Showing papers on "Method of matched asymptotic expansions published in 2003"
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TL;DR: In this article, the convergence of solutions to the Navier-Stokes equations to the stationary one was proved using a direct method and a Razumikhin-type method.
Abstract: Some results on the asymptotic behaviour of solutions to Navier–Stokes equations when the external force contains some hereditary characteristics are proved. We show two different approaches to prove the convergence of solutions to the stationary one, when this is unique. The first is a direct method, while the second is based on a Razumikhin–type method.
97 citations
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TL;DR: For a class of second order nonlinear differential equations, sufficient conditions are presented in this paper to ensure that some, respectively all solutions are asymptotic to lines, where lines are defined as a line connecting two points.
Abstract: For a class of second order nonlinear differential equations, sufficient conditions are presented to ensure that some, respectively all solutions are asymptotic to lines.
75 citations
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TL;DR: The rate at which blowup occurs is investigated in settings with certain symmetries, using the method of matched asymptotic expansions to identify a range of blowup scenarios in both finite and infinite time, including degenerate cases.
Abstract: The harmonic map heat flow is a model for nematic liquid crystals and also has origins in geometry. We present an analysis of the asymptotic behavior of singularities arising in this flow for a special class of solutions which generalizes a known (radially symmetric) reduction. Specifically, the rate at which blowup occurs is investigated in settings with certain symmetries, using the method of matched asymptotic expansions. We identify a range of blowup scenarios in both finite and infinite time, including degenerate cases.
66 citations
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TL;DR: This survey paper contains a surprisingly large amount of material on singularly perturbed partial differential equations and indeed can serve as an introduction to some of the ideas and methods of the singular perturbation theory.
64 citations
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TL;DR: A turning-point theory is developed for the second-order difference equation where the coefficients An and Bn have asymptotic expansions of the form θ≠0 being a real number and it is shown how the Airy functions arise in the uniform asym PT expansion of the solutions to this three-term recurrence relation.
Abstract: A turning-point theory is developed for the second-order difference equation
$$$$
where the coefficients An and Bn have asymptotic expansions of the form
$$$$
θ≠0 being a real number. In particular, it is shown how the Airy functions arise in the uniform asymptotic expansions of the solutions to this three-term recurrence relation. As an illustration of the main result, a uniform asymptotic expansion is derived for the orthogonal polynomials associated with the Freud weight exp(−x4), xℝ.
53 citations
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TL;DR: In this paper, the second and third-order structure functions were determined from the isotropic Karman-Howarth [Proc. R. London, Ser. A 164, 192 (1938)] equation.
Abstract: The Kolmogorov [Dokl. Akad. Nauk. SSSR 30, 299 (1941), hereafter K41] inertial range theory is derived from first principles by analysis of the Navier–Stokes equation using the method of matched asymptotic expansions without assuming isotropy or homogeneity and the Kolmogorov (K62) [J. Fluid Mech. 13, 82 (1962)] refined theory is analyzed. This paper is an extension of Lundgren [Phys. Fluids 14, 638 (2002)], in which the second- and third-order structure functions were determined from the isotropic Karman–Howarth [Proc. R. Soc. London, Ser. A 164, 192 (1938)] equation. The starting point for the present analysis is an equation for the difference in velocity between two points, one of which is a Lagrangian fluid point and the second, slaved to the first by a fixed separation r, is not Lagrangian. The velocity difference, so defined, satisfies the Navier–Stokes equation with spatial variable r. The analysis is carried out in two parts. In the first part the physical hypothesis is made that the mean dissipat...
50 citations
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TL;DR: In this paper, a conditional limit theorem and conditional asymptotic expansions are considered based on the Malliavin calculus, and the problem of lifting limit theorems to their conditional counterparts is treated.
43 citations
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TL;DR: A method of reduction of order is proposed for solving singularly perturbed two-point boundary value problems with a boundary layer at one end point and several linear and non-linear singular perturbation problems have been solved.
34 citations
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TL;DR: In this paper, a central difference scheme for the numerical solution of a system of coupled reaction-diffusion equations is proposed, and it is shown that the scheme is almost second-order convergent, uniformly in the perturbation parameter.
Abstract: Abstract We consider a central difference scheme for the numerical solution of a system of coupled reaction-diffusion equations. We show that the scheme is almost second-order convergent, uniformly in the perturbation parameter. We present the results of numerical experiments to confirm our theoretical results.
31 citations
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TL;DR: In this paper, a singularly perturbed system of linear differential equations with a small delay is considered, and estimates of blocks of the fundamental matrix solution to this system uniformly valid for all sufficiently small values of the parameter of singular perturbations are obtained in the cases of time independent and time dependent coefficients of the system.
26 citations
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01 Jan 2003TL;DR: In this article, the asymptotic matching principle is applied to the case where a small parameter multiplies the highest derivative in a differential equation and there occurs a sharp change in the dependent variable in a certain region of the domain of the independent variable.
Abstract: In cases where a small parameter multiplies the highest derivative in a differential equation, there occurs a sharp change in the dependent variable in a certain region of the domain of the independent variable. In constructing a solution to the differential equation through uniformly-valid expansions, one characterizes the sharp changes by a magnified scale that is different from the scale characterizing the behavior of the dependent variable outside the “boundary-layer” regions. In other words, one represents the solution by two different asymptotic expansions using the independent variables x and x/e say. Since they are different asymptotic representations of the same function, they should be related to each other in a rational manner in an overlapping region where both are valid (Friedrichs, 1955); this leads to the asymptotic matching principle (the latter makes the two representations completely determinate).
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TL;DR: In this paper, the authors considered singularly perturbed boundary value problems (BVPs) for fourth-order ODEs with a small positive parameter multiplying the highest derivative of the form −ey iv (x)+b(x)y″(x),−c(x,y)y(x)=−f (x), x∈D≔(0,1), y(0)=p, y(1)=q, y
Abstract: Singularly perturbed boundary value problems (BVPs) for fourth-order ordinary differential equations (ODES) with a small positive parameter multiplying the highest derivative of the form −ey iv (x)+b(x)y″(x)−c(x)y(x)=−f(x), x∈D≔(0,1) , y(0)=p, y(1)=q, y″(0)=−r, y″(1)=−, 0≤e≪1 , are considered. The given fourth-order BVP is transformed into a system of weakly coupled systems of two second-order ODEs, one without the parameter and the other with the parameter e multiplying the highest derivative, and suitable boundary conditions. In this paper, computational methods for solving this system are presented. In these methods, we first find the zero-order asymptotic approximation expansion of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero-order asymptotic approximation expansion of the solution in the second equation. Then the second equation is solved by the fitted operator method, fitted mesh method, and boundary value technique. Error estimates are derived and examples are provided to illustrate the methods.
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TL;DR: In this article, a computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions.
Abstract: A computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions. In this method, first we construct a zeroth order asymptotic expansion for the solution of the given boundary-value problem. Then, this problem is integrated to get an equivalent initial-value problem for first-order ordinary differential equations. This initial-value problem is solved by either a classical method or a fitted operator method after approximating some of the terms in the differential equations by using the zeroth order asymptotic expansion. This method is effective and easy to implement. An error estimate is derived for the numerical solution. Examples are given to illustrate the method.
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TL;DR: In this article, the authors extend singular perturbation theory in ordinary differential equations to delay differential equations with a fixed lag, and give an explicit sufficient condition so that the solution of a class of singularly perturbed delay differential equation can be asymptotically expanded.
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TL;DR: In this paper, the authors studied the integrability and stability of the zero solution of the Ito-Volterra equation under Lipschitz conditions on the state-dependent functions, and showed that any solution which is p-th mean integrable for some p ≥ 2 is p -th mean-asymptotically stable.
Abstract: This paper studies the pathwise asymptotic stability and integrability of the zero solution of a finite dimensional Ito-Volterra equation. Under Lipschitz conditions on the state-dependent functions, and with continuity and integrability required of the kernels, it is shown that any solution which is p-th mean integrable for some p ≥ 2 is p-th mean-asymptotically stable, and also p-th mean integrable and asymptotically stable, almost surely. If there is no delay-dependent term in the volatility, the same result can be shown for p ≥ 1. Examples which illustrate the usefulness of these results are presented, and extensions to other classes of functional differential equations discussed.
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09 Dec 2003
TL;DR: In this paper, singularly impulsive or generalized impulsive dynamical systems are defined as the class of hybrid systems, where algebraic equations represent constraints that differential and difference equations need to satisfy.
Abstract: In this paper we present singularly impulsive or generalized impulsive dynamical systems. Dynamics of this system is characterized by the set of differential, difference, and algebraic equations. They represent the class of hybrid systems, where algebraic equations represent constraints that differential and difference equations need to satisfy. Generalized term have as source generalized systems theory where singular system theory can be viewed as generalization of regular system theory. For this class of system we develop Lyapunov and asymptotic stability theorems.
01 Jan 2003
TL;DR: In this article, an asymptotic representation for a fundamental solution matrix for scalar linear dynamic systems on time scales is given, which is a generalization of the usual exponential function.
Abstract: We consider linear dynamic systems on time scales, which contain as special cases linear differential systems, difference systems, or other dynamic systems. We give an asymptotic representation for a fundamental solution matrix that reduces the study of systems in the sense of asymptotic behavior to the study of scalar dynamic equations. In order to understand the asymptotic behavior of solutions of scalar linear dynamic equations on time scales, we also investigate the behavior of solutions of the simplest types of such scalar equations, which are natural generalizations of the usual exponential function.
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TL;DR: In this paper, a comparison of results predicted by the method of matched asymptotic expansions, by the Child-Langmuir model, and by patching in three problems of the theory of space-charge sheaths is performed.
Abstract: A comparison is performed of results predicted by the method of matched asymptotic expansions, by the Child-Langmuir model, and by patching in three problems of the theory of space-charge sheaths: a collisionless steady-state sheath, a matrix sheath, and a collisionless RF sheath. In each problem, results are compared between themselves and with the exact solution. In all the cases, the Child-Langmuir model and patching provide results which are accurate to the first approximation in the sheath voltage but not to the second one, irrespective of details of patching. For a steady-state sheath and a matrix sheath, accuracy of the asymptotic solution is exponential. The asymptotic solution for an RF sheath is accurate to the second approximation and can be further improved, if necessary. Comparison of numerical values shows that the method of matched asymptotic expansions indeed provides a considerably higher accuracy.
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TL;DR: In this paper, the authors investigate a system of two first-order differential equations that appears when averaging nonlinear systems over fast one-frequency oscillations and find the asymptotic behavior of a two-parameter family of solutions with an infinitely growing amplitude.
Abstract: We investigate a system of two first-order differential equations that appears when averaging nonlinear systems over fast one-frequency oscillations. The main result is the asymptotic behavior of a two-parameter family of solutions with an infinitely growing amplitude. In addition, we find the asymptotic behavior of another two-parameter family of solutions with a bounded amplitude. In particular, these results provide the key to understanding autoresonance as the phenomenon of a considerable growth of forced nonlinear oscillations initiated by a small external pumping.
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TL;DR: In this article, a singularly perturbed system of second-order differential equations describing steady state of a chemical process was considered, and a formal asymptotic expansion of the solution was constructed in the case when solution exhibits a corner-type behavior.
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TL;DR: In this paper, the relation between the Magnus and the product of exponentials expansions can be expressed in terms of a system of first-order differential equations in the parameters of the two expansions.
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TL;DR: In this paper, the authors studied the Dirichlet problem in half-space for the equation ∆u+ g(u)|∇u|2 = 0, where g is continuous or has a power singularity (in the latter case positive solutions are considered).
Abstract: We study the Dirichlet problem in half-space for the equation ∆u+ g(u)|∇u|2 = 0, where g is continuous or has a power singularity (in the latter case positive solutions are considered). The results presented give necessary and sufficient conditions for the existence of (pointwise or uniform) limit of the solution as y → ∞, where y denotes the spatial variable, orthogonal to the hyperplane of boundary-value data. These conditions are given in terms of integral means of the boundary-value function. Introduction The phenomenon called stabilization is well known for parabolic equations both in linear (see e.g. [1] and references therein) and non-linear (see e.g. [2] and references therein) cases; it means the existence of a finite limit of the solution as t → ∞. However, there are well-posed non-isotropic elliptic boundary-value problems in unbounded domains (see e.g. [3]) for which we can talk about stabilization in the following sense: the solution has a finite limit as a selected spatial variable tends to infinity. This paper is devoted to the Dirichlet problem in half-space for elliptic equations. We present necessary and sufficient conditions for the stabilization of its solution; here the spatial variable, orthogonal to the hyperplane of boundary-value data, plays the role of time. In Section 1, the linear case is presented; Sections 2 and 3 are devoted to quasi-linear equations with the so-called Burgers-Kardar-ParisiZhang non-linearity type (see e.g. [4], [5]). Equations with such non-linearities arise, for example, in modeling of directed polymers and interface growth. They also present an independent theoretical interest because they contain second powers of the first derivatives (see e.g. [6] and references therein). Note that we deal with the stabilization problem in cylindrical domains with an unbounded base (in particular, here the base of the cylinder is the whole E ). As in the parabolic case, this problem is principally different (this refers both to the results and to the methods of research) from the stabilization problem in cylindrical domains with a bounded base. The latter problem has been investigated Received by the editors March 6, 2002. 2000 Mathematics Subject Classification. Primary 35J25; Secondary 35B40, 35J60.
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TL;DR: In this paper, the existence of a corresponding Lyapunov function is shown to be a necessary and sufficient condition for the uniform asymptotic stability of the zero solution of a system of ordinary differential equations with impulse action at fixed moments of time.
Abstract: A system of ordinary differential equations with impulse action at fixed moments of time is considered. The system is assumed to have the zero solution. It is shown that the existence of a corresponding Lyapunov function is a necessary and sufficient condition for the uniform asymptotic stability of the zero solution. Restrictions on perturbations of the right-hand sides of differential equations and impulse actions are obtained under which the uniform asymptotic stability of the zero solution of the "unperturbed" system implies the uniform asymptotic stability of the zero solution of the "perturbed" system.
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TL;DR: Some numerical methods presented for singularly perturbed two-point boundary value problems for second order ordinary differential equations with two small parameters multiplying the derivatives are well suited for parallel computing.
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TL;DR: The method of matched asymptotic expansions is used to derive the dynamics in bulk quasi-neutral plasma, transition, and sheath regions and a constructive existence theorem is presented for solutions of the system governing sheath dynamics.
Abstract: This paper considers the dynamics of a radio-frequency driven plasma consisting of ions and electrons. The method of matched asymptotic expansions is used to derive the dynamics in bulk quasi-neutral plasma, transition, and sheath regions. Furthermore, a constructive existence theorem is presented for solutions of the system governing sheath dynamics.
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TL;DR: In this article, the authors change differential equations on the half line to singular problems on finite interval, and then use a specific Jacobi approximation for solving the resulting problems numerically.
Abstract: In this paper, we change differential equations on the half line to certain singular problems on finite interval, and then use a specific Jacobi approximation for solving the resulting problems numerically. Theoretical analysis and numerical results demonstrate the efficiency of this method.
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TL;DR: In this article, conditions for the existence of a continuum of solutions, the graphs of which remain within a given prescribed set, are formulated for the general systems of discrete equations Δu ( k ) = F ( k, u (k )).
Abstract: The second Liapunov method serves as a powerful tool for the investigation of the stability of the trivial solution of ordinary differential equations systems and discrete equations systems. In the presented paper, a Liapunov-type qualitative approach is used for the investigation of asymptotic behaviour of the solutions of systems of discrete equations. Conditions for the existence of continuum of solutions, the graphs of which remain within a given prescribed set, are formulated for the general systems of discrete equations Δu ( k ) = F ( k , u ( k )). An additional advantage of the presented approach consists of the fact that no assumption concerning the existence of the trivial solution (or the existence of an equilibrium point) of systems considered is assumed. Moreover, the asymptotic behaviour of solutions of some classes of linear difference systems is given by means of concrete asymptotic formulae. Several illustrative examples are considered, too.
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TL;DR: In this paper, a two-stages strategy was used with singular perturbations in control theory to compute a balancing form of non-linear singularly perturbed systems and an approximate balancing form was derived from the balancing forms of the slow and fast subsystems both computed separately.
Abstract: In this work, a balancing method is investigated for a class of non-linear singularly perturbed systems. The main result presented here shows that the well-known 'two-stages' strategy used with singular perturbations in control theory can be extended to compute a balancing form of non-linear singularly perturbed systems. So, an approximate balancing form is derived from the balancing forms of the slow and fast subsystems both computed separately. This two-stage method avoids the difficult task of solving high dimensional and ill-conditioned Hamilton-Jacobi equations due to the presence of the small singular perturbation parameter.