Showing papers on "Method of matched asymptotic expansions published in 2006"
Book•
01 Jan 2006TL;DR: Asymptotic analysis of exponential integrals has been studied extensively in the literature, see as discussed by the authors for a survey of asymptotics of exponential and weakly nonlinear waves.
Abstract: Fundamentals: Themes of asymptotic analysis The nature of asymptotic approximations Asymptotic analysis of exponential integrals: Fundamental techniques for integrals Laplace's method for asymptotic expansions of integrals The method of steepest descents for asymptotic expansions of integrals The method of stationary phase for asymptotic analysis of oscillatory integrals Asymptotic analysis of differential equations: Asymptotic behavior of solutions of linear second-order differential equations in the complex plane Introduction to asymptotics of solutions of ordinary differential equations with respect to parameters Asymptotics of linear boundary-value problems Asymptotics of oscillatory phenomena Weakly nonlinear waves Appendix: Fundamental inequalities Bibliography Index of names Subject index.
343 citations
TL;DR: The velocity boundary condition that must be imposed at an interface between a porous medium and a free fluid is investigated in this article, where a heterogeneous transition zone characterized by rapidly varying properties is introduced between the two homogeneous porous and free fluid regions.
158 citations
TL;DR: This series of two articles considers the propagation of a time harmonic wave in a medium made of the junction of a half-space (containing possibly scatterers) with a thin slot, and performs a complete asymptotic expansion of the solution of this problem with respect to the small parameter $\varepsilon/\lambda$.
Abstract: In this series of two articles, we consider the propagation of a time harmonic wave in a medium made of the junction of a half-space (containing possibly scatterers) with a thin slot. The Neumann boundary condition is considered along the boundary on the propagation domain, which authorizes the propagation of the wave inside the slot, even if the width of the slot is very small. We perform a complete asymptotic expansion of the solution of this problem with respect to the small parameter $\varepsilon/\lambda$, the ratio between the width of the slot, and the wavelength. We use the method of matched asymptopic expansions which allows us to describe the solution in terms of asymptotic series whose terms are characterized as the solutions of (coupled) boundary value problems posed in simple geometrical domains, independent of $\varepsilon/\lambda$: the (perturbed) half-space, the half-line, a junction zone. In this first article, we derive and analyze, from the mathematical point of view, these boundary valu...
72 citations
TL;DR: In this paper, a second order monotone numerical method is constructed for a singularly perturbed ordinary differential equation with two small parameters affecting the convection and diffusion terms, and an asymptotic error bound in the maximum norm is established theoretically whose error constants are shown to be independent of both singular perturbation parameters.
66 citations
TL;DR: First, two schemes to integrate initial-value problem (IVP) for system of two first-order differential equations are developed, and then by using these schemes the authors solve the BVP.
62 citations
TL;DR: In this article, a phase-field model is extended to the consideration of edges by an appropriate regularization of the underlying mathematical model, using the method of matched asymptotic expansions.
Abstract: In the presence of sufficiently strong surface energy anisotropy, the equilibrium shape of an isothermal crystal may include corners or edges. Models of edges have, to date, involved the regularization of the corresponding free-boundary problem resulting in equilibrium shapes with smoothed out edges. In this paper, we take a new approach and consider how a phase-field model, which provides a diffuse description of an interface, can be extended to the consideration of edges by an appropriate regularization of the underlying mathematical model. Using the method of matched asymptotic expansions, we develop an approximate solution which corresponds to a smoothed out edge from which we are able to determine the associated edge energy.
58 citations
TL;DR: In this article, a new technique is proposed for the analysis of shape optimization problems using the asymptotic analysis of boundary value problems in singularly perturbed geometrical domains.
Abstract: A new technique is proposed for the analysis of shape optimization problems. The technique uses the asymptotic analysis of boundary value problems in singularly perturbed geometrical domains. The asymptotics of solutions are derived in the framework of compound and matched asymptotics expansions. The analysis involves the so–called interior topology variations. The asymptotic expansions are derived for a model problem, however the technique applies to general elliptic boundary value problems. The self–adjoint extensions of elliptic operators and the weighted spaces with detached asymptotics are exploited for the modelling of problems with small defects in geometrical domains. The error estimates for proposed approximations of shape functionals are provided.
46 citations
TL;DR: In this paper, the Navier boundary condition is replaced by a boundary condition which attempts to account for boundary slip due to the tangential shear at the boundary, where the slip length l is made dimensionless with respect to the corresponding radius.
Abstract: For micro- and nanoscale problems, boundary surface roughness often means that the usual no-slip boundary condition of fluid mechanics does not apply. Here we examine the steady low-Reynolds-number flow past a nanosphere and a circular nanocylinder in a Newtonian fluid, with the no-slip boundary condition replaced by a boundary condition which attempts to account for boundary slip due to the tangential shear at the boundary. We apply the so-called Navier boundary condition and use the method of matched asymptotic expansions. This model possesses a single parameter to account for the slip, the slip length l, which is made dimensionless with respect to the corresponding radius, which is assumed to be of the same order of magnitude as the slip length. Numerical results are presented for the two extreme cases, l = 0 corresponding to classical theory, and l → ∞ corresponding to complete slip. The streamlines for l > 0 are closer to the body than for l = 0, while the frictional drag for l > 0 is reduced below the values for l = 0, as might be expected. For the circular cylinder, results corresponding to l → ∞ are in complete accord with certain low-Reynolds-number experimental results, and this excellent agreement is much better than that predicted by the no-slip boundary condition.
42 citations
TL;DR: A novel technique unifies different approaches to asymptotic integration and addresses a new type of asymPTotic behavior in a class of second-order nonlinear differential equations locally near infinity.
35 citations
01 Apr 2006
33 citations
TL;DR: In this article, it was shown that the existence of a nonoscillatory solution of the perturbed solution of a functional differential equation often implies a real eigenvalue of the limiting equation.
TL;DR: In this article, a construction of regular solutions to the Navier-Stokes equations was proposed for the study of their asymptotic expansions, and sufficient conditions for the convergence of those expansions were given.
Abstract: We introduce a construction of regular solutions to the Navier-Stokes equations that is specifically designed for the study of their asymptotic expansions. Using this construction, we give sufficient conditions for the convergence of those expansions. We also construct suitable normed spaces in which they converge. Moreover, in these spaces, the normal form of the Navier-Stokes equations associated with the terms of the asymptotic expansions [9] is a well-behaved infinite system of differential equations.
TL;DR: In this paper, matched asymptotic expansions are applied to analyze whether cosmological variations in physical constants and scalar fields are detectable locally on the surface of local gravitationally bound systems such as planets and stars, or inside virialized systems like galaxies and clusters.
Abstract: We apply the method of matched asymptotic expansions to analyze whether cosmological variations in physical ``constants'' and scalar fields are detectable, locally, on the surface of local gravitationally bound systems such as planets and stars, or inside virialized systems like galaxies and clusters. We assume spherical symmetry and derive a sufficient condition for the local time variation of the scalar fields that drive varying constants to track the cosmological one. We calculate a number of specific examples in detail by matching the Schwarzschild spacetime to spherically symmetric inhomogeneous Tolman-Bondi metrics in an intermediate region by rigorously constructing matched asymptotic expansions on cosmological and local astronomical scales which overlap in an intermediate domain. We conclude that, independent of the details of the scalar-field theory describing the varying constant, the condition for cosmological variations to be measured locally is almost always satisfied in physically realistic situations. The proof of this statement provides a rigorous justification for using terrestrial experiments and solar system observations to constrain or detect any cosmological time variations in the traditional constants of nature.
15 Feb 2006
TL;DR: The derived expression for the electrostatic potential is compared to numerical solutions of the Poisson-Boltzmann equation and it is shown that the agreement is excellent for capillaries with radii greater or equal to four times the electrical double layer thickness.
Abstract: The electrostatic potential in a capillary filled with electrolyte is derived by solving the nonlinear Poisson–Boltzmann equation using the method of matched asymptotic expansions. This approach allows obtaining an analytical result for arbitrary high wall potential if the double layer thickness is smaller than the capillary radius. The derived expression for the electrostatic potential is compared to numerical solutions of the Poisson–Boltzmann equation and it is shown that the agreement is excellent for capillaries with radii greater or equal to four times the electrical double layer thickness. The knowledge of the electrostatic potential distribution inside the capillary enables the derivation of the electroosmotic velocity flow profile in an analytical form. The obtained results are applicable to capillaries with radii ranging from nanometers to micrometers depending on the ionic strength of the solution.
TL;DR: In this article, the asymptotic stability of the weak solution of the critical and supercritical dissipative quasi-geostrophic equation in the Serrin-type class ∇θ Lr(0, ∞;Lp(R2)) is examined.
Abstract: The asymptotic stability for the weak solution θ of the critical and supercritical dissipative quasi-geostrophic equation in the Serrin-type class ∇θ Lr(0, ∞;Lp(R2)) is examined. This equation is perturbed by large initial data and external functions. It is shown that every weak perturbed solution has the same asymptotic behaviour as that of θ. More precisely, the difference decays in the norm of L2(R2).
TL;DR: In this paper, the existence of positive solutions concentrating on higher dimensional manifolds near the boundary of the domain for a nonlinear singularly perturbed elliptic problem was proved, where the positive solutions were concentrated on some higher dimensional manifold.
Abstract: We prove the existence of positive solutions concentrating on some higher dimensional manifolds near the boundary of the domain for a nonlinear singularly perturbed elliptic problem.
01 Dec 2006
TL;DR: A geometrically invariant concept of singularly perturbed systems of ordinary differential equations (singularly perturbation vector fields) is proposed in this article, where the authors focus on possible ways of fast and slow directions/manifolds evaluations.
Abstract: A geometrically invariant concept of singularly perturbed systems of ordinary differential equations (singularly perturbed vector fields) is proposed in this paper. Singularly perturbed vector fields can be represented locally as singularly perturbed systems (for corresponding coordinate system choice. The paper focuses on possible ways of fast and slow directions/manifolds evaluations. A special algorithm for the evaluation is proposed. The algorithm is called as a global quasi-linearization procedure. A practical application of the proposed algorithm for numerical simulations is the main issue of the paper.
TL;DR: The Wavelet-Galerkin method is shown to be a very effective tool in numerically solving singularly perturbed parabolic problems and a comparison with the method of reduction order yields better results.
TL;DR: A computational method for solving Singularly perturbed two-point boundary value problems (SPBVPs) of convection–diffusion type for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative with a discontinuous source term is presented.
TL;DR: In this article, the authors consider a linear dynamic equation on a time scale together with a perturbation term and show that, if certain exponential dichotomy conditions are satisfied, then for any solution of the perturbed equation there exists a solution of an unperturbed equation that asymptotically differs from the solution of a perturbed one no more than the order of the term.
16 Jan 2006
TL;DR: In this article, a robust stability analysis of nonlinear singularly perturbed systems with vanishing uncertainties, of which the upper norm bounds only are available, is presented and a stabilising controller is proposed.
Abstract: The singularly perturbed system is easily analysed by two time-scale systems, each of which with a lower dimension than the original system. However, if uncertainties are added, then the analysis is very difficult because uncertainties change the slow manifold, the boundary layer model and the reduced model of the nominal system. Robust stability analysis of nonlinear singularly perturbed systems with vanishing uncertainties, of which the upper norm bounds only are available, is presented. The stability condition, under which the zero state equilibrium of the singularly perturbed system is exponentially stable for the sufficiently small perturbation parameter e, is found and a stabilising controller is proposed.
TL;DR: An infinite horizon linear-quadratic optimal control problem for a singularly perturbed system with multiple point-wise and distributed small delays in the state variable is considered and the zero-order asymptotic solution is constructed.
Abstract: An infinite horizon linear-quadratic optimal control problem for a singularly perturbed system with multiple point-wise and distributed small delays in the state variable is considered. The set of Riccati-type equations, associated with this problem by the control optimality conditions, is studied. Since the system in the control problem is singularly perturbed, the equations of this set are also perturbed by a small parameter of the singular perturbations. The zero-order asymptotic solution to this set of equations is constructed and justified. Based on this asymptotic solution, parameter-free sufficient conditions for the existence and uniqueness of solution to the original optimal control problem are established.
TL;DR: In this article, a non-linear one-dimensional model for thin-walled rods with open strongly curved cross-section, obtained by asymptotic methods, is presented.
Abstract: In this paper, we present a non-linear one-dimensional model for thin-walled rods with open strongly curved cross-section, obtained by asymptotic methods. A dimensional analysis of the non-linear three-dimensional equilibrium equations lets appear dimensionless numbers which reflect the geometry of the structure and the level of applied forces. For a given force level, the order of magnitude of the displacements and the corresponding one-dimensional model are deduced by asymptotic expansions.
TL;DR: In this article, asymptotic properties of solutions to first order linear neutral differential equations with variable coefficients and constant delays are studied. And the results are stated in terms of the solution to a characteristic equation.
TL;DR: In this article, it was shown that if 0 is a turning point of order p, then any solution y not exponentially large has, in some sector centred at 0, an asymptotic behaviour of the form ∑ Y n (x / e ) e ǫ n, where e p + 1 = e, for x = e ′ X with X large enough, but independent of e.
27 Mar 2006
TL;DR: Tordeux et al. as discussed by the authors consider the Laplace-Dirichlet equation in a polygonal domain which is perturbed at the scale e near one of its vertices.
TL;DR: In this paper, singularly perturbed Fredholm equations of the second kind were studied and sufficient conditions for existence and uniqueness of solutions were given and the asymptotic behavior of the solutions was examined.
TL;DR: This paper introduces an exponential fitting parameter to the standard finite difference scheme which reflects the singularly perturbed nature of differential operator and derives an e-uniform convergent scheme based on fitting operator for boundary value problems for singularlyperturbed differential-difference equations with small shifts.
Abstract: In this paper, we continue the study of singularly perturbed differential-difference equations with small shifts, which is motivated by the problem of determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in the dendrites [1]. We consider a more general boundary-value problem which contains both convection and reaction terms with both type of shifts (negative as well as positive) than the problem discussed in paper [2,7]. We consider the case when the solution of such type of boundary-value problem exhibits boundary layer behavior.
An e-uniform convergent scheme based on fitting operator is derived for boundary value problems for singularly perturbed differential-difference equations with small shifts. We introduce an exponential fitting parameter to the standard finite difference scheme which reflects the singularly perturbed nature of differential operator. The method is analyzed for convergence. Several numerical experiments are carried in support of theoretical results and to show the effect of small shifts on the boundary layer solution.
TL;DR: In this article, the authors derived the transverse force acting on a hydrodynamic vortex in the presence of a sound wave from a global solution of the scattering problem, using the method of matched asymptotic expansions.
Abstract: We present a derivation of the transverse force acting on a hydrodynamic vortex in the presence of a sound wave from a global solution of the scattering problem, using the method of matched asymptotic expansions. The solution presented includes a detailed treatment of the interaction of the incident wave with the vortex core, and is free from the singularities in the momentum exchange between vortex and sound wave which have led to contradictory results for the value of the transverse force in the literature.