Showing papers on "Method of matched asymptotic expansions published in 1995"
01 Jan 1995
TL;DR: In this article, the boundary function method has been applied in the theory of semiconductor devices, and a mathematical model of combustion process in the case of autocatalytic reaction has been proposed.
Abstract: 1. Basic Ideas Regular and singular perturbations Asymptotic approximations Asymptotic and convergent series Examples of asymptotic expansions for solutions of regularly and singularly perturbed problems 2. Singularly perturbed ordinary differential equations Initial value problem The critical case Boundary value problems Spike-type solutions and other contrast (dissipative) structures 3. Singularly perturbed partial differential equations The method of Vishik-Lyusternik Corner boundary functions The smoothing procedure Systems of equations in critical cases Periodic solutions Hyperbolic systems 4. Applied problems Mathematical model of combustion process in the case of autocatalytic reaction Heat conduction in thin Bodies Application of the boundary function method in the theory of semiconductor devices Relaxation waves in the FitzHugh-Nagumo system On some other applied problems Index.
330 citations
Book•
01 Mar 1995
TL;DR: When you read more every page of this impulse differential equations asymptotic properties of the solutions, what you will obtain is something great.
Abstract: Read more and get great! That's what the book enPDFd impulse differential equations asymptotic properties of the solutions will give for every reader to read this book. This is an on-line book provided in this website. Even this book becomes a choice of someone to read, many in the world also loves it so much. As what we talk, when you read more every page of this impulse differential equations asymptotic properties of the solutions, what you will obtain is something great.
251 citations
TL;DR: In this paper, the authors studied the internal layer behavior associated with the following viscous shock problem in the limit e 0: ==================\/\/\/\/\/\/£££€££ £££•££'€£• ££ £•£•€£ £'££ • ££•$££·££
Abstract: Using formal asymptotic methods, we study the internal layer behavior associated with the following viscous shock problem in the limit e 0:
The convex nonlinearity f(u) satisfies f(α) = f(–α) For the steady problem, we show that the method of matched asymptotic expansions fails to uniquely determine the location of the equilibrium shock layer solution This indeterminacy, resulting from neglecting certain exponentially small effects, is eliminated by using the projection method, which exploits certain properties of the spectrum associated with the linearized operator For the time dependent problem, we show that the viscous shock, which is formed from initial data, drifts towards the equilibrium solution on an exponentially long time interval of the order O(eC/e), for some C > 0 This exponentially slow behavior is analyzed by deriving an equation of motion for the location of the viscous shock For Burgers equation (f(u) = u2/2), the results give an analytical characterization of the slow shock layer motion observed numerically in Kreiss and Kreiss; see [11] We also show that the shock layer behavior is very sensitive to small changes in the boundary operator In addition, using a WKB-type method, the slow viscous shock motion is studied numerically for small e, the results comparing favorably with corresponding analytical results Finally, we relate the slow viscous shock motion to similar slow internal layer motion for the Allen-Cahn equation
91 citations
TL;DR: The use of matched asymptotic expansions to illuminate the description of functions exhibiting Stokes phenomenon and highlights the way in which the local structure and the possibility of finding Stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymPTotic expansions.
Abstract: This paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting Stokes phenomenon. In particular the approach highlights the way in which the local structure and the possibility of finding Stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions.
85 citations
Book•
01 Oct 1995
TL;DR: Mathematical preliminaries, basic definitions and theorems from the theory of boundary value problems and semigroups of operators in Banach spaces formal asymptotic expansion and general theory of singularly perturbed evolution equations.
Abstract: Mathematical preliminaries, basic definitions and theorems from the theory of boundary value problems and semigroups of operators in Banach spaces formal asymptotic expansion, singularly perturbed evolution equations of the resonance type standard asymptotic expansion general theory of singularly perturbed evolution equations with bounded operators, bounded generators applications to the kinetic theory, general properties of linear kinetic equations, existence theorems, equations with bounded operators of resonance type, equations with unbounded operators of resonance type nonlinear equations, selected examples of nonlinear kinetic equations.
63 citations
TL;DR: In this paper, the first boundary value problem is considered in a strip for a system of two parabolic equations in which the parameter multiplying the highest derivatives takes arbitrary values from the half-interval (0, 1).
Abstract: The first boundary-value problem is considered in a strip for a system oftwo parabolic equations in which the parameter multiplying the highest derivatives takes arbitrary values from the half-interval (0, 1]. When the parameter is zero, the system of parabolic equations degenerates into a system of hyperbolic equations which contains no derivatives with respect to the space variables. Difference schemes for the problem, which converge uniformly with respect to the parameter, are constructed using the clustering mesh method. Schemes for the Dirichlet problem in the case ofa system of singularly perturbed elliptic equations which degenerate into equations of zero order are also considered.
62 citations
TL;DR: In this article, a new theoretical formulation is developed for the effects of surfactants on mass transport across the dynamic interface of a bubble which undergoes spherically symmetric volume oscillations.
Abstract: A new theoretical formulation is developed for the effects of surfactants on mass transport across the dynamic interface of a bubble which undergoes spherically symmetric volume oscillations. Owing to the presence of surfactants, the Henry's law boundary condition is no longer applicable; it is replaced by a flux boundary condition that features an interfacial resistance that depends on the concentration of surfactant molecules on the interface. The driving force is the disequilibrium partitioning of the gas between free and dissolved states across the interface. As in the clean surface problem analysed recently (Fyrillas & Szeri 1994), the transport problem is split into two parts: the smooth problem and the oscillatory problem. The smooth problem is treated using the method of multiple scales. An asymptotic solution to the oscillatory problem, valid in the limit of large Peclet number, is developed using the method of matched asymptotic expansions. By requiring that the outer limit of the inner approximation match zero, the splitting into smooth and oscillatory problems is determined unambiguously in successive powers of [weierp ]−1/2, where [weierp ] is the Peclet number. To leading order, the clean surface solution is recovered. Continuing to higher order it is shown that the concentration field depends on RI[weierp ]−1/2, where RI is the (dimensionless) interfacial resistance associated with the presence of surfactants. Although the influence of surfactants appears at higher order in the small parameter, surfactants are shown to have a very significant effect on bubble growth rates owing to the fact that the magnitude of RI is approximately the same as the magnitude of [weierp ]1/2 at conditions of practical interest. Hence the higher-order ‘corrections’ happen numerically to be of the same magnitude as the leading-order, clean surface problem. This is the fundamental reason for major increases in the bubble growth rates associated with the addition of surfactants. This is in contrast to the case of a still, surfactant-covered bubble, in which the first-order correction to the growth rate is of order RI[weierp ]−1 and presents a [weierp ]−1/2 correction. Finally, although existing experimental results have shown only enhancement of bubble growth in the presence of a surfactant the present theory suggests that it is possible for a surfactant, characterized by weak dependence of interfacial resistance on surface concentration, to inhibit rather than enhance the growth of bubbles by rectified diffusion.
59 citations
TL;DR: In this paper, a numerical approach for the calculation of an acoustic field is applied to the case of a spinning vortex pair to investigate the sound generation by quadrupole sources in unsteady vortical flows.
Abstract: A numerical approach for the calculation of an acoustic field is applied to the case of a spinning vortex pair to investigate the sound generation by quadrupole sources in unsteady vortical flows. Based on the unsteady hydrodynamic information from the known incompressible flowfield, the perturbed compressible acoustic terms derived from the Euler equations are calculated. Nonreflecting boundary conditions are developed to obtain highly stable solutions. Calculated results are compared with analytical solutions obtained by the method of matched asymptotic expansions. The possibility of predicting the effect of convective mean flow is also tested. It is concluded that the sound generated by the quadrupole sources of unsteady vortical flows without a sound-generating body or surface can be calculated by using the source terms of hydrodynamic flow fluctuations.
55 citations
TL;DR: In this paper, the authors derived a nonlinear-diffusion porous-Fisher's equation for population dynamics using explicit traveling wave solutions, initially-separated, expanding populations are studied as they first coalesce.
50 citations
01 Aug 1995
TL;DR: In this paper, the authors developed an asymptotic theory for the thermoelastic analysis of anisotropic inhomogeneous plates subject to general temperature variations and under the action of lateral loads.
Abstract: On the basis of three-dimensional elasticity without a priori assumptions, we develop an asymptotic theory for the thermoelastic analysis of anisotropic inhomogeneous plates subject to general temperature variations and under the action of lateral loads. The inhomogeneities considered are in the thickness direction, and the laminated plate represents an important special case. Through reformulation of the basic equations and nondimensionalization of the field variables, we find that the method of asymptotic expansions is well suited for the problem. Upon using the asymptotic expansion, we obtain sets of recurrence equations that can be integrated successively to determine the solution for a problem. We show that the classical laminated plate theory (CLT) is merely the leading-order approximation in the asymptotic theory. Furthermore, the higher-order equations are essentially the same as the CLT equations, only with nonhomogeneous terms that are completely determined from the lower-order solutions. As a r...
Book•
01 Jan 1995
TL;DR: This paper presents an asymptotic approximation of the Navier-Stokes model for small and large mean free path and some models of this approximation are compared to the Boltzmann model.
Abstract: Preface The basics of asymptotics Perturbation theory Model examples Models of asymptotic approximation of the Navier-Stokes model Asymptotic approximation of the Boltzmann model for small and large mean free path Other models of asymptotic approximation References Index
TL;DR: In this article, the radiation properties of an asymmetric cylinder, formed by slicing a circular cylinder with a plane making an angle π 2 − Λ with the cylinder axis, are investigated as a model problem of relevance to noise emission by novel aeroengine intakes.
TL;DR: In this article, Radially symmetric stationary points of the functional were studied under the Euler-Lagrange equation and the assumption of small energy, that is, for small e, and the existence of precisely two solutions for the corresponding Euler Lagrangian equation was proved.
Abstract: We study radially symmetric stationary points of the functionalwhere u denotes the density of a fluid confined to a container Ω, W(u) is the course-grain free energy and e accounts for surface energy Under the further assumption of small energy, that isfor small e, we prove existence of precisely two solutions for the corresponding Euler-Lagrange equation Each of these solutions is monotone in the radial direction and converges as e→0 to one of two possible radially symmetric single interface minimizers of E0 Our main tool is the method of matched asymptotic expansions from which we construct exact solutions
TL;DR: In this article, the convergence of extended implicit Pouzet-Volterra-Runge-Kutta methods applied to singularly perturbed systems of Volterra integro-differential equations and to the associated integrodifferential-algebraic systems is analyzed.
TL;DR: In this paper, the thermal bar is analyzed as a laminar free-convection boundary layer, following the example of Kuiken & Rotem (1971) for the plume above a line source of heat.
Abstract: The thermal bar, a descending plane plume of fluid at the temperature of maximum density (3.98° C in water), is analysed as a laminar free-convection boundary layer, following the example of Kuiken & Rotem (1971) for the plume above a line source of heat. Numerical integration of the similarity form of the boundary-layer equations yields values of the vertical velocity and temperature gradient on the centre line and the horizontal velocity induced outside the thermal bar as functions of Prandtl number σ. The asymptotic behaviour of these parameters for both large and small σ is also obtained; in these cases, the thermal bar has a two-layer structure, and the method of matched asymptotic expansions is used. For the intermediate case σ= 1, an analytical calculation using approximate velocity and temperature profiles in the integrated boundary-layer equations yields good agreement with the numerical results. The applicability of the results to naturally occurring thermal bars (e. g. in lakes) is limited, but the laminar-flow analysis is likely to relate more closely to the phenomenon on a laboratory scale.
TL;DR: In this article, the authors address and clarify a number of issues related to matched asymptotic expansion analysis of skip trajectories or any class of problems that give rise to inner layers that are not associated directly with satisfying boundary conditions.
Abstract: In this paper we address and clarify a number of issues related to matched asymptotic expansion analysis of skip trajectories or any class of problems that give rise to inner layers that are not associated directly with satisfying boundary conditions. The procedure for matching inner and outer solutions and using the composite solution to satisfy boundary conditions is developed and rigorously followed to obtain a set of algebraic equations for the problem of inclination change with minimum energy loss. Repeated solution of these algebraic equations along the trajectory, treating each current state as an initial state, constitutes a feedback guidance algorithm. The solution is uniformly valid to zero order in an expansion parameter.
TL;DR: In this paper, a uniformly valid asymptotic expansion for the solution to a class of singularly perturbed Volterra integral equations displaying exponential boundary layer behavior is established, and a model for population growth with attrition is briefly discussed.
Abstract: A uniformly valid asymptotic expansion for the solution to a class of singularly perturbed Volterra integral equations displaying exponential boundary layer behavior is established. Certain quasilinear ordinary differential equations are a noted special case, and a model for population growth with attrition is briefly discussed.
TL;DR: In this article, an asymptotic analytic solving method for a differential equation with complex function, small nonlinearity and a slow variable parameter is developed, which is an extension of the well known Bogolubov-Mitropolski method.
Abstract: In this paper an asymptotic analytic solving method for a differential equation with complex function, small nonlinearity and a slow variable parameter is developed. The procedure is an extension of the well known Bogolubov-Mitropolski method. The correctness of the procedure is proved by an example. The vibrations of a rotor on which a thin band is wound and on which a small linear damping acts are obtained. The analytical solutions are compared with numerical ones. They are in good agreement.
TL;DR: The linear response of a thin superconductor strip subjected to an applied perpendicular time-dependent magnetic field is treated analytically using the method of matched asymptotic expansions to treat the response of thin superconducting disks and thin strips with an applied current.
Abstract: The linear response of a thin superconducting strip subjected to an applied perpendicular time-dependent magnetic field is treated analytically using the method of matched asymptotic expansions. The calculation of the induced current density is divided into two parts: an ``outer`` problem, in the middle of the strip, which can be solved using conformal mapping; and an ``inner`` problem near each of the two edges, which can be solved using the Wiener-Hopf method. The inner and outer solutions are matched together to produce a solution which is uniformly valid across the entire strip, in the limit that the effective screening length {lambda}{sub eff} is small compared to the strip width 2{ital a}. From the current density it is shown that the perpendicular component of the magnetic field inside the strip has a weak logarithmic singularity at the edges of the strip. The linear Ohmic response, which would be realized in a type-II superconductor in the flux-flow regime, is calculated for both a sudden jump in the magnetic field and for an ac magnetic field. After a jump in the field the current propagates in from the edges at a constant velocity {ital v}=0.772{ital D}/{ital d} (with {ital D} the diffusion constantmore » and {ital d} the film thickness), rather than diffusively, as it would for a thick sample. The ac current density and the high-frequency ac magnetization are also calculated. The long time relaxation of the current density after a jump in the field is found to decay exponentially with a time constant {tau}{sub 0}=0.255{ital ad}/{ital D}. The method is extended to treat the response of thin superconducting disks and thin strips with an applied current. There is generally excellent agreement between the results of the asymptotic analysis and the recent numerical calculations by Brandt [Phys. Rev. B 49, 9024; 50, 4034 (1994)].« less
TL;DR: In this article, an application of the asymptotic method of nonlinear mechanics to the construction of an approximate solution of the Klein-Gordon equation was considered, and the authors considered an application to nonlinear nonlinear problems.
Abstract: We consider an application of the asymptotic method of nonlinear mechanics to the construction of an approximate solution of the Klein-Gordon equation.
TL;DR: In this paper, the divergence of the asymptotic expansions is investigated and a new perturbation theory is proposed, in which a convergent series corresponds to any physical quantity represented by a functional integral.
Abstract: Asymptotic expansions, employed in quantum physics as series of perturbation theory, appear as a result of the representation of functional integrals by power series with respect to coupling constant. To derive these series one has to change the order of functional integration and infinite summation. In general, this procedure is incorrect and is responsible for the divergence of the asymptotic expansions. In the present work, we suggest a method of construction of a new perturbation theory. In the framework of this perturbation theory, a convergent series corresponds to any physical quantity represented by a functional integral. The relations between the coefficients of these series and those of the asymptotic expansions are established.
TL;DR: In this article, a guidance algorithm for aero-assisted orbit transfer based on the method of matched asymptotic expansions (MAEs) is presented. But it is not shown that the original problem can be reduced to a problem of 6 implicit equations in 6 unknowns.
Abstract: In this paper we perform an evaluation of a guidance algorithm for aeroassisted orbit transfer based on the method of matched asymptotic expansions (MAEs). It is shown that, by exploiting the structure of the matched asymptotic expansion solution procedure, the original problem, which requires the solution of a set of 20 implicit algebraic equations, can be reduced to a problem of 6 implicit equations in 6 unknowns. Guidance law implementation entails treating the current state as a new initial state and repetitively solving the MAE problem to obtain the feedback controls.
TL;DR: In this article, the authors apply desingularization techniques to a singularly perturbed boundary value problem and show that the problem has exactly three solutions, two of which reach negative values.
Abstract: The paper deals with the boundary value problem &x+xj*-x2 = 0, with x(0) = A, x(T) = B for A, B, T > 0 and e > 0 close to zero. It is shown that for T sufficiently big, the problem has exactly three solutions, two of which reach negative values. Solutions reaching negative values occur for T > T(e) > 0 and we show that asymptotically for e -+ 0, T(e) -ln(e), i.e. lim6,o T(e) = 1 . The main tools are transit time analysis in the Lienard plane and normal form techniques. As such the methods are rather qualitative and useful in other similar problems. INTRODUCTION In this paper, the authors will apply desingularization techniques to a singularly perturbed boundary value problem. By doing so, we will be able to explain apparent transition phenomena in a geometric way. The equation we study is given by