforated sheet. The proposed method decouples the effects that are dominant on different scales: (a) viscous and incompressible flow at the scale of one hole, (b) inviscid and incompressible flow at the scale of the hole pattern, and (c) inviscid and compressible flow at the scale of the wave-length. With the method of matched asymptotic expansions we couple the different scales and eventually obtain effective impedance conditions on the macroscopic scale. For this the effective Rayleigh conductivity results by numerical solution of an instationary Stokes problem in frequency domain around one hole with prescribed pressure at infinite distance to the aperture. It depends on hole shape, frequency, mean density and viscosity divided by the area of the periodicity cell. This enables us to estimate dissipation losses and transmission properties, that we compare with acoustic measurements in a duct acoustic test rig with a circular cross-section by the German Aerospace Center in Berlin. A precise and reasonable definition of an effective Rayleigh conductivity at the scale of one hole is proposed and impedance conditions for the macroscopic pressure or velocity are derived in a systematic procedure. The comparison with experiments show that the derived impedance conditions give a good prediction of the dissipation losses.

The Movement of an Ion-exchange Microparticle in a Weak External Electric Field

Abstract: The electrokinetic motion of spherical particle suspended in the electrolyte solution under influence of external electric field is studied. Due to impermeability of particle’s surface for one kind of ion species the particle exhibit behavior different to well investigated dielectric particles. Under an assumption of a weak external electric field, we derive the analytical estimation of the particle’s velocity by means of a method of matched asymptotic expansions. The analytical analysis is complemented by numerical solution, which gives the distribution of ion’s concentrations, electric potential profiles and flows streamlines. The analytical results are successfully compared with the results of numerical simulation.

Application of the Method of Matched Asymptotic Expansions to Solve a Nonlinear Pseudo-Parabolic Equation: The Saturation Convection-Dispersion Equation

Abstract: In this work, we apply the method of matched asymptotic expansions to solve the onedimensional saturation convection-dispersion equation, a nonlinear pseudo-parabolic partial differential equation. This equation is one of the governing equations for twophase flow in a porous media when including capillary pressure effects, for the specific initial and boundary conditions arising when injecting water in an infinite radial piecewise homogeneous horizontal medium containing oil and water. The method of matched asymptotic expansions combines inner and outer expansions to construct the global solution. In here, the outer expansion corresponds to the solution of the nonlinear first-order hyperbolic equation obtained when the dispersion effects driven by capillary pressure became negligible. This equation has a monotonic flux function with an inflection point, and its weak solution can be found by applying the method of characteristics. The inner expansion corresponds to the shock layer, which is modeled as a traveling wave obtained by a stretching transformation of the partial differential equation. In the transformed domain, the traveling wave solution is solved using regular perturbation theory. By combining the solution for saturation with the so-called Thompson-Reynolds steady-state theory for obtaining the pressure, one can obtain an approximate analytical solution for the wellbore pressure, which can be used as the forward solution which analyzes pressure data by pressure-transient analysis.

Small obstacle asymptotics for a 2D semi-linear convex problem

Abstract: 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.

Perturbations of the scattering resonances of an open cavity by small particles. Part II: The transverse electric polarization case

Abstract: This paper is concerned with the scattering resonances of open cavities. It is a follow-up of "Perturbation of the scattering resonances of an open cavity by small particles. Part I" where the transverse magnetic polarization was assumed. In that case, using the method of matched asymptotic expansions, the leading-order term in the shifts of scattering resonances due to the presence of small particles of arbitrary shapes was derived and the effect of radiation on the perturbations of open cavity modes was characterized. The derivations were formal. In this paper, we consider the transverse electric polarization and prove a small-volume formula for the shifts in the scattering resonances of a radiating dielectric cavity perturbed by small particles. We show a strong enhancement in the frequency shift in the case of plasmonic particles. We also consider exceptional scattering resonances and perform small-volume asymptotic analysis near them. Our method in this paper relies on pole-pencil decompositions of volume integral operators.

Asymptotic Analysis for the Singularly Perturbed Initial Value Problem with Degenerate Equation Having Triple Root

Abstract: In this paper, we study the asymptotic behavior of the initial value problem for the singularly perturbed first order nonlinear differential equation with degenerate equation having triple root. In order to obtain a more precise description of the boundary layer, the modified method of boundary layer function is chosen to construct the boundary layer function possessing the behavior of exponential decay characteristic, and then the asymptotic solution is constructed and used to prove that the formal asymptotic solution is uniformly valids.

A combined analytical and numerical analysis of the flow-acoustic coupling in a cavity-pipe system

Abstract: The generation of sound by flow through a closed, cylindrical cavity (expansion chamber) accommodated with a long tailpipe is investigated analytically and numerically. The sound generation is due to self-sustained flow oscillations in the cavity. These oscillations may, in turn, generate standing (resonant) acoustic waves in the tailpipe. The main interest of the paper is in the interaction between these two sound sources. An analytical, approximate solution of the acoustic part of the problem is obtained via the method of matched asymptotic expansions. The sound-generating flow is represented by a discrete vortex method, based on axisymmetric vortex rings. It is demonstrated through numerical examples that inclusion of acoustic feedback from the tailpipe is essential for a good representation of the sound characteristics.

The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow

Abstract: The free surface and flow field structure generated by the uniform acceleration (with dimensionless acceleration σ) of a rigid plate, inclined at an angle α ∈ (0, π/2) to the exterior horizontal, as it advances (σ > 0) or retreats (σ < 0) from an initially stationary and horizontal strip of inviscid, incompressible fluid under gravity, are studied in the small-time limit via the method of matched asymptotic expansions. This work generalises the case of a uniformly accelerating plate advancing into a fluid as studied in Needham et al. (2008). Particular attention is paid to the innermost asymptotic regions encompassing the initial interaction between the plate and the free surface. We find that the structure of the solution to the governing initial boundary value problem is characterised in terms of the parameters α and μ (where μ = 1+σ tan α), with a bifurcation in structure as μ changes sign. This bifurcation in structure leads us to question the well-posedness and stability of the governing initial boundary value problem with respect to small perturbations in initial data in the innermost asymptotic regions, the discussion of which will be presented in the companion paper Gallagher et al. (2016) . In particular, when (α, μ) ∈ (0, π/2) × R+, the free surface close to the initial contact point remains monotone, and encompasses a swelling jet when (α, μ) ∈ (0, π/2)×[1,∞), or a collapsing jet when (α, μ) ∈ (0, π/2) × (0, 1). However, when (α, μ) ∈ (0, π/2) × R−, the collapsing jet develops a more complex structure, with the free surface close to the initial contact point now developing a finite number of local oscillations, with near resonance type behaviour occurring close to a countable set of critical plate angles α = α∗n ∈ (0, π/2) (n = 1, 2, . . .).

Asymptotic Modeling of the Electromagnetic Scattering by Small Spheres Perfectly Conducting

Abstract: In this paper, we develop a method of matched asymptotic expansions for solving the time-harmonic electromagnetic scattering problem by a small sphere perfectly conducting. This method consists in defining an approximate solution using multi-scale expansions over far and near fields, related in a matching area. We make explicit the asymptotics up to the second order of approximation for the near-field expansion and up to the fifth order of approximation for the far-field expansion. We illustrate the results with numerical experiments which make evident the performance of the asymptotic models. The reference solution is an analytical solution computed thanks to Montjoie code.