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This paper is about solutions to PDEs in which the ratio between the largest and smallest values of the coefficient is very large. Such problems are referred to as high-contrast ones. A full asympt...

Most theoretical models for the thermal response of vertical geothermal boreholes assume the mean azimuthal borehole wall temperature to be constant along the borehole. This fundamental simplification, closely related to the $g$-functions introduced by Eskilson in 1987, has dominated the research field for the past 30 years and has allowed the analysis of entire borehole fields in reasonable amounts of time. However, the assumption is not physically correct, which inevitably hinders the attainable accuracy. In the present work the steady-state thermal response of a single vertical geothermal borehole is analyzed using matched asymptotic expansion techniques, which exploit the extreme slenderness of real world boreholes. The obtained solutions do not require the aforementioned simplification and deliver accuracy levels that are on par with the best performing theoretical/numerical model developed so far, the superposition borehole model.

We derive a set of modified Poisson-Nernst-Planck (mPNP) equations for ion transport from the variation of the free energy functional which includes the many-body Coulomb correlation in media of variable dielectric coefficient. The correlation effects are considered through the Debye charging process in which the self energy of an ion is governed by the generalized Debye-H\"uckel equation. We develop the asymptotic expansions of the self energy taking the ion radius as the small parameter such that the multiscale model can be solved efficiently by numerical methods. It is shown that the variations of the energy functional give the self-energy-modified PNP equations which satisfy a proper weak formulation. We present numerical results from different asymptotic expansions with a semi-implicit conservative numerical method and investigate the effect of the Coulomb correlation., Comment: 20 pages; 2 figures; version to be published in SIAM J. Appl. Math

Saturation of a Kerr-type nonlinearity in the nonlinear Schroźdinger equation (NLSE) can be regarded as a singular perturbation which regularizes the well-known blow-up phenomenon in the cubic NLSE. An asymptotic expansion is proposed which takes into account multiple scale behavior both in the longitudinal and transverse directions. In one dimension, this leads to a free boundary problem reduction where the solitary wave acts solely to reflect impinging waves and is accelerated by an elastic transfer of momentum. In two dimensions, we find that interaction of a solitary wave and an adjacent wave field is governed by behavior of certain eigenfunctions of the linearized fast-scale operator. This leads to an outer solution with a free logarithmic singularity, whose position evolves by virtue of a large transfer of momentum between the ambient and solitary waves. However, for a certain value of wave power, we find there is essentially no interaction and the solitary wave is asymptotically transparent to the ambient field. We test our results by numerical simulation of both the full equation and free boundary reductions.

General higher-order rogue waves of a vector nonlinear Schrodinger equation (Manakov system) are derived using a Darboux-dressing transformation with an asymptotic expansion method. The $N$th-order semirational solutions containing 3N free parameters are expressed in separation-of-variables form. These solutions exhibit rogue waves on a multisoliton background. They demonstrate that the structure of rogue waves in this two-component system is richer than that in a one-component system. Our results would be of much importance in understanding and predicting rogue wave phenomena arising in nonlinear and complex systems, including optics, fluid dynamics, Bose--Einstein condensates, and finance.

The objective of this research is to investigate the upscaling of flow and transport models including both an evolving solid-liquid interface and a quite general potentially even oscillating interaction potential. Starting from a comprehensive pore-scale model, formal, two-scale, asymptotic expansion in a level set framework is applied. In doing so, the interplay between flow, transport, interaction potential, and evolving geometry becomes evident. As a result of an averaging procedure, a fully coupled micro-macro model is established with new main variables. Moreover, time- and space-dependent coefficient functions are explicitly characterized by means of supplementary, fully coupled cell problems. The theoretical results obtained are complemented by the numerical computations of a heterogeneous multiscale scenario with a focus on the development of anisotropic transport. Because of the general framework considered in this research, numerous application are expected, such as in biology or colloidal dynamics.

The main purpose of this paper is to study the structure of the well-known non-iterative MUltiple SIgnal Classification (MUSIC) algorithm for identifying the shape of extended electromagnetic inclusions of small thickness located in a two-dimensional homogeneous space. We construct a relationship between the MUSIC-type imaging functional for thin inclusions and the Bessel function of integer order of the first kind. Our construction is based on the structure of the left singular vectors of the collected multistatic response matrix whose elements are the measured far-field pattern and the asymptotic expansion formula in the presence of thin inclusions. Some numerical examples are shown to support the constructed MUSIC structure., Comment: 19 pages, 7 figures

We consider incompressible viscous flows between two transversely vibrating solid walls and construct an asymptotic expansion of solutions of the Navier--Stokes equations in the limit when both the amplitude of the vibration and the thickness of the oscillatory boundary layers (the Stokes layers) are small and have the same order of magnitude. Our asymptotic expansion is valid up to the flow boundary. In particular, we derive equations and boundary conditions for the averaged flow. At leading order, the averaged flow is described by the stationary Navier--Stokes equations with an additional term which contains the leading-order Stokes drift velocity. The same equations had been derived by Craik and Leibovich in 1976 when they proposed their model of Langmuir circulations in the ocean [A. D. D. Craik and S. Leibovich, J. Fluid Mech., 73 (1976), pp. 401--426]. In the context of flows induced by an oscillating conservative body force, these equations had also been derived by Riley [Ann. Rev. Fluid Mech., 33 ...

An asymptotic expansion in powers of small dimensionless dispersion coefficient is sought to solve an initial value problem posed for an advection diffusion equation modeling orientation of pulp fibers in a steady fully turbulent flow. The regular expansion is shown to be nonuniform in a small neighborhood of $\phi=0$. Although the highest order derivative with respect to the orientation angle $\phi$ is multiplied by the small parameter, application of matched asymptotic expansions to obtain the inner solution in a small neighborhood of $\phi=0$ matchable with the regular expansion turned out to be unsuccessful. The multiple scales do not lead to the solution either. The problem is solved by matching two asymptotic expansions, one solving the initial value problem in a small neighborhood of the initial point, while another one solves the equation at large distances from the initial point. Thus, this is an example of using the method of matched asymptotic expansions to satisfy the given initial condition b...

A uniform rational approximation of the Mittag-Leffler function is derived which serves as a global approximant, accounting for both the Taylor series for small arguments and asymptotic series for ...

The steady, gravity-driven flow of a liquid film over a topographically structured substrate is investigated. The analysis is based on a model nonlinear equation for the film thickness derived on the basis of long-wave asymptotics. The free-surface shape is expanded in a regular asymptotic expansion in powers of the topography amplitude, and solutions are obtained up to second order. Solutions are constructed for downward steps, upward steps, and rectangular trenches, and the results are compared favorably with numerical solutions of the nonlinear model equation. The results indicate that all of the salient features previously found for film flows over steps and into trenches are captured by the small-step asymptotics, including the capillary ridge formed just above a downward step and oscillations upstream of an upward step. We derive analytical expressions for the period and amplitude of these oscillations. The effect of a normal electric on the film surface shape is also investigated on the assumption ...

We consider the inverse problem to reconstruct the number and the positions of a collection of finitely many small perfectly conducting scatterers buried within the lower halfspace of an unbounded two-layered background medium from near field measurements of time harmonic electromagnetic fields. For this purpose we first study the direct scattering problem and derive an asymptotic expansion of the scattered field, as the size of the scatterers tends to zero. Integral equation methods and a factorization of the corresponding near field measurement operator are applied to prove this expansion. In the second part of this work we use the asymptotic expansion to justify a noniterative reconstruction algorithm, which is a combination of factorization methods and MUSIC-type methods. We illustrate the feasibility of this method by a numerical example.

In this paper, a stochastic model for the turbulent fiber lay-down in the industrial production of nonwoven materials is extended by including a moving conveyor belt. In the hydrodynamic limit corr...

In this paper we describe a semianalytical approach to computing the temperature and thermal stress inside a III-V compound grown with the Czochralski technique. An analysis of the growing conditions indicates that the crystal growth occurs on the conductive time scale. A perturbation method for the temperature field is developed for an arbitrary crystal profile using the Biot number as a (small) expansion parameter. The zeroth order solution is one-dimensional in the axial direction. Explicit solutions are obtained for a cylindrical and a conical crystal. Under typical growth conditions, a parabolic temperature profile in the radial direction is shown to arise naturally as the first order correction. As a result, the thermal stress is obtained explicitly and its magnitude is shown to depend on the zeroth order temperature and Biot number. Both the axial temperature gradient and crystal profile are shown to be important for controlling thermal stress and defect density. Some issues relevant to growth cond...

We consider wave propagation in a perturbed open waveguide. We provide a new asymptotic expansion for the scattered wave when the inclusion is of small diameter. We design a MUSIC (multiple signal classification) type of algorithm for locating the inclusion and illustrate its viability in numerical examples.

We consider the lubrication model for a thin film driven by competing gravitational forces and thermal gradients on an inclined plane. We are interested in the general traveling wave problem when the Navier slip boundary condition is used. We contrast (1) gravity dominated flow, (2) Marangoni dominated flow, and (3) flow in which the two driving effects balance. For a "singular slip" model we show that when Marangoni forces are present the resulting traveling wave ODE reduces locally near the contact line to a case not considered previously in the literature. We compute an asymptotic expansion of the solution near the contact line and compare with numerical simulations of the full problem. Using numerical simulations and phase space analysis involving Poincare sections, we show that for all three problems there is a finite range of admissible contact angles for which traveling wave solutions exist. Even in the well-studied case (1), this is a new observation that has ramifications for the use of constitut...

Formal closed models for vehicular traffic flow are obtained based on the novel equilibrium solution of the Prigogine--Herman equation. To that effect, Hilbert and Chapman--Enskog asymptotic series expansions are employed, obtaining the Euler and Navier--Stokes equivalent equations for traffic flow.

The Bremmer coupling series solution of the wave equation, in generally inhomogeneous media, requires the introduction of pseudodifferential operators. Such operators appear in the diagonalization process of the acoustic system's matrix of partial differential operators upon extracting a principal direction of (one-way) propagation. In this paper, in three dimensions, uniform asymptotic expansions of the Schwartz kernels of these operators are derived. Also, we derive a uniform asymptotic expansion of the one-way propagator appearing in the series. We focus on designing closed-form representations, valid in the high-frequency limit, taking into account critical scattering-angle phenomena. The latter phenomena are not dealt with in the standard calculus of pseudodifferential operators. Our expansion is not limited by propagation angle. In principle, the uniform asymptotic expansion of a kernel follows by matching its asymptotic behaviors away and near its diagonal.

We introduce the notion of (generalized) anisotropic polarization tensors and prove that they are symmetric and positive-definite. We also estimate the eigenvalues of the anisotropic polarization tensor in terms of the volume of the given domain. We apply these properties to an electrical impedance tomography problem to detect a single anisotropic inhomogeneity of small size. The goal is to detect an unknown inclusion when anisotropic conductivities of the background and the inclusion are known. Three results of computational experiments for the detection of an inclusion are given. One experiment is to assure the validity of the algorithm, while the other two are to see how anisotropy plays a role in the detection procedure.

This paper studies systems of partial differential equations modelling laser percussion drilling. The particular phenomenon considered in detail is the ejection of the thin layer of molten material. This thin layer is modelled as an inviscid flow between the fluid surface and fluid/solid interface, both of which are unknown moving boundaries. Through a regular asymptotic expansion, the governing equations are reduced to a combination of the shallow water equations in the zero gravity limit and a two-phase Stefan problem; the key small parameter is the square of the aspect ratio. These leading-order problems exhibit shocks which represent a possible mechanism for the previously unexplained fluid clumping. Approximate formulas and a parameter grouping are derived to predict the rate of melt solidification during ejection. Finally, weak formulations of the convection-diffusion equation for energy conservation are presented. These weak formulations are novel because the fluid is moving across a solid surface....

We study the short time behavior of the early exercise boundary for American style put options in the Black--Scholes theory. We develop an asymptotic expansion which shows that the simple lower bound of Barles et al. is a more accurate approximation to the actual boundary than the more complex upper bound. Our expansion is obtained through iteration using a boundary integral equation. This integral equation is derived from the time derivative of the option value function, which closely resembles the classical Stefan free boundary value problem for melting ice. Our analytical results are supported by numerical computations designed for very short times.

We consider an isotropic Lame system for an elastic medium consisting of finitely many imperfections of small diameter, embedded in a homogeneous reference medium. The Lame coefficients of the imperfections are different from those of the background medium. First, we establish an asymptotic formula for the displacement vector in terms of the reference Lame coefficients, the location of the imperfections, and their geometry. Second, we use this asymptotic expansion to establish continuous dependence estimates for certain characteristics of the imperfections in terms of the boundary data and to derive integral boundary formulae that yield to a very effective identification procedure.

Deflagrations in porous energetic materials are characterized by regions of two-phase flow, where, for sufficiently large flow velocities, temperature-nonequilibration effects can significantly affect the overall burning rate. In the present work, we analyze a two-temperature model of deflagrations in confined porous propellants that exhibit a bubbling melt layer at their surfaces. For appropriately scaled rates of interphase heat transfer, the problem reduces to a nontrivial eigenvalue calculation in the thin reaction region where final conversion of the liquid to gaseous products occurs. For realistically small values of the gas-to-liquid thermal-conductivity ratio, solutions in the reaction zone take on a singular-perturbation character that can be exploited to derive an asymptotic expansion of the burning-rate eigenvalue. The resulting problem requires a rather sophisticated application of techniques in matched asymptotic expansions (asymptotics beyond all orders) stemming from the appearance of an in...

The problem of fixed-point smoothing of a scalar diffusion process consists of estimating the initial value of the process, given its noisy measurements as a function of time. An asymptotic expansion of the joint filtering-smoothing conditional density function is constructed in the limit of small measurements noise. The approximate optimal nonlinear fixed-point smoother of Part I [SIAM J. Appl. Math., 54 (1994), pp. 833--853] is rederived from the expansion. A detailed analysis of the conditional mean square estimation error (CMSEE) of the optimal fixed-point smoother and of its leading-order approximation is presented. It is shown that if the initial error is small, e.g., if asymptotically optimal filtering is used first, the leading-order approximation to the optimal smoother is three dimensional and thus simpler than the four-dimensional extended Kalman smoother. Furthermore, nonlinear fixed-point smoothing can reduce the CMSEE relative to that of filtering by a factor of $\frac{1}{2}$ within smoothin...

The Bremmer coupling series solution of the wave equation, in generally inhomogeneous media, requires the introduction of pseudodifferential operators. In this paper, in two dimensions, uniform asymptotic expansions of the Schwartz kernels of these operators are derived. Also, we derive a uniform asymptotic expansion of the one-way propagator appearing in the series. We focus on designing closed-form representations, valid in the high-frequency limit, taking into account critical scattering-angle phenomena. Our expansion is not limited by propagation angle. In principle, the uniform asymptotic expansion of a kernel follows by matching its asymptotic behaviors away from and near its diagonal. The Bremmer series solver consists of three steps: directional decomposition into up- and downgoing waves, one-way propagation, and interaction of the counter-propagating constituents. Each of these steps is represented here by a kernel for which a uniform asymptotic expansion is found. The associated algorithm provid...

A formal asymptotic expansion is constructed for the joint probability density function (pdf) of a stationary ergodic Markov chain, Xn, and its averages, Yn. Since the pair (Xn,Yn)$is Markovian, the joint pdf satisfies a forward Kolmogorov equation whose solution is expanded asymptotically for large n. An algorithm is proposed for the calculation of the full asymptotic series, but only the three leading terms are found explicitly. It is found that for small values of the average, the asymptotic expansion coincides with the appropriate version of the central limit theorem. The ideas and methods are generalized to a large class of averages and to vector valued Markov chains.

The inviscid limit of vorticity in steady Navier--Stokes flows in two dimensions containing regions of closed streamlines (eddies) is studied. Prandtl [International Mathematical Congress, Heidelberg, Teubner, Leipzig, 1904, pp. 484--491; see also Gesammelte Abhandlungen zur angewandten Mechanik, Hydro- und Aerodynamik, vol. 2, Springer-Verlag, Berlin, 1961, pp. 575--584] and Batchelor [J. Fluid Mech., 1 (1956), pp. 177--190] showed that, in the limit of large Reynolds number R, the vorticity in such a region becomes constant. For the circular domain, Batchelor and Wood [J. Fluid Mech., 2 (1957), pp. 77--87] gave a formula which computes the constant vorticity. To study the effect of finite Reynolds number R on this formula, a matched asymptotic expansion of the velocity in two parameters, $ \epsilon $ (the perturbation parameter) and R, is calculated up to order of $O(\eps^2 /\sqrt{R})$ and compared with the numerical results. For the resulting vorticity, including the correction for finite R, $\omega_{f...

This paper deals with the low frequency scattering problem by a conducting object in a chiral medium. Using boundary integral equation methods, we obtain the limit of the scattered field solution of the Drude--Born--Fedorov equations as frequency tends to zero. Making use of a Hodge decomposition of the tangent fields shows that the asymptotics strongly depend on the topological properties of the domains under consideration. The asymptotic expansion's dependence on the chirality parameter is shown completely.

We consider a set of N independent sources, each of which alternates between "on" and "off" states. When a source is on, it generates a Poisson arrival stream to a finite-capacity queue with a general server. We derive the balance equations satisfied by the joint steady-state distribution of the queue length and the number of "on" sources. Then we analyze the problem in the heavy traffic limit where $N\rightarrow\infty$ and the average arrival rate is nearly equal to the mean service rate. The capacity is scaled to be $O(\sqrt{N})$. The first two terms in the asymptotic series are characterized as solutions to elliptic partial differential equations (PDEs) with appropriate boundary conditions. We then develop numerical and asymptotic methods for solving these PDEs. The analysis makes use of singular perturbation techniques and special functions.

Asymptotic expansion series for the moment Lyapunov exponent and stability index are constructed and justified for the two-dimensional linear stochastic system close to a harmonic oscillator. As an example, a one-degree-of-freedom mechanical system parametrically excited in stiffness and damping is considered and several terms of the expansion are obtained.

We study dynamical hysteresis in a simple class of nonlinear ordinary differential equations, namely first-order equations subject to sinusoidal forcing. The assumed nonlinearities are such that the area of the hysteresis loop vanishes as the forcing frequency tends to zero; in other words, there is no static hysteresis. Using regular and singular perturbation techniques, we derive the first term in the asymptotic expansion for the loop area as a function of the driving frequency, in the limit of both large and small frequency. Although the theory was originally motie have expanded ents on bistable semiconductor lasers, it is applied here to explain (and in some cases, to correct) the scaling laws that were recently reported in numerical studies of mean-field kinetic Ising models of ferromagnets.

A model problem in the scattering of a time-harmonic wave by an obstacle coated with a thin penetrable shell is examined. In previous studies, the contrast coefficients of the thin shell are assumed to tend to infinity in order to compensate for the thickness considered. In this paper, these coefficients are assumed to remain finite. Such a treatment leads to a singular perturbation term that creates a typical difficulty for the asymptotic analysis of the problem with respect to the thickness of the coating. As a result, the asymptotic analysis is essentially based on a suitable handling of the stability of the solution relative to the thickness. As a consequence, it is shown how effective boundary conditions which can be substituted to the thin shell can then be obtained and analyzed in a simple way.

We derive asymptotic properties of transition densities for singularly perturbed diffusion processes with fast and slow components. Our study focuses on the Kolmogorov–Fokker–Planck equations. The model can be viewed as a diffusion process having two time scales and is motivated by a wide variety of applications involving singularly perturbed Markov processes in manufacturing systems, homogenization, reliability analysis, queueing networks, statistical physics, population biology, financial economics, and many other related fields. By virtue of the methods of matched singular perturbation, asymptotic expansion is constructed for the transition density. The expansion includes both regular part and boundary layer corrections. Detailed justification of the asymptotic expansion is given, and error bounds are also provided.

We derive limit theorems for the transition densities of diffusion processes and develop asymptotic expansions for solutions of a class of singularly perturbed Kolmogorov–Fokker–Planck equations. The model under consideration can be viewed as a Markov process having two time scales. One of them is a rapidly changing scale, and the other is a slowly varying one. The study is motivated by a wide range of applications involving singularly perturbed Markov processes in manufacturing systems, reliability analysis, queueing networks, statistical physics, population biology, financial economics, and many other related fields. In this work, the asymptotic expansion is constructed explicitly. It is shown that the initial layer terms in the expansion decay at an exponential rate. Error bounds on the remainder terms also are obtained. The validity of the expansion is rigorously justified.

We derive an evolution equation for three-dimensional thermal convection in an imposed shear flow between horizontal boundaries that conduct heat poorly. Through a long-wavelength asymptotic expansion the governing equations are reduced to a single partial differential equation for the depth-averaged temperature field. A linear stability analysis of this equation indicates that the first instability of the quiescent state is to convective rolls whose axes are aligned with the direction of the shear flow. This result complements previous numerical solutions of the linear stability problem for a fluid between perfectly conducting horizontal boundaries, where a similar preference for such rolls has been found.

The time-dependent Ginzburg–Landau model of superconductivity is examined in the high-$\kappa $, high magnetic field setting. This work generalizes the previous result for the steady-state model with a constant applied magnetic field. The significance of this generalization lies in the ability to incorporate the effects of both the applied magnetic field and applied current or voltage. Thus, it is possible to use the simplified setting obtained in this paper to study the motion and “pinning” of vortices in the presence of an applied current and a variable applied field. The results within are obtained via a formal asymptotic expansion of the Ginzburg–Landau equations in terms of $\kappa$, which yields a simplified system of leading-order equations. The formal asymptotic expansion is then justified by showing the solution to the full time-dependent Ginzburg-Landau equations converges to the solution of the leading-order equations as $\kappa \to \infty $. Computational results are also given that show the s...

The one-dimensional Fokker–Planck equation for a bistable potential is considered in the “underdamped” limit. Using singular perturbation analysis, an asymptotic series solution is constructed in the parameter associated with the small viscosity. This series solution is not uniformly valid: rather, it consists of four different types of asymptotic series valid in different time intervals and/or portions of phase space. By matching these series representations across overlapping regions, a reduced problem for the long term evolution of the distribution is extracted.

A class of singularly perturbed time-varying systems with a small parameter $\varepsilon > 0$ is considered in this paper. The importance of the study stems from the fact that many problems arise in various applications involve a rapidly fluctuating Markov chain. To investigate the limit behavior of such systems, it is necessary to consider the corresponding singular-perturbation problems. Existing results in singular perturbation of ordinary differential equations cannot be applied since the coefficient matrix of the equation is a generator of a finite-state Markov chain, and as a result it is singular. Asymptotic properties of the aforementioned systems are developed via matched asymptotic expansion in this paper. Thanks to the specific structure and the properties of Markovian generators, it is established that the solution of the system can be approximated “as close as possible” by a series expansion in terms of the small parameter $\varepsilon > 0$.

The method of induced forces is employed to evaluate the Oseen drag on a sphere. The accuracy of the values obtained for the drag coefficient for large Reynolds numbers is increased by application of Shanks’s transformation. On the basis of the improved values we obtain as numerical approximations for the exponent of the first term in the asymptotic expansion values approaching $- 0.660 $ for Reynolds numbers between $10^2 $ and $10^6 $.

The linear stability theory of rapidly rotating, self-gravitating polytropes is developed by an asymptotic, shallow-layer method. This reduces the general three-dimensional stability problem to an integro-differential eigenvalue problem (a Fredholm integral equation of the second kind) for normal modes. At leading order, the asymptotic analysis produces familiar, zero-thickness disk equations. In subsequent orders, stabilizing effects due to compressibility enter. We solve the stability equations numerically and construct approximate solutions using short-wavelength arguments. The eigenspectrum of a disk can have various forms; instabilities of a pressure-less configuration form a continuous piece of the spectrum, but polytropic disks can have discrete, unstable eigenvalues. A further example is provided by the rigidly rotating disk, which, at leading order, can be solved exactly.

The use of limit process expansions and asymptotic matching is illustrated for a simple problem of homogenization. Rayleigh’s problem of calculating the effective thermal conductivity $k_{eff} $ of a regular cubic array of spheres of conductivity $k_I $ in a matrix of conductivity $k_M $ is considered for small volume fractions $f_I $. An inner expansion is found near each sphere. This is matched to an outer expansion in which the sphere shrinks to a singularity. Rayleigh’s result that the expansion for $k_{eff} $ contains terms $f_I ,f_I^2 ,f_I^3 ,f_I^4 ,f^{13/3} $ is reconstructed. In addition an independent verification of Rayleigh’s summability method for an infinite array of dipoles is given.Comments are made on related problems and the random case where similar summability difficulties appear.