Showing papers on "Method of matched asymptotic expansions published in 2004"

Asymptotic behaviour and expansions of solutions of an ordinary differential equation

Abstract: An ordinary differential equation of quite general form is considered. It is shown how to find the following near a finite or infinite value of the independent variable by using algorithms of power geometry: (i) all power-law asymptotic expressions for solutions of the equation; (ii) all power-logarithmic expansions of solutions with power-law asymptotics; (iii) all non-power-law (exponential or logarithmic) asymptotic expressions for solutions of the equation; (iv) certain exponentially small additional terms for a power-logarithmic expansion of a solution that correspond to exponentially close solutions. Along with the theory and algorithms, examples are presented of calculations of the above objects for one and the same equation. The main attention is paid to explanations of algorithms for these calculations.

Inclusion Theorems for Non-linear Difference Equations with Applications

Abstract: For non-linear difference equations an inclusion theorem is set up which guarantees the existence of solutions in a given asymptotic stripe. This theorem is the limit case of an inclusion theorem for a boundary value problem in a finite interval. Different applications to rational difference equations of order two are given.

Initial stage of flat plate impact onto liquid free surface

Abstract: The liquid flow and the free surface shape during the initial stage of flat plate impact onto liquid half-space are investigated. Method of matched asymptotic expansions is used to derive equations of motion and boundary conditions in the main flow region and in small vicinities of the plate edges. Asymptotic analysis is performed within the ideal and incompressible liquid model. The liquid flow is assumed potential and two dimensional. The ratio of the plate displacement to the plate width plays the role of a small parameter. In the main region the flow is given in the leading order by the pressure-impulse theory. This theory provides the flow field around the plate after a short acoustic stage and predicts unbounded velocity of the liquid at the plate edges. In order to resolve the singular flow caused by the normal impact of a flat plate, the fine pattern of the flow in small vicinities of the plate edges is studied. It is shown that the initial flow close to the plate edges is self-similar in the leading order and is governed by nonlinear boundary-value problem with unknown shape of the free surface. The Kutta conditions are imposed at the plate edges, in order to obtain a nonsingular inner solution. This boundary-value problem is solved numerically by iterations. At each step of iterations the “inner” velocity potential is calculated by the boundary-element method. The asymptotics of the inner solution in both the far field and the jet region are obtained to make the numerical algorithm more efficient. The numerical procedure is carefully verified. Agreement of the computed free surface shape with available experimental data is fairly good. Stability of the numerical solution and its convergence are discussed.

Asymptotic properties of solutions to wave equations with time-dependent dissipation

Abstract: problems of the form utt +Au+ b(t)ut = 0 for a function u(t) taking values in a Hilbert space H and with a positive closed operator A : H ⊇ D(A) → H can be treated by the same arguments in terms of a spectral calculus for the operator A. For the corresponding damped problem with b(t) ≡ 1 R. Ikehata and K. Nishihara investigated in [IN03] a corresponding diffusion phenomenon towards an abstract parabolic problem. A scattering theory for abstract Cauchy problems with time-dependent operator A(t) was developed by A. Arosio in [Aro84]. The treatment is closely related to our approach of Section 3.1. Coefficients depending on both variables seem to be a closely related problem. Nevertheless, there arise essential problems in dealing with utt −∆u+ b(t, x)ut = 0, u(0, ·) = u1, Dtu(0, ·) = u2.. The main point is that one has to control all frequencies in order to deduce sharp operator estimates. By means of the pseudo-differential calculus and a diagonalization/decoupling procedure J. Rauch and M. Taylor obtained in [RT75] estimates of the solution and the energy in the Calkin algebra L(L)/K(L2) of bounded modulo compact operators. The obtained pseudo-differential representations are closely related to our results restricted to the hyperbolic part. In case of non-effective dissipation their results transfer to estimates in the operator algebra. Our considerations show that for effective dissipation terms essential properties of the solutions are lost in this way. A different approach to handle coefficients depending on t and x are so-called weighted energy inequalities. By means of this technique the cited results of A. Matsumura, [Mat77], H. Uesaka, [Ues80], K. Mochizuki, [Moc77], [MN96] and F. Hirosawa / H. Nakazawa, [HN03], are obtained. All these results are estimates in L2-scale and provide no further structural information on the representation of solutions. For coefficients depending on x only and under the strong effectivity assumption b(x) ≥ c0 > 0 for large values of |x|, M. Nakao, [Nak01], has proven Lp–Lq estimates related to damped waves. His approach works on general exterior domains with further effectivity assumptions near parts of the boundary and is based on L2-estimates for the local energy. It is an interesting question to weaken the above effectivity assumption for large x and to consider coefficients estimated from below like b(x) ≥ c0 〈x〉−α for some α ∈ (0, 1). For the upper estimate |b(x)| ≤ 〈x〉−1−2 it is known from the scattering results of K. Mochizuki, [Moc77], that the solutions are asymptotically free. One may conjecture that in this case the same Lp–Lq estimates like for the free wave equation are valid.

Accurate Solution of a System of Coupled Singularly Perturbed Reaction-diffusion Equations

Abstract: We study a system of coupled reaction-diffusion equations. The equations have diffusion parameters of different magnitudes associated with them. Near each boundary, their solution exhibit two overlapping layers. A central difference scheme on layer-adapted piecewise uniform meshes is used to solve the system numerically. We show that the scheme is almost second-order convergent, uniformly in both perturbation parameters, thus improving previous results [5]. We present the results of numerical experiments to confirm our theoretical results.

Generalized thermoelastic problem for an infinitely long hollow cylinder for short times

Abstract: In this work, we consider the one-dimensional problem for an infinitely long hollow cylinder in the context of the theory of generalized thermoelasticity with one relaxation time. The Laplace transform technique is used to solve the problem. The solution in the transformed domain is obtained by a direct approach. The inverse transforms are obtained in an approximate analytical manner using asymptotic expansions valid for small values of time. The temperature, displacement, and stress are computed and represented graphically.

Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications

Abstract: We establish some asymptotic expansions for infinite weighted convolution of distributions having regular varying tails. Various applications to statistics and probability are developed.

Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I. Reduction to Spatially One-Dimensional Equations

Abstract: We consider equations of nonrelativistic quantum mechanics in thin three-dimensional tubes (nanotubes). We suggest a version of the adiabatic approximation that permits reducing the original three-dimensional equations to one-dimensional equations for a wide range of energies of longitudinal motion. The suggested reduction method (the operator method for separating the variables) is based on the Maslov operator method. We classify the solutions of the reduced one-dimensional equation. In Part I of this paper, we deal with the reduction problem, consider the main ideas of the operator separation of variables (in the adiabatic approximation), and derive the reduced equations. In Part II, we will discuss various asymptotic solutions and several effects described by these solutions.

Multi-point Taylor Expansions of Analytic Functions

Abstract: Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.

A boundary value technique for boundary value problems for singularly perturbed fourth-order ordinary differential equations

Abstract: Singularly perturbed two-point boundary value problems (BVPs) for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative are considered. A numerical method is suggested in this paper to solve such problems. In this method, the given BVP is transformed into a system of two ODEs subject to suitable boundary conditions. Then, the domain of definition of the differential equation (a closed interval) is divided into two nonoverlapping subintervals, which we call ''inner region'' (boundary layer) and ''outer region''. Then, the DE is solved in these intervals separately. The solutions obtained in these regions are combined to give a solution in the entire interval. To obtain terminal boundary conditions (boundary values inside this interval) we use mostly zero-order asymptotic expansion of the solution of the BVP. First, linear equations are considered and then nonlinear equations. To solve nonlinear equations, Newton's method of quasilinearization is applied. The present method is demonstrated by providing examples. The method is easy to implement.

Rigorous asymptotic expansions for Lagerstrom's model equation—a geometric approach

Abstract: The present work is a continuation of the geometric singular perturbation analysis of the Lagerstrom model problem which was commenced in J. Differential Equations (199 (2) (2004) 290–325). We establish the same framework here, reinterpreting Lagerstrom's equation as a dynamical system which is subsequently analyzed by means of methods from dynamical systems theory as well as of the blow-up technique. We show how rigorous asymptotic expansions for the Lagerstrom problem can be obtained using geometric methods, thereby establishing a connection to the method of matched asymptotic expansions. We explain the structure of these expansions and demonstrate that the occurrence of the well-known logarithmic (switchback) terms therein is caused by a resonance phenomenon.

Global Existence and Asymptotic Behavior of Solutions of Nonlinear Differential Equations

Abstract: The paper addresses two open problems related to global existence of solutions with a "linear-like" behavior at infinity. For a class of second-order nonlinear differential equations, we establish global existence of solutions under milder assumption on the rate of decay of the coefficient. Furthermore, as opposed to results reported in the literature, we prove for another class of second-order nonlinear differential equations that the region of the initial data for the solutions with desired asymptotic behavior is unbounded and proper.

A model for the condensation of a dusty plasma

Abstract: A model for the condensation of a dusty plasma is constructed by considering the spherical shielding layers surrounding a dust grain test particle. The collisionless region less than a collision mean free path from the test particle is shown to separate into three concentric layers, each having distinct physics. The method of matched asymptotic expansions is invoked at the interfaces between these layers and provides equations which determine the radii of the interfaces. Despite being much smaller than the Wigner–Seitz radius, the dust Debye length is found to be physically significant because it gives the scale length of a precipitous cut-off of the shielded electrostatic potential at the interface between the second and third layers. Condensation is predicted to occur when the ratio of this cut-off radius to the Wigner–Seitz radius exceeds unity and this prediction is shown to be in good agreement with experiments.

Global asymptotic stability of solutions of cubic stochastic difference equations

Abstract: Global almost sure asymptotic stability of solutions of some nonlinear stochastic difference equations with cubic-type main part in their drift and diffusive part driven by square-integrable martingale differences is proven under appropriate conditions in ℝ1. As an application of this result, the asymptotic stability of stochastic numerical methods, such as partially drift-implicit θ-methods with variable step sizes for ordinary stochastic differential equations driven by standard Wiener processes, is discussed.

Inverse factorial-series solutions of difference equations

Abstract: We obtain inverse factorial-series solutions of second-order linear difference equations with a singularity of rank one at infinity. It is shown that the Borel plane of these series is relatively simple, and that in certain cases the asymptotic expansions incorporate simple resurgence properties. Two examples are included. The second example is the large $a$ asymptotics of the hypergeometric function ${}_2F_1(a,b;c;x)$. AMS 2000 Mathematics subject classification: Primary 34E05; 39A11. Secondary 33C05

Asymptotic Behavior Of Solutions Of Dynamic Equations

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.

On the calculation of the transmission line parameters for long tubes using the method of multiple scales.

Abstract: The present paper deals with the classical problem of linear sound propagation in tubes with isothermal walls. The perturbation technique of the method of multiple scales in combination with matched asymptotic expansions is applied to derive the first-order solutions and, in addition, the second-order solutions representing the correction due to boundary layer attenuation. The propagation length is assumed to be so large that in order to obtain asymptotic solutions which extend over the whole spatial range the first-order corrections to the classical attenuation rates of the different modes come into play as well. Starting with the case of the characteristic wavelength being large compared to the characteristic dimension of the duct, the analysis is then extended to the case where both of these quantities are of the same order of magnitude. Furthermore, the transmission line parameters and the transfer functions relating the sound pressures at the ends of the duct to the axial velocities are calculated.

Singular perturbation methods for a class of initial and boundary value problems in multi-parameter classical digital control systems

Abstract: A stable linear time-invariant classical digital control system with several widely different small coefficients multiplying the lowest functions is considered. It is formulated as a multi-parameter singularly perturbed system. Perturbation methods are developed for both initial and boundary value problems based on asymptotic expansions of the perturbation parameters. The approximate solution consists of an outer solution and a number of boundary layer correction solutions equal to the number of initial conditions lost in the process of degeneration. An example is provided for illustration.