Showing papers on "Method of matched asymptotic expansions published in 1986"
TL;DR: The asymptotic stability of self-similar solutions of the Cauchy problem for the semilinear parabolic equation has been studied in this article, where conditions for the stability of these solutions are established.
Abstract: The asymptotic () behavior of solutions of the Cauchy problem is studied for the semilinear parabolic equation where and as . The existence is established of an infinite collection (a continuum) of distinct self-similar solutions of the form , , where the function satisfies an ordinary differential equation. Conditions for the asymptotic stability of these solutions are established. It is shown that for there exist solutions of the problem whose behavior as is described by approximate self-similar solutions (ap.s.-s.s.'s) which in the case coincide with a family of self-similar solutions of the heat equation , while for and the ap.s.-s.s. has the form where .Figures: 2. Bibliography: 78 titles.
107 citations
TL;DR: For reaction-diffusion equations with Neumann boundary conditions, it was shown in this article that the solutions are asymptotic to the solutions of an ordinary differential equation if the diffusivity is large.
102 citations
TL;DR: In this paper, the authors performed an analysis for a fully-developed, forced convective flow through a packed-sphere bed between concentric cylinders maintained at different temperatures using the Brinkman model with variable permeability.
97 citations
TL;DR: In this paper, Stein's method is used to derive asymptotic expansions for expectations of smooth functions of sums of independent random variables, together with Lyapounov estimates of the error in the approximation.
Abstract: Stein's method is used to derive asymptotic expansions for expectations of smooth functions of sums of independent random variables, together with Lyapounov estimates of the error in the approximation.
66 citations
TL;DR: In this paper, the stability of the in-phase and out-of-phase modes of vibration is shown to be governed by the behavior of a linear variational equation with periodic coefficients.
Abstract: This paper concerns the dynamics of a pair of identical, linearly coupled van der Pol relaxation oscillators. We study the stability of the in-phase and out-of-phase modes of vibration. The stability of both modes is shown to be governed by the behavior of a linear Variational equation with periodic coefficients. Approximate analytical solutions are obtained by the method of matched asymptotic expansions. These analytical results are supplemented by numerical integrations based on Floquet theory.It is shown that previous work based on the sinusoidal (nonrelaxation) limit fails to predict a significant region of instability for both modes.
45 citations
TL;DR: By using the method of matched asymptotic expansions analytical equations for the shapes of a sessile (pendant) bubble or drop and the external meniscus around it are derived.
44 citations
TL;DR: In this paper, a theoretical solution of the channeling effect is presented and an approximate method for obtaining the temperature distribution is presented, which has a number of different applications in energy-related problems, such as fixed-bed catalytic reactors, metal processing, underground coal gasification, oil shale, chemical reaction engineering, drying and packed-bed heat exchangers.
Abstract: A theoretical solution of the channeling effect is presented. The details of the channeling production are investigated in detail and an approximate method for obtaining the temperature distribution is presented. These results complete a theoretical solution of the channeling effect which has a number of different applications in energy-related problems, such as fixed-bed catalytic reactors, metal processing, underground coal gasification, oil shale, chemical-reaction engineering, drying and packed-bed heat exchangers. These differential permeability problems are also encountered in several energy resource extraction applications related to underground coal conversion, vertical modified in-situ (VSIS) oil shale retortion, steam flooding and oil recovery from tar sands. The method of matched asymptotic expansions is used to obtain the theoretical solution. The effects of using the singular perturbation solution in obtaining the temperature distribution are discussed. The existence and the concept of the triple momentum boundary layer in variable porosity media is analyzed in detail. The theoretical results are found to be in good agreement with the numerical and the available experimental results.
41 citations
TL;DR: The main tool for the analysis is the theory of boundary value problems on long intervals as mentioned in this paper, and conditions are derived which ensure the existence of a locally unique solution, which can be approximated by an asymptotic expansion.
Abstract: Singularly perturbed systems ordinary differential equations for which the reduced system has a manifold of solutions are called singular singularly perturbed. Boundary value problems for a general class of such systems are examined. Conditions are derived which ensure the existence of a locally unique solution, which can be approximated by an asymptotic expansion. The main tool for our analysis is the theory of boundary value problems on long intervals.
38 citations
TL;DR: In this paper, a matched asymptotic expansion (MSAE) method is proposed to analyze two-dimensional uniform streaming flow at low Reynolds number past a straight cylinder (or cylinders) of arbitrary cross-sectional shape.
Abstract: A numerical implementation of the method of matched asymptotic expansions is proposed to analyse two-dimensional uniform streaming flow at low Reynolds number past a straight cylinder (or cylinders) of arbitrary cross-sectional shape. General solutions for both the Stokes and Oseen equations in two dimensions are expressed in terms of a boundary distribution of fundamental single- and double-layer singularities. These general solutions are then converted to integral equations for the unknown distributions of singularity strengths by application of boundary conditions at the cylinder surface, and matching conditions between the Stokes and Oseen solutions. By solving these integral equations, using collocation methods familiar from three-dimensional application of ‘boundary integral’ methods for solutions of Stokes equation, we generate a uniformly valid approximation to the solution for the whole domain.
We demonstrate the method by considering, as numerical examples, uniform flow past an elliptic cylinder, uniform flow past a cylinder of rectangular cross-section, and uniform flow past two parallel cylinders which may be either equal in radius, or of different sizes.
35 citations
TL;DR: In this article, a differentially heated rectangular cavity of small aspect ratio A was examined and it was shown that when the capillary number is C=O(A3), the interface undergoes an O(1) deformation from its flat position and the flow inside the cavity becomes nonparallel everywhere.
Abstract: Steady thermocapillary convection is examined in a differentially heated rectangular cavity of small aspect ratio A. It is shown that when the capillary number is C=O(A3), the interface undergoes an O(1) deformation from its flat position and the flow inside the cavity becomes nonparallel everywhere. The velocity and temperature profiles and the shape of the deformed interface are derived using the method of matched asymptotic expansions.
29 citations
TL;DR: In this article, the authors derived the zeroth and first order terms of the matched asymptotic expansion of the solutions, which are the sums of uniformly smooth outer terms (reduced solutions) and the exponentially varying inner terms (layer solutions).
Abstract: : In this paper the basic semiconductor device equations modelling a symmetric one-dimensional voltage-controlled diode are formulated as a singularly perturbed two point boundary value problem. The perturbation parameter is the normed Debye-length of the device. The authors derive the zeroth and first order terms of the matched asymptotic expansion of the solutions, which are the sums of uniformly smooth outer terms (reduced solutions) and the exponentially varying inner terms (layer solutions). The main result of the paper is that, if the perturbation parameter is sufficiently small then there exists a solution of the semiconductor device problem which is approximated uniformly by the zeroth order term of the expansion, even for large applied voltages. This result shows the validity of the asymptotic expansions of the solutions of the semiconductor device problem in physically relevant high-injection conditions.
TL;DR: In this article, a method for suppressing these oscillations and evoking more accurate approximations is introduced, based on the Edgeworth type asymptotic expansions for the distributions of statistics.
Abstract: Approximate formulae using a large number of terms of Edgeworth type asymptotic expansions for the distributions of statistics often produce spurious oscillations and give poor fits to the exact distribution functions in parts of the tails. A general method for suppressing these oscillations and evoking more accurate approximations is introduced here.
TL;DR: In this article, asymptotic expansions for a general class of linear parabolic initial boundary value problems in semi-infinite spatial domains were derived for numerical computations and used to derive accurate boundary conditions at an artificial boundary.
Abstract: We derive asymptotic expansions for a general class of linear parabolic initial boundary value problems in semi-infinite spatial domains. The expansions are used to derive accurate boundary conditions at an artificial boundary which are appropriate for numerical computations. The asymptotic solutions, which must be constructed numerically, represent the decay of disturbances as they propagate to infinity and admit a simple physical description.
TL;DR: In this paper, first-order correction terms, developed by using the method of matched asymptotic expansions, are incorporated in the feedback solution of a class of singularly perturbed nonlinear optimal control problems frequently encountered in aerospace applications.
Abstract: In this paper first-order correction terms, developed by using the method of matched asymptotic expansions, are incorporated in the feedback solution of a class of singularly perturbed nonlinear optimal control problems frequently encountered in aerospace applications This improvement is based on an explicit solution of the integrals arising from the first-order matching conditions and leads to correct the initial values of the slow costate variables in the boundary layer Consequently, a uniformly valid feedback control law, corrected to the first-order, can be synthesized The new method is applied to an example of a constant speed minimum-time interception problem Comparison of the zeroth- and first-order feedback control laws to the exact optimal solution demonstrates that first-order corrections greatly extend the domain of validity of the approximation obtained by singular perturbation methods
TL;DR: In this article, an asymptotic analysis of the lower hybrid wave propagation in toroidal geometry is presented which relies on the difference in magnitude between the parallel and perpendicular wave vectors when the wave frequency is of the order of the ion plasma frequency.
Abstract: An asymptotic analysis of the lower hybrid wave propagation in toroidal geometry is presented which relies on the difference in magnitude between the parallel and perpendicular wave vectors when the wave frequency is of the order of the ion plasma frequency. Using the method of matched asymptotic expansions, the equation for the radial position of the ray is solved and the effects of the toroidal corrections are discussed. Moreover, the equation for the parallel wave vector is derived and analytically solved by using a multiple scale analysis. The latter solution is asymptotically matched to a local exact solution near the plasma edge where the multiple scale analysis breaks down. The result of the asymptotic analytical treatment is compared with the findings of a numerical integration of the starting set of equations. Finally, using the previous results, the equation for the variation of the electric field amplitude along the trajectory is analytically solved.
TL;DR: In this paper, the plane steady problem of the flow of a viscous wall jet past a smoothed break in the contour of a body is considered, and the flow in the neighborhood of the junction between two flat plates inclined at an angle to each other is chosen for study.
Abstract: The plane steady problem of the flow of a viscous wall jet past a smoothed break in the contour of a body is considered. For convenience, the flow in the neighborhood of the junction between two flat plates inclined at an angle to each other is chosen for study. As a result of the small extent of the region investigated the flow field is divided into two layers: the main part of the jet, which undergoes inviscid rotation, and a thin sublayer at the wall, which ensures the satisfaction of the no-slip condition. Particular interest attaches to the flow regime in which the solution in the sublayer satisfies the Prandtl boundary layer equations with a given pressure gradient. A similar problem was studied in [1–4]. The present case is distinguished by the structure of the free interaction region in a small neighborhood of the point of zero surface friction stress. By means of the method of matched asymptotic expansions, applied to the analysis of the Navier-Stokes equations, it is established that the interaction mechanism is that described in [5–7]. As a result, an integrodifferential equation describing the behavior of the surface friction stress function is obtained. A numerical solution of this equation is presented. The range of plate angles on which solutions of the equation obtained exist and, therefore, flows of this general type are realized is determined. The essential nonuniqueness of the possible solutions is established, and in particular attention is drawn to the possible existence of six permissible friction distributions.
TL;DR: In this paper, the Titchmarsh-Weyl m-coefficient for the differential equation y + q(x))y = 0 on [0,∞] was derived, where the potential q is integrable near 0.
Abstract: This article is concerned with the Titchmarsh-Weyl m-coefficient for the differential equation y″+(λ−q(x))y=0 on [0,∞), whee the potential q is integrable near 0. The m-coefficient, which is a function of the complex variable λ∈C, depends on an initialization parameter α∈(−π, π]. A uniform as well as various non-uniform asymptotic expansions of m are derived, which are valid as λ → ∞ in any sector of the complex plane that does not contain the real axis. The results reported here extend those of Kaper and Kwong [Proc. Roy. Soc. Edinburgh Sect. A. to appear].
TL;DR: In this article, a Galerkin method is presented for weak self-adjoint differential equations with regular singular points, such as the ideal MHD equation for zero-frequency displacements about a finite beta, cylindrical plasma equilibrium (the Newcomb equation).
TL;DR: Theoretical solutions for diffraction of oblique waves by an infinite vertical barrier with a small slit in deep water were obtained in this paper, where the method of matched asymptotic expansions was used to solve two problems: (1) thin barrier where the thickness of the barrier is small in comparison with wavelength, and (2) wide barrier where both the wavelength and the thickness are in the same order of magnitude.
01 Dec 1986
TL;DR: Sufficient conditions for finite difference equations with a solution behaving in a precisely specified way like a given polynomial are given in this paper, and sufficient conditions for polynomials with a similar solution are also given.
Abstract: Sufficient conditions for somem-th order finite difference equations are presented which have a solution behaving in a precisely specified way like a given polynomial.
TL;DR: In this article, matched asymptotic expansions were used to investigate the vibration of a highly prestressed rectangular thin plate exhibiting natural material orthotropy, when the bending rigidity is small compared to the applied in-plane loading.
Abstract: The method of matched asymptotic expansions is used in investigating the vibration of a highly prestressed rectangular thin plate exhibiting natural material orthotropy. When the bending rigidity is small compared to the applied in-plane loading, analytical results which are correct toO(e2) (where e2 denotes a normalised bending rigidity) are presented for various boundary conditions including the fully clamped case. To leading order solution in e, the eigenvalues of an ideal orthotropic membrane are obtained. The first order solutions in e show the influence of bending stiffness and material orthotropy on the eigenvalue, while torsional rigidity affects the eigenvalues to second order in e. In particular, Hutter and Olunloyo's results are recovered for the special case of isotropic material properties.
TL;DR: The Asymptotic Finite Element method for improvement of standard finite element solutions of perturbation equations by the addition of asymptotics to the right hand side terms is presented in this paper.
Abstract: The Asymptotic Finite Element method for improvement of standard finite element solutions of perturbation equations by the addition of asymptotic corrections to the right hand side terms is presented It is applied here to 1-D and 2-D diffusion–convection equations and to non-linear similarity equations Excellent results were obtained without the a priori use of special trial and test functions Theoretical expectations were confirmed
TL;DR: In this paper, a general formulation of the axisymmetric vortex sound is given, and three typical problems are considered in detail, in which the acoustic wave is generated by a vortex ring interacting with either a sphere, a circular disk, or a circular aperture of a plane wall in an axisymetric manner.
Abstract: A general formulation of the axisymmetric vortex sound is given, and three typical problems are considered in detail, in which the acoustic wave is generated by a vortex ring interacting with either a sphere, a circular disk, or a circular aperture of a plane wall in an axisymmetric manner. The velocity potentials induced by the vortex in the presence of the body are determined by the method of dual integral equations or the image vortex. From the vortex trajectories, the acoustic wave fields are determined. The method of matched asymptotic expansions yields the result that the force exerted on the body is related to the profile of the dipole component. It is also found that the wave amplitude is expressed in terms of the value of a streamfunction at the vortex position, which represents a hypothetical potential flow around the body and is defined appropriately in each problem.
TL;DR: In this paper, the trajectories of a singularly perturbed autonomous system are shown to be extensions of the Tikhonov Theorem in the neighborhood of a point where a certain stability assumption ceases to be valid.
Abstract: In singular perturbation analysis two stages can generally be distinguished: First, by some method a sequence of formal approximations is constructed which are supposed to form together an asymptotic approximation to the solution of the singular perturbation problem. Second, it should be proved that the sequence provides a correct approximation in a rigorous mathematical sense.For this second stage analytical results providing error estimates are needed. A classical one is due to A. N. Tikhonov (see e.g. [10]). We quote this theorem, and state and prove results for the trajectories of a singularly perturbed autonomous system that are extensions of the Tikhonov Theorem in the neighborhood of a point where a certain stability assumption ceases to be valid.
TL;DR: In this paper, the Fokker-Planck equation is equivalent to an interesting Schrodinger equation, and the runaway current is extracted from the smallest eigenvalue, after the equation has been solved by the method of matched asymptotic expansions.
Abstract: Diffusion driven by a constant field, and opposed by a velocity‐dependent diffusion coefficient that decreases to zero at large velocity, leads to the phenomenon of ‘‘runaway.’’ It is studied here in the case of a one‐dimensional velocity space, when the Fokker–Planck equation is equivalent to an interesting Schrodinger equation. The runaway current is extracted from the smallest eigenvalue, after the equation has been solved by the method of matched asymptotic expansions. There is discussion of connections between our approach, the conventional approach, and the classical two‐dimensional results.
TL;DR: In this article, a simple example related to the exponential integral is made simple, and special application is made to derive and clarify the connection formulas (to all orders) in the JWKB method.
Abstract: Asymptotic expansions occur widely in quantum physics. The Rayleigh-Schrodinger perturbation theory for hydrogen in an electrostatic field (the LoSurdo—Stark effect) is one example. The 1/R expansion for the hydrogen molecule ion H is a second. The quantum defect theory and the JWKB method are two more. It is not so widely known that the sum of such real asymptotic expansions may be complex, while the sum of complex asymptotic expansions may be real. The key to this nonintuitive behavior is Borel summation. By examining a simple example related to the exponential integral, the nature of this real-iscomplex, complex-is-real phenomenon is made simple. Then special application is made to derive and clarify the connection formulas (to all orders) in the JWKB method.
TL;DR: In this paper, the authors analyzed wave reflection and transmission by an infinite array of cylinders of arbitrary crosssection and water particle velocities through the slits of the array using matched asymptotic expansions.
Abstract: Wave reflection and transmission by an infinite array of cylinders of arbitrary crosssection and water particle velocities through the slits of the array are analyzed as a boundary-value problem. The method of matched asymptotic expansions is employed. For the case of flat plates, calculation were done for two different types of the inner solution, and the difference of the final solutions is discussed. The comparison of the results with experimental data shows that the present linear theory, which does not account for the quadratic energy loss effect at slits, predicts well the reflection and the water particle velocities, while it overestimates the transmission due to not taking the energy loss effect into account. In the case of flat plates and the case of circular cylinders, the results coincide with existing theoretical results.
TL;DR: In this paper, the initial phase of transient natural convection between two horizontal concentric cylinders is investigated, and short-time solutions to the momentum and energy equations are obtained using the method of matched asymptotic expansions.