Showing papers on "Method of matched asymptotic expansions published in 2008"

An introduction to singular perturbations

Abstract: . In this article we discuss the asymptotic analysis of so-called singularly perturbed differential equations. We begin by discussing some very simple nonmathematical examples, and then proceed to two mathematical examples, namely, the well-known example of Friedrichs and a problem of John Mahony's. The presentation is addressed primarily to those who may have had little or no previous experience with this particular type of problem. However, the author hopes others will find something of interest in the somewhat unusual approach we take.

Difference Methods for Singular Perturbation Problems

Abstract: Preface Part I: Grid Approximations of Singular Perturbation Partial Differential Equations Introduction Boundary Value Problems for Elliptic Reaction-Diffusion Equations in Domains with Smooth Boundaries Boundary Value Problems for Elliptic Reaction-Diffusion Equations in Domains with Piecewise-Smooth Boundaries Generalizations for Elliptic Reaction-Diffusion Equations Parabolic Reaction-Diffusion Equations Elliptic Convection-Diffusion Equations Parabolic Convection-Diffusion Equations Part II: Advanced Trends in epsilon Uniformly Convergent Difference Methods Grid Approximations of Parabolic Reaction-Diffusion Equations with Three Perturbation Parameters Application of Widths for Construction of Difference Schemes for Problems with Moving Boundary Layers High-Order Accurate Numerical Methods for Singularly Perturbed Problems A Finite Difference Scheme on a priori Adapted Grids for a Singularly Perturbed Parabolic Convection-Diffusion Equation On Conditioning of Difference Schemes and Their Matrices for Singularly Perturbed Problems Approximation of Systems of Singularly Perturbed Elliptic Reaction-Diffusion Equations with Two Parameters Survey References

Classical two-phase Stefan problem for spheres

Abstract: The classical Stefan problem for freezing (or melting) a sphere is usually treated by assuming that the sphere is initially at the fusion temperature, so that heat flows in one phase only. Even in this idealized case there is no (known) exact solution, and the only way to obtain meaningful results is through numerical or approximate means. In this study, the full two-phase problem is considered, and in particular, attention is given to the large Stefan number limit. By applying the method of matched asymptotic expansions, the temperature in both the phases is shown to depend algebraically on the inverse Stefan number on the first time scale, but at later times the two phases essentially decouple, with the inner core contributing only exponentially small terms to the location of the solid–melt interface. This analysis is complemented by applying a small-time perturbation scheme and by presenting numerical results calculated using an enthalpy method. The limits of zero Stefan number and slow diffusion in the inner core are also noted.

Asymptotic Expansions of I-V Relations via a Poisson–Nernst–Planck System

Abstract: We investigate higher order matched asymptotic expansions of a steady-state Poisson-Nernst-Planck (PNP) system with particular attention to the I-V relations of ion channels. Assuming that the Debye length is small relative to the diameter of the narrow channel, the PNP system can be viewed as a singularly perturbed system. Special structures of the zeroth order inner and outer systems make it possible to provide an explicit derivation of higher order terms in the asymptotic expansions. For the case of zero permanent charge, our results concerning the I-V relation for two oppositely charged ion species are (i) the first order correction to the zeroth order linear I-V relation is generally quadratic in V; (ii) when the electro-neutrality condition is enforced at both ends of the channel, there is NO first order correction, but the second order correction is cubic in V. Furthermore (Theorem 3.4), up to the second order, the cubic I-V relation has (except for a very degenerate case) three distinct real roots that correspond to the bistable structure in the FitzHugh-Nagumo simplification of the Hodgkin-Huxley model.

On the Global Asymptotic Stability of a Second-Order System of Difference Equations

Abstract: A sufficient condition is obtained for the global asymptotic stability of the following system of difference equations 𝑧𝑛

Asymptotic analysis of autoresonance models

Abstract: In recent decades new problems have arisen in oscillation theory which are related to the investigation of a?physical phenomenon known as autoresonance. This paper presents a?survey of results on the asymptotic analysis of such problems. Systems of differential equations corresponding to non-linear non-autonomous oscillators with variable excitation frequency are considered. For their solution asymptotic formulae are constructed with respect to a?small parameter or with respect to an independent variable.

Hydrodynamic loads during early stage of flat plate impact onto water surface

Abstract: The hydrodynamic loads during the water entry of a flat plate are investigated. Initially the water is at rest and the plate is floating on the water surface. Then the plate starts suddenly its vertical motion. The analysis is focused on the early stage during which the highest hydrodynamic loads are generated. The liquid is assumed ideal and incompressible; gravity and surface tension effects are not taken into account. The flow generated by the impact is two dimensional and potential. The penetration depth is either a given function of time or calculated by using the equation of the body motion. A theoretical estimate of the loads during the early stage of the water impact is derived with the help of the method of matched asymptotic expansions. The ratio of the plate displacement to the plate half-width plays the role of a small parameter. The second-order uniformly valid solution of the problem is derived. In order to evaluate the hydrodynamic loads, the second-order pressure distribution is asymptotically integrated along the plate. It is shown that the initial asymptotics of the loads involve a logarithmic term and a negative noninteger power of the nondimensional plate displacement, the latter contribution is related to the inner solution. In addition to the theoretical estimate, a numerical model of the unsteady free-surface flow generated by plate impact is developed. The hydrodynamic loads are numerically evaluated and compared to their asymptotic estimates. A fairly good agreement between the theoretical and numerical predictions of the hydrodynamic loads just after the impact has been found. In the case of constant velocity of the body, it is shown that the relative difference between the theoretical and numerical predictions of the hydrodynamic force is less than 5% when the nondimensional plate displacement is one-fifth and rises to 20% when the nondimensional plate displacement is equal to unity. Similar results are found in the free fall case when the comparison is established in terms of hydrodynamic loads. The theoretical and numerical predictions are remarkably close to each other, even for moderate displacements of the plate, if the comparison is established in terms of the entry velocity.

Asymptotic Solutions of Hamilton-Jacobi Equations with State Constraints

Abstract: We study Hamilton-Jacobi equations in a bounded domain with the state constraint boundary condition. We establish a general convergence result for viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations with the state constraint boundary condition to asymptotic solutions as time goes to infinity.

Homogenization of a boundary‐value problem with a nonlinear boundary condition in a thick junction of type 3:2:1

Abstract: We consider a boundary-value problem for the Poisson equation in a thick junction Ωe, which is the union of a domain Ω0 and a large number of e-periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition ∂νue + eκ(ue)=0 is given on the lateral surfaces of the thin cylinders. The asymptotic analysis of this problem is performed as e 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove the convergence theorem and show that the nonlinear Robin boundary condition is transformed (as e 0) in the blow-up term of the corresponding ordinary differential equation in the region that is filled up by the thin cylinders in the limit passage. The convergence of the energy integral is proved as well. Using the method of matched asymptotic expansions, the approximation for the solution is constructed and the corresponding asymptotic error estimate in the Sobolev space H1(Ωe) is proved. Copyright © 2007 John Wiley & Sons, Ltd.

High-precision computations of divergent asymptotic series and homoclinic phenomena

Abstract: We study asymptotic expansions for the exponentially small splitting of separatrices of area preserving maps combining analytical and numerical points of view. Using analytic information, we conjecture the basis of functions of an asymptotic expansion and then extract actual values of the coefficients of the asymptotic series numerically. The computations are performed with high-precision arithmetic, which involves up to several thousands of decimal digits. This approach allows us to obtain information which is usually considered to be out of reach of numerical methods. In particular, we use our results to test that the asymptotic series are Gevrey-1 and to study positions and types of singularities of their Borel transform. Our examples are based on generalisations of the standard and Henon maps.

Pressure-gradient-dependent logarithmic laws in sink flow turbulent boundary layers

Abstract: Experiments were done on sink flow turbulent boundary layers over a wide range of streamwise pressure gradients in order to investigate the effects on the mean velocity profiles. Measurements revealed the existence of non-universal logarithmic laws, in both inner and defect coordinates, even when the mean velocity descriptions departed strongly from the universal logarithmic law (with universal values of the Karman constant and the inner law intercept). Systematic dependences of slope and intercepts for inner and outer logarithmic laws on the strength of the pressure gradient were observed. A theory based on the method of matched asymptotic expansions was developed in order to explain the experimentally observed variations of log-law constants with the non-dimensional pressure gradient parameter (Δp=(ν/ρU3τ)dp/dx). Towards this end, the system of partial differential equations governing the mean flow was reduced to inner and outer ordinary differential equations in self-preserving form, valid for sink flow conditions. Asymptotic matching of the inner and outer mean velocity expansions, extended to higher orders, clearly revealed the dependence of slope and intercepts on pressure gradient in the logarithmic laws.

Matching of asymptotic expansions for waves propagation in media with thin slots II: The error estimates

Abstract: We are concerned with a 2D time harmonic wave propagation problem in a medium including a thin slot whose thickness $\epsilon$ is small with respect to the wavelength. In Part I [P. Joly and S. Tordeux, Multiscale Model. Simul. 5 (2006), no. 1, 304--336 (electronic); MR2221320 (2007e:35041)], we derived formally an asymptotic expansion of the solution with respect to $\epsilon$ using the method of matched asymptotic expansions. We also proved the existence and uniqueness of the terms of the asymptotics. In this paper, we complete the mathematical justification of our work by deriving optimal error estimates between the exact solutions and truncated expansions at any order.

Asymptotic representation of the solutions of linear volterra difference equations

Abstract: This article analyses the asymptotic behaviour of solutions of linear Volterra difference equations. Some sufficient conditions are presented under which the solutions to a general linear equation converge to limits, which are given by a limit formula. This result is then used to obtain the exact asymptotic representation of the solutions of a class of convolution scalar difference equations, which have real characteristic roots. We give examples showing the accuracy of our results.

Stationary Solutions of Driven Fourth- and Sixth-Order Cahn–Hilliard-Type Equations

Abstract: New types of stationary solutions of a one-dimensional driven sixth-order Cahn–Hilliard-type equation that arises as a model for epitaxially growing nanostructures, such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially nonmonotone solutions in the limit of small driving force strength, which is the deposition rate in case of epitaxial growth. These solutions extend the family of the monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert W function. Using phase-space analysis for the corresponding fifth-order dynamical system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input. Additionally, our approach is first demonstrated for the related but simpler driven f...

Approximate similarity reduction for singularly perturbed Boussinesq equation via symmetry perturbation and direct method

Abstract: We investigate the singularly perturbed Boussinesq equation in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions and similarity reduction equations of different orders display formal coincidence for both methods. Series reduction solutions are consequently derived. For the approximate symmetry perturbation method, similarity reduction equations of the zero order are equivalent to the Painleve IV, Painleve I, and Weierstrass elliptic equations. For the approximate direct method, similarity reduction equations of the zero order are equivalent to the Painleve IV, Painleve II, Painleve I, or Weierstrass elliptic equations. The approximate direct method yields more general approximate similarity reductions than the approximate symmetry perturbation method.

Asymptotic Solution of Activator Inhibitor Systems for Nonlinear Reaction Diffusion Equations

Abstract: A nonlinear reaction diffusion equations for activator inhibitor systems is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the variables of multiple scales and the expanding theory of power series the formal asymptotic expansions of the solution are constructed, and finally, using the theory of differential inequalities the uniform validity and asymptotic behavior of the solution are studied.

Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters

Abstract: This work is concerned with the two-fluid Euler-Maxwell equations for plasmas with small parameters. We study, by means of asymptotic expansions, the zero-relaxation limit, the non-relativistic limit and the combined non-relativistic and quasi-neutral limit. For each limit with well-prepared initial data, we show the existence and uniqueness of an asymptotic expansion up to any order. For general data, an asymptotic expansion up to order 1 of the non-relativistic limit is constructed by taking into account the initial layers. Finally, we discuss the justification of the limits.

Asymptotic behaviour of two-dimensional time-delayed navier-stokes equations

Abstract: We consider the two-dimensional Navier-Stokes equations with a time-delayed convective term and a forcing term which contains some hereditary features. Some results on existence and uniqueness of solutions are established. We discuss the asymptotic behaviour of solutions and we also show the exponential stability of stationary solutions.

A new algorithm appropriate for solving singular and singularly perturbed autonomous initial-value problems

Abstract: A nonlinear explicit scheme is proposed for numerically solving first-order singular or singularly perturbed autonomous initial-value problems (IVP) of the form y'=f(y). The algorithm is based on the local approximation of the function f(y) by a second-order Taylor expansion. The resulting approximated differential equation is then solved without local truncation error. For the true solution the method has a local truncation error that behaves like either O(h3) or O(h4) according to whether or not some parameter vanishes. Some numerical examples are provided to illustrate the performance of the method. Finally, an application of the method for detecting and locating singularities is outlined.

Exponential asymptotics for nonlinearly perturbed renewal equation with non-polynomial perturbations

Abstract: In this thesis we investigate a model of nonlinearly perturbed continuous-time renewal equation. Some characteristics of the renewal equation are assumed to have non-polynomial perturbations, more specifically they can be expanded with respect to a non-polynomial asymptotic scale. The main result of the present study is exponential asymptotic expansions for the solution of the perturbed renewal equation. These asymptotic results are also applied to various applied probability models like perturbed risk processes, perturbed M/G/1 queues and perturbed dam/storage processes. The thesis is based on five papers where the model described above is successively studied.

Analytical study of turbulent Poiseuille flow with wall transpiration

Abstract: An incompressible, pressure-driven, fully developed turbulent flow between two parallel walls, with an extra constant transverse velocity component, is considered. A closure condition is formulated, which relates the shear stress to the first and the second derivatives of the longitudinal mean velocity. The closure condition is derived without invoking any special hypotheses on the nature of turbulent motion, only taking advantage of the fact that the flow depends on a finite number of governing parameters. By virtue of the closure condition, the momentum equation is reduced to the boundary-value problem for a second-order differential equation, which is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction velocity. There are three characteristic flow regions in the channel: the core region and two wall regions near injection and suction walls. For each region, the solution is constructed. The asymptotic matching gives formulas for the wall shear stress and the maximum mean velocity. A limit transpiration velocity is obtained, such that the shear stress at the injection wall vanishes, while the maximum point on the velocity profile approaches the suction wall. In this case, a sublayer near the suction wall appears where the mean velocity is proportional to the square root of the distance from the wall. A friction law for Poiseuille flow with transpiration is found, which makes it possible to describe the relation between the wall shear stress, the Reynolds number, and the transpiration velocity by a function of one variable. A velocity defect law, which generalizes the classical law for the core region in a channel with impermeable walls to the case of transpiration, is also established. In similarity variables, the mean velocity profiles across the whole channel width outside viscous sublayers can be described by a one-parameter family of curves. The theoretical results obtained are in good agreement with available direct numerical simulation data.

Singularly Perturbed Solution of Coupled Model in Atmosphere-ocean for Global Climate

Abstract: A box model of the interhemispheric thermohaline circulation (THC) in atmosphere-ocean for global elimate is considered. By using the multi-scales method, the asymptotic solution of a simplified weakly nonlinear model is discussed. Firstly, by introducing first scale, the zeroth order approximate solution of the model is obtained. Secondly, by using the multi-scales, the first order approximate equation of the model is found. Finally, second order approximate equation is formed to eliminate the secular terms, and a uniformly valid asymptotic expansion of solution is decided. The multi-scales solving method is an analytic method which can be used to analyze operation sequentially. And then we can also study the diversified qualitative and quantitative behaviors for corresponding physical quantities. This paper aims at providing a valid method for solving a box model of the nonlinear equation.

Asymptotic Expansions of Mellin Convolution Integrals

Abstract: We present a new method for deriving asymptotic expansions of $\int_0^\infty f(t)h(xt)dt$ for small $x$. We only require that $f(t)$ and $h(t)$ have asymptotic expansions at $t=\infty$ and $t=0$, respectively. Remarkably, it is a very general technique that unifies a certain set of asymptotic methods. Watson's lemma and other classical methods, Mellin transform techniques, McClure and Wong's distributional approach, and the method of analytic continuation turn out to be simple corollaries of this method. In addition, the most amazing thing about it is that its mathematics are absolutely elemental and do not involve complicated analytical tools as the aforementioned methods do: it consists of simple “sums and subtractions." Many known and new asymptotic expansions of important integral transforms are trivially derived from the approach presented here.