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Showing papers on "Method of matched asymptotic expansions published in 1998"


Journal ArticleDOI
TL;DR: In this article, a modeling framework is proposed for two-time-scale chemical processes modeled by nonlinear ordinary differential equations (ODEs) with large parameters of the form 1 e, to obtain a standard singularly perturbed representation where the slow and fast variables are explicitly separated.

87 citations


Journal ArticleDOI
TL;DR: In this paper, the authors analyzed singularly perturbed telegraph systems applying the newly developed compressed asymptotic method and showed that the diffusion equation of the singularized telegraph system is a limit of the system of equations.
Abstract: In the paper we analyze singularly perturbed telegraph systems applying the newly developed compressed asymptotic method and show that the diffusion equation is an asymptotic limit of singularly perturbed telegraph system of equations. The results are applied to the random walk theory for which the relationship between correlated and uncorrelated random walks is explained in asymptotic terms.

83 citations


Journal ArticleDOI
TL;DR: In this paper, the bilinear finite element method on a Shishkin mesh for singularly perturbed elliptic boundary value problem was considered and it was shown that the method achieves almost second-order uniform convergence rate in L2-norm.
Abstract: We consider the bilinear finite element method on a Shishkin mesh for the singularly perturbed elliptic boundary value problem −ϵ 2 ( ι 2 u ιx 2 + ι 2 u ιy 2 ) + a(x,y)u = f(x,y) in two space dimensions. By using a very sophisticated asymptotic expansion of Han et al. [1] and the technique we used in [2], we prove that our method achieves almost second-order uniform convergence rate in L2-norm. Numerical results confirm our theoretical analysis.

66 citations


Journal ArticleDOI
TL;DR: In this article, a generalisation of the Mullins-Sekerka problem to model phase separation in multi-component systems is proposed, which includes equilibrium equations in bulk, the Gibbs-Thomson relation on the interfaces, Young's law at triple junctions, together with a dynamic law of Stefan type.
Abstract: We propose a generalisation of the Mullins–Sekerka problem to model phase separation in multi-component systems. The model includes equilibrium equations in bulk, the Gibbs–Thomson relation on the interfaces, Young's law at triple junctions, together with a dynamic law of Stefan type. Using formal asymptotic expansions, we establish the relationship to a transition layer model known as the Cahn-Hilliard system. We introduce a notion of weak solutions for this sharp interface model based on integration by parts on manifolds, together with measure theoretical tools. Through an implicit time discretisation, we construct approximate solutions by stepwise minimisation. Under the assumption that there is no loss of area as the time step tends to zero, we show the existence of a weak solution.

52 citations


Journal ArticleDOI
TL;DR: In this paper, an analytical and numerical study was carried out for the steady incompressible vapor and liquid flow in an asymmetrical flat plate heat pipe, where the boundary layer approximation was employed to describe the vapor flow under conditions including strong flow reversal and the method of matched asymptotic expansions to incorporate the non-Darcian effects for the liquid flow through porous wicks.

51 citations


Journal Article
TL;DR: In this paper, the solitary-wave solutions of Benjamin's model were investigated for a class of equations that include Benjamin's equation, which feature conflicting contributions to dispersion from dynamic effects on the interface and surface tension.
Abstract: Benjamin recently put forward a model equation for the evolution of waves on the interface of a two-layer system of fluids in which surface tension effects are not negligible. In this case, the fluid motion η on the interface of these two fluids can be approximately described by an equation ηt + ηx + ηηx − αLηx ± βηxxx = 0, where η depends on saptial variable x and time variable t, and L = H∂x is the composition of the Hilbert transform and the spatial derivative in the direction of primary propagation, or, equivalently, L is a Fourier multiplier operator with symbol |ξ|. It is our purpose here to investigate the solitary-wave solutions of Benjamin’s model. For a class of equations that include Benjamin’s equation, which feature conflicting contributions to dispersion from dynamic effects on the interface and surface tension, we establish existence of travelling-wave solutions. This is complished by using P.L. Lions concentrated-compactness principle. Using the recently developed theory of Li and Bona, we are also able to determine rigorously the spatial asymptotics of these solutions. Department of Mathematics, The University of Texas at Austin, Austin, TX 78712. Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin, Austin, TX 78712.

50 citations


Journal ArticleDOI
TL;DR: In this paper, the authors describe the solution of various Becker-Doring models, together with the large-time behaviour of solutions for the case of constant monomer concentration, and solve a weak fragmentation, constant mass, case by using matched asymptotic expansions.
Abstract: We describe the solution of various Becker-Doring models. First, we analyse equilibrium and steady-state solutions, together with the large-time behaviour of solutions for the case of constant monomer concentration. Then we solve a weak fragmentation, constant mass, case by using matched asymptotic expansions. The methods are applied both to the full Becker-Doring equations and to a coarse-grained, or contracted, system. Comparison of the results show good qualitative agreement; all the phenomena present in the full model are reproduced in the contracted system and no anomalous effects are introduced. However, the quantitative agreement depends strongly on the choice of parameter values in the contracted system.

45 citations


Journal ArticleDOI
TL;DR: A computational method is suggested in which exponentially fitted difference schemes are combined with classical numerical methods to obtain numerical solution of singularly perturbed turning point problems for second order ordinary differential equations exhibiting twin boundary layers.

43 citations


Journal ArticleDOI
TL;DR: In this article, the asymptotic stability of theoretical and numerical solutions for neutral multidelay-differential equations (NMDEs) is dealt with, and sufficient conditions on the stability of the theoretical solutions for NMDEs are obtained.
Abstract: The asymptotic stability of theoretical and numerical solutions for neutral multidelay-differential equations (NMDEs) is dealt with. A sufficient condition on the asymptotic stability of theoretical solutions for NMDEs is obtained. On the basis of this condition, it is proved that A-stability of the multistep Runge-Kutta methods for ODEs is equivalent to NGPk-stability of the induced methods for NMDEs.

39 citations


Journal ArticleDOI
TL;DR: In this paper, the authors proposed a method to find a composite scale that matches the dissimilar scales, in lieu of the asymptotic solutions, in their applicable domains, thus reducing the number of independent scales to two.
Abstract: For a class of boundary-value problems involving damped oscillations that occur at three or more dissimilar scales, both matched asymptotic and multiple scale expansions can fail to provide uniformly valid solutions. A novel approach is introduced in this paper that suggests determining a composite scale that matches the dissimilar scales, in lieu of the asymptotic solutions, in their applicable domains. Information contained in the dissimilar scales is condensed into one composite scale, thus reducing the number of independent scales to two. For that purpose, a procedure is presented herein that consists of: 1) identifying the form and location of prevalent characteristic scales, 2) determining a composite scale that matches the stretched or contracted scales in their respective intervals, and 3) invoking a two-variable multiple scale method that employs the composite scale as one of its independent variables. This procedure is applied successfully to a problem that eludes conventional perturbation methods. The corresponding boundary-layer equation pertains to the separable transversely-dependent component of the rotational momentum equation used in modeling oscillatory flows in low aspect ratio rectangular channels where blowing is present at the walls. An expansion series is constructed in the parameter associated with small viscosity. A uniformly valid expression is extracted that captures the physical effects of unsteady inertia, viscous diffusion, and transverse convection of unsteady vorticity, while clearly showing that spatial attenuation of rotational waves is controlled by a single similarity parameter. Analytical results are numerically verified for a wide range of physical parameters and test cases.

33 citations


Journal ArticleDOI
TL;DR: In this paper, the existence of asymptotic expansions of the error of Galerkin methods with splines of arbitrary degree for the approximate solution of integral equations with logarithmic kernels is analyzed.
Abstract: In this paper we analyse the existence of asymptotic expansions of the error of Galerkin methods with splines of arbitrary degree for the approximate solution of integral equations with logarithmic kernels. These expansions are obtained in terms of an interpolation operator and are useful for the application of Richardson extrapolation and for obtaining sharper error bounds. We also present and analyse a family of fully discrete spline Galerkin methods for the solution of the same equations. Following the analysis of Galerkin methods, we show the existence of asymptotic expansions of the error.

Journal ArticleDOI
TL;DR: In this article, the boundary value problem for the set of functional-differential equations with partial derivatives of Riccati type, associated with a singularly perturbed linear-quadratic optimal control problem with delay in state, is considered.
Abstract: The boundary-value problem for the set of functional-differential equations with partial derivatives of Riccati type, associated with a singularly perturbed linear-quadratic optimal control problem with delay in state, is considered. The expression for a solution of the problem, which transforms it to the explicit singular perturbation form, is proposed. An asymptotic solution of this problem is constructed.


Journal ArticleDOI
31 Aug 1998-Chaos
TL;DR: The construction of the exponential asymptotic expansions of the unstable and stable manifolds of the area-preserving Henon map enables one to capture the exponentially small effects that result from the Stokes phenomenon in the analytic theory of equations with irregular singular points.
Abstract: The subject of this paper is the construction of the exponential asymptotic expansions of the unstable and stable manifolds of the area-preserving Henon map. The approach that is taken enables one to capture the exponentially small effects that result from what is known as the Stokes phenomenon in the analytic theory of equations with irregular singular points. The exponential asymptotic expansions were then used to obtain explicit functional approximations for the stable and unstable manifolds. These approximations are compared with numerical simulations and the agreement is excellent. Several of the main results of the paper have been previously announced in A. Tovbis, M. Tsuchiya, and C. Jaffe [“Chaos-integrability transition in nonlinear dynamical systems: exponential asymptotic approach,” Differential Equations and Applications to Biology and to Industry, edited by M. Martelli, K. Cooke, E. Cumberbatch, B. Tang, and H. Thieme (World Scientific, Singapore, 1996), pp. 495–507, and A. Tovbis, M. Tsuchiya, and C. Jaffe, “Exponential asymptotic expansions and approximations of the unstable and stable manifolds of the Henon map,” preprint, 1994].

Journal ArticleDOI
TL;DR: In this article, an initial-value technique is presented for solving singularly perturbed two-point boundary-value problems for linear and semilinear second-order ordinary differential equations arising in chemical reactor theory.
Abstract: An initial-value technique is presented for solving singularly perturbed two-point boundary-value problems for linear and semilinear second-order ordinary differential equations arising in chemical reactor theory. In this technique, the required approximate solution is obtained by combining solutions of two terminal-value problems and one initial-value problem which are obtained from the original boundary-value problem through asymptotic expansion procedures. Error estimates for approximate solutions are obtained. Numerical examples are presented to illustrate the present technique.

Journal ArticleDOI
TL;DR: In this article, the qualitative properties of solutions of the perfect-fluid Einstein field equations in the case of spherical symmetry were investigated, and exact solutions were obtained and the asymptotic behaviour of the solutions were fully studied in these important subcases.
Abstract: The perfect-fluid Einstein field equations in the case of spherical symmetry reduce to an autonomous system of ordinary differential equations when a spacetime is assumed to admit a kinematic self-similarity (of either the second or zeroth kind). The qualitative properties of solutions of this system of equations, and in particular their asymptotic behaviour, are investigated. The geodesic subcase and a subcase containing the static models are examined in detail. In particular, exact solutions are obtained and the asymptotic behaviour of the solutions is fully studied in these important subcases. Exact solutions admitting a homothetic vector are found to play an important role in describing the asymptotic behaviour of the kinematic self-similar models.

Journal ArticleDOI
TL;DR: In this article, the authors explore the dynamics of a narrow electron stream embedded in a magnetized plasma and use matched asymptotic expansions to describe the linear stage of the instability for a nonuniform drift profile.
Abstract: In this analytical and computer simulation study we explore the dynamics of a narrow electron stream embedded in a magnetized plasma. The transverse dimension of the stream is envisioned to be on the order of the electron skin depth, as is appropriate to several problems of current interest (e.g., auroral beams, reconnection). Within the layer the drift velocity exceeds the thermal velocity, and thus the Buneman instability is excited. The method of matched asymptotic expansions is used to describe the linear stage of the instability for a nonuniform drift profile. It predicts a lowering of the growth rate and a rapid decrease in wave amplitude at the spatial location where the beam mode resonance is encountered. A particle-in-cell (PIC) simulation is used to verify the predictions of the analysis and to illustrate the important nonlinear behavior. It is found that the rapid flash of the Buneman instability excites a lower-hybrid wave, causes strong perpendicular ion acceleration, and results in a region ...

Book
01 Jan 1998
TL;DR: In this paper, asymptotic solutions on Manifolds with Cusp-Type Singularities and Asymptotics with Corner-type Singularity are presented.
Abstract: GENERALITIES Structure Rings on Singular Manifolds Interaction of Asymptotic Expansions Resurgent Analysis of Functions of Polynomial Growth ELLIPTIC EQUATIONS Asymptotic Solutions on Manifolds with Conical Singularities Asymptotic Solutions on Manifolds with Cusp-Type Singularities Asymptotic Solutions on Manifolds with Corner-Type Singularities General Asymptotic Theory Finiteness Theorems HYPERBOLIC EQUATIONS Equations of Borel-Fuchs Type Vibration of Elastic Shells with Conical Points Appendices Bibliography Index.

Journal ArticleDOI
TL;DR: It is shown that it is the exception rather than the rule for the integration process to be stable, particularly so for implicit solutions, and boundary-value methods are developed to overcome these instabilities.
Abstract: An investigation is made of the nature of asymptotic solutions of homogeneous linear differential equations of arbitrary order in the neighborhood of a singularity of unit rank. We introduce a classification of the solutions into two types, explicit and implicit. For the former there exists a sector on which the solution is dominated by all independent solutions as the singularity is approached. No such sector exists for implicit solutions. In consequence, the two types of solution have different uniqueness properties. Another difference is that error bounds for the asymptotic expansions of explicit solutions are generally stronger than those for implicit solutions. We also investigate the computation of the solutions by numerical integration of the differential equation. It is shown that it is the exception rather than the rule for the integration process to be stable, particularly so for implicit solutions. To overcome these instabilities we develop boundary-value methods, complete with error analysis. Numerical examples illustrate the computation of both explicit and implicit solutions, and also the associated Stokes multipliers.

Journal ArticleDOI
TL;DR: In this article, an asymptotic theory that describes the kinetics of first-order phase transitions is presented. But the main difference between the two is that the Lifshits-Slezov theory uses for the first integral of the kinetic equation an approximate solution of the characteristic equation, which is valid in the entire range of sizes except for the blocking point, i.e., it uses a nonuniformly applicable approximation.
Abstract: We construct an asymptotic theory that describes the kinetics of first-order phase transitions. The theory is a considerable refinement of the well-known Lifshits-Slezov theory. The main difference between the two is that the Lifshits—Slezov theory uses for the first integral of the kinetic equation an approximate solution of the characteristic equation, which is valid in the entire range of sizes except for the blocking point, i.e., it uses a nonuniformly applicable approximation. At the same time, the behavior of the characteristic solution near the blocking point determines the asymptotic behavior of the size distribution function of the nuclei for the new phase. Our theory uses a uniformly applicable solution of the characteristic equation, a solution valid at long times over the entire range of sizes. This solution is used to find the asymptotic behavior of all basic properties of first-order phase transitions: the size distribution function, the average nucleus size, and the nucleus density.


Journal ArticleDOI
TL;DR: In this paper, the case where the vertical end walls are isothermal and the horizontal boundaries are adiabatic is considered for which the flow and temperature fields in the cavity are influenced by nonlinear effects.

Journal ArticleDOI
TL;DR: In this article, it was shown that there are no two-peaked solutions of singularly perturbed elliptic equations in a strictly convex domain, and that these conditions are related to the geometry of the domain.
Abstract: We obtain necessary conditions for the existence of two-peaked solutions of singularly perturbed elliptic equations. These conditions are related to the geometry of the domain. In particular, we prove there are no two-peaked solutions in a strictly convex domain.

Journal ArticleDOI
TL;DR: In this paper, the Richardson extrapolation method and two defect correction schemes by an interpolation post-processing technique, namely, interpolation correction and iterative correction for the numerical solution of a Volterra integral equation by iterated finite element methods, were analyzed.
Abstract: On the basis of asymptotic expansions, we study the Richardson extrapolation method and two defect correction schemes by an interpolation post-processing technique, namely, interpolation correction and iterative correction for the numerical solution of a Volterra integral equation by iterated finite element methods. These schemes are of higher accuracy than the postprocessing method and analyzed in a recent paper [5] by Brunner, Q. Lin and N. Yan. Moreover, we give a positive answer to a conjecture in [5].

Journal Article
TL;DR: In this article, the authors studied the asymptotic behavior of solutions to the Korteweg-deVries-Burgers equation in the case when the initial data has different scaling factors at different scales.
Abstract: In this work we study the asymptotic behaviour of solutions to the Korteweg--deVries--Burgers equation in the case when the initial data has different asymptotic limits at $\pm\infty $ The method used is the one developed by Kawashima and Matsumura to discuss the asymptotic behaviour of travelling-wave solutions to Burgers equation

Journal ArticleDOI
TL;DR: In this paper, the asymptotic behavior of solutions to the generalized Becker-Doring equations is studied and it is proved that solutions converge strongly to a unique equilibrium if the initial density is sufficiently small.
Abstract: The asymptotic behaviour of solutions to the generalized Becker-Doring equations is studied. It is proved that solutions converge strongly to a unique equilibrium if the initial density is sufficiently small.

Journal ArticleDOI
TL;DR: In this paper, the effect of Brinkmann boundary layer on the onset of convection driven by surface tension gradients, called Marangoni convection, in a thin horizontal fluid-saturated sparsely packed porous layer bounded by adiabatic free boundaries is studied analytically by means of linear stability analysis.
Abstract: The effect of Brinkmann boundary layer on the onset of convection driven by surface tension gradients, called Marangoni convection, in a thin horizontal fluid-saturated sparsely packed porous layer bounded by adiabatic free boundaries is studied analytically by means of linear stability analysis. The single-term Galerkin expansion technique is shown to be convenient and instructive to establish eigenvalues. The convergence of the results obtained by this approximate technique is checked by comparing the results with those of exact solutions obtained using a regular perturbation technique. By comparing the results of the two techniques, we found that the single-term Galerkin expansion is accurate only for small values of the porous parameter σ(<10). The effect of large values of σ on the onset of Marangoni convection is determined using a method of matched asymptotic expansions. The effect of a boundary layer that exists for large values of σ is shown to increase the critical Marangoni number by an amount of 2 σ compared to that for small values of σ. This uniformly valid solution permits a unified treatment of Marangoni convection and provides the means for a deeper explanation of the physical phenomena. The results obtained are valid for highly porous materials of current practical importance.

Journal ArticleDOI
Igor Boglaev1
TL;DR: In this article, an iterative algorithm for domain decomposition applied to the solution of a singularly perturbed reaction-diffusion problem is presented, and convergence properties of the algorithm are established.

Journal ArticleDOI
TL;DR: In this article, a bilinear Finite Element Method (FEM) was proposed for singularly perturbed elliptic problems with two small parameters. Butuzov asymptotic expansion was used to prove that the FEM on a special piecewise uniform mesh converges independently of small parameters, and the results showed that their method perform much better than the classical FEM.
Abstract: In this paper, we consider a bilinear Finite Element Method (FEM) for a singularly perturbed elliptic problem with two small parameters. By using Butuzov asymptotic expansion [1] and the technique we developed in [2–4], we prove that our FEM on a special piecewise uniform mesh converges independently of small parameters. Numerical results show that our method perform much better than the classical FEM. Published by Elsevier Science Ltd.

01 Jan 1998
TL;DR: In this paper, the authors considered the problem of subject classification and determined the probability that all solutions of the problem tend to zero as t → ∞, where k is a given increasing sequence of positive numbers, and k − tk−1 are independent random variables uniformly distributed on interval [0, 1].
Abstract: We consider the equation x ′′ + a2(t)x = 0, a(t) := ak if tk−1 ≤ t < tk, for k = 1, 2, . . . , where {ak} is a given increasing sequence of positive numbers, and {tk} is chosen at random so that {tk − tk−1} are totally independent random variables uniformly distributed on interval [0, 1]. We determine the probability of the event that all solutions of the equation tend to zero as t → ∞. AMS Subject Classification. 34F05, 34D20, 60K40