Showing papers on "Method of matched asymptotic expansions published in 1988"
TL;DR: In this paper, an efficient numerical recursive algorithm with a quadratic rate of convergence is developed for solving the algebraic equations comprising the Chang transformation for singularly perturbed linear-quadratic control problems.
Abstract: Under stability-observability conditions imposed on a singularly perturbed system, an efficient numerical method for solving the corresponding matrix differential Riccati equation is obtained in terms of the reduced-order problems. The order reduction is achieved via the use of the Chang transformation applied to the Hamiltonian matrix of a singularly perturbed linear-quadratic control problem. An efficient numerical recursive algorithm with a quadratic rate of convergence is developed for solving the algebraic equations comprising the Chang transformation. >
79 citations
01 Jan 1988
TL;DR: In this paper, the authors studied the dynamical behavior of the initial value problem for the equation u t = u xx + f(u, u x ), x ∈ S 1 = R/Z, t > 0.
Abstract: We study the dynamical behavior of the initial value problem for the equation u t = u xx + f(u, u x ), x ∈ S 1 =R/Z, t > 0. One of our main results states that any C 1-bounded solution approaches either a single periodic solution or a set of equilibria as t → ∞. We also consider the case where the solution blows up in a finite time and prove that under certain conditions on f the blow-up set of any solution with nonconstant initial data is a finite set.
74 citations
TL;DR: In this article, the problem of a thermally developing forced convective flow in a packed channel heated asymmetrically is analyzed and a closed-form solution based on the method of matched asymptotic expansions is obtained for the axial velocity distribution.
Abstract: The problem of a thermally developing forced convective flow in a packed channel heated asymmetrically is analyzed in this paper. The flow in the packed channel is assumed to be hydrodynamically fully developed and is governed by the Brinkman-Darcy-Ergun equation with variable porosity taken into consideration. A closed-form solution based on the method of matched asymptotic expansions is obtained for the axial velocity distribution, and the wall effect on pressure drop is illustrated. The energy equation for the thermally developing flow, with transverse thermal dispersion and variable stagnant thermal conductivity taken into consideration, was solved numerically. To match the predicted temperature distribution with existing experimental data, it is found that a wall function must be introduced to model the transverse thermal dispersion process in order to account for the wall effect on the lateral mixing of fluid. The variations of the local Nusselt number along the streamwise direction in terms of the appropriate parameters are illustrated. The thermal entrance length effect on forced convection in a packed channel is discussed.
64 citations
TL;DR: In this article, the matched asymptotic expansion method was applied to several long-wave problems including the scattering of acoustic waves by a grating of cylinders and the scatter of water waves incident on horizontal cylinders.
Abstract: The method of matched asymptotic expansions is applied to several long-wave problems including the scattering of acoustic waves by a grating of cylinders and the scattering of water waves incident on horizontal cylinders. It is shown that a naive application of the method can lead to incorrect results. A modified expansion procedure is developed and applied to a number of problems.
60 citations
TL;DR: In this article, the relationship between the regulator problem for a singularly perturbed system and the analogous one for a descriptor system is investigated, and a decomposition approach is provided for solution of the linear quadratic regulator problem.
Abstract: The relationship between the regulator problem for a singularly perturbed system and the analogous one for a descriptor system is investigated. Meanwhile, a decomposition approach is provided for solution of the linear quadratic regulator problem for singularly perturbed systems. This approach is a significant advance over the two-stage method. >
56 citations
TL;DR: In this article, the idea of adding harbor walls to improve the performance of wave-energy devices is studied for the case when the harbours are set into a reflecting wall and an approximate theory is derived based on a simple application of the method of matched asymptotic expansions.
41 citations
TL;DR: The study of singularly perturbed initial value problems for nonlinear systems of ordinary differential equations parallels the analysis underlying the development of numerical algorithms for obtaining solutions to systems of stiff differential equations as discussed by the authors.
Abstract: The study of singularly perturbed initial value problems for nonlinear systems of ordinary differential equations parallels the analysis underlying the development of numerical algorithms for obtaining solutions to systems of stiff differential equations. This paper seeks to emphasize the advantages of combining these two substantial research efforts. It develops insight and intuition based on a sequence of solvable model problems, and it relates a variety of literature scattered throughout asymptotic and numerical analyses, stability and control theory and specific topics in applied mathematical modeling.
39 citations
TL;DR: In this paper, a new method of asymptotic integration is developed -the method of regularization - in the case when the spectrum of the variable limit operator is zero at isolated points.
Abstract: A new method of asymptotic integration is developed - the method of regularization - in the case when the spectrum of the variable limit operator is zero at isolated points. To describe the singular dependence of a solution on the perturbation, additional independent variables are introduced; the space of resonance-free solutions is introduced, in which the coefficients of regularized kind (the solution of the extended problem) are defined. Asymptotic convergence of the series thus obtained to the exact solution of the original singularly perturbed problem is proved.Bibliography: 14 titles.
17 citations
16 citations
TL;DR: The concept of asymptotic proportionality and conditional equality was introduced in this paper to make global stability statements for time-heterogeneous difference and differential equations, and it has been applied to a general sublinear difference equation in an ordered Banach space and, for illustration, several well-known differential equations modelling the spread of gonorrhea in a heterogeneous population.
14 citations
TL;DR: An adaptation of the decomposition method (Adomian, 1986) allows asymptotic solutions for differential and partial differential equations as discussed by the authors, which is a special case of the Adomian decomposition.
TL;DR: In this article, special methods with interface point are presented to deal with some singularly perturbed initial value and boundary value problems in ordinary differential equations, and good numerical results have been obtained and compared with other methods.
Abstract: Special methods with interface point are presented to deal with some singularly perturbed initial—value and boundary—value problems in ordinary differential equations. Good numerical results have been obtained and compared with other methods.
TL;DR: A new mathematical technique to describe population density functions by way of matched asymptotic expansions and the method of multiple scales is introduced and illustrated by application to population densities at both the metropolitan and regional levels.
Abstract: This paper introduces a new mathematical technique to describe population density functions. Two length scales which characterize the variation of these density functions within a region are identified. A differential equation is derived and asymptotic solutions obtained. Two specific techniques the method of matched asymptotic expansions and the method of multiple scales are introduced and illustrated by application to population densities at both the metropolitan and regional levels. (EXCERPT)
TL;DR: The laminar natural convective heat transfer near a rectangular corner formed by the intersection of two vertical quarter-infinite flat plates is considered in this paper, where the boundary-layer equations in the corner layer are derived and appropriate boundary conditions are determined using the method of matched asymptotic expansions.
01 Jan 1988
TL;DR: In this article, the behavior of metal oxide semiconductor field effect transistors (MOSFETs) with small aspect ratio and large doping levels is analyzed using formal perturbation techniques.
Abstract: The behavior of metal oxide semiconductor field effect transistors (MOSFETs) with small aspect ratio and large doping levels is analyzed using formal perturbation techniques. Formally, we will show that in the limit of small aspect ratio there is a region in the middle of the channel under the control of the gate where the potential is one-dimensional. The influence of interface and internal layers in the one-dimensional potential on the averaged channel conductivity is closely examined in the large doping limit. The interface and internal layers that occur in the one-dimensional potential are resolved in the limit of large doping using the method of matched asymptotic expansions. The asymptotic potential in the middle of the channel is constructed for various classes of variable doping models including a simple doping model for the built-in channel device. Using the asymptotic one-dimensional potential, the asymptotic mobile charge, needed for the derivation of the long-channel I-V curves, is found by using standard techniques in the asymptotic evaluation of integrals. The formal asymptotic approach adopted not only provides a pointwise description of the state variables, but by using the asymptotic mobile charge, the lumped long-channel current-voltage relations, which vary uniformly across the various bias regimes, can be found for various classes of variable doping models.
Using the explicit solutions of some free boundary problems solved by Howison and King (1988), the two-dimensional equilibrium potential near the source and drain is constructed asymptotically in strong inversion in the limit of large doping. From the asymptotic potential constructed near the source and drain, a uniform analytical expression for the mobile charge valid throughout the channel is obtained. From this uniform expression for the mobile charge, we will show how it is possible to find the I-V curve in a particular bias regime taking into account the edge effects of the source and drain. In addition, the asymptotic potential for a two-dimensional n+-p junction is constructed.
TL;DR: In this paper, the singularly perturbed boundary value problem is considered and an asymptotic expansion of the solution for vanishing viscosity is developed, which is an example of a singular-singular perturbation problem.
TL;DR: In this paper, the general expansion produced in the companion paper (J. Math. Anal. Appl. 133 (1988)) is specialized to obtain expansions in terms of some special functions.
TL;DR: In this article, an alternative approach to the method of asymptotic expansions for the study of a singularly perturbed linear system with multiparameters and multiple time scales is developed.
TL;DR: In this paper, three term asymptotic expansions for R far from the origin were obtained for distributions in linear boundary crossing problems, where the renewal measure R = ΣP*n where n denotes convolution.
Abstract: Let P be a distribution in the plane and define the renewal measure R=ΣP*nwhere * denotes convolution. The main results of this paper are three term asymptotic expansions for R far from the origin. As an application, expansions are obtained for distributions in linear boundary crossing problems.
TL;DR: In this article, the authors show that the results obtained in this way do not reliably determine whether a spacetime is stable to perturbations which evolve according to the full and exact Einstein equations.
Abstract: Barrow and Sonoda (1986) have investigated the stability, at large time, of certain Bianchi universes. Their method involves studying a set of first-order differential equations which governs the evolution of the three principal expansion rates, and the conservation equations. This set of equations is not in general equivalent to Einstein's equations; in some cases it may be a subset, or an asymptotic approximation. The results given by Barrow and Sonoda refer to the stability of exact solutions of the Einstein equations with respect to perturbations which are governed by this set of first-order equations, rather than by Einstein's equations. The authors show that the results obtained in this way do not reliably determine whether a spacetime is stable to perturbations which evolve according to the full and exact Einstein equations.
07 Dec 1988
TL;DR: In this paper, a class of uncertain singularly perturbed control systems described by ordinary differential equations is considered and the uncertainties are characterized deterministically: the singular perturbation is characterized by a real nonnegative system parameter mu.
Abstract: A class of uncertain singularly perturbed control systems described by ordinary differential equations is considered. The uncertainties are characterized deterministically: the singular perturbation is characterized by a real nonnegative system parameter mu . For mu =0, the system order is lower than that for mu >0. Based only on information available on the uncertain reduced-order ( mu =0), controllers are proposed which assure that the behavior of the feedback-controlled reduced-order system is close to that of global uniform asymptotic stability about zero. Subject to the same controllers, the full-order system ( mu >0) has the same qualitative behavior, provided mu 0 can be computed from the available information. >
01 Oct 1988
TL;DR: In this paper, a parallel algorithm for the efficient solution of a time dependent reaction convection diffusion equation with small parameter on the diffusion term is presented, which is based on a domain decomposition that is dictated by singular perturbation analysis.
Abstract: A parallel algorithm for the efficient solution of a time dependent reaction convection diffusion equation with small parameter on the diffusion term will be presented. The method is based on a domain decomposition that is dictated by singular perturbation analysis. The analysis is used to determine regions where certain reduced equations may be solved in place of the full equation. Parallelism is evident at two levels. Domain decomposition provides parallelism at the highest level, and within each domain there is ample opportunity to exploit parallelism. Run-time results demonstrate the viability of the method. * Research conducted while in residence at the Center for Supercomputing Research and Development, University of Illinois supported in part by the National Science Foundation under Grant No. US NSF PIP-8410110, the U.S. Department of Energy under Grant No. US DOE-DE-FG02-85ER25001, the Air Force Office of Scientific Research under Grant No. AFOSR-85-0211, the IBM Donation to CSRD_ and by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. W-7405-Eng48. Research was also partially supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18605 while in residence at the Institute for Computer Applications in Science and Engineering (ICASE}. t Work supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy under Contracts W-31-109-Eng-38, DE-AC05-840R21400.
TL;DR: In this article, the authors derived asymptotic expansions of the distributions of test statistics for over-identifying restrictions in a system of simultaneous equations under the null and the non-null hypotheses.
Abstract: We derive asymptotic expansions of the distributions of test statistics for over-identifying restrictions in a system of simultaneous equations under the null and the non-null hypotheses. We investigate the effects of the normality assumption for disturbances on the test statistics based on their asymptotic expansions. We also study the power functions of test statistics based on their asymptotic expansions. After modifying their critical regions to the same significance level, the power function of Basmann's statistic is larger than that of the likelihood ratio test when the variance of disturbances is sufficiently small. However, the difference in powers of the two test statistics disappears as the sample size grows larger.
TL;DR: In this article, the effect of a thermal gradient on the transverse vibration of a prestressed rectangular plate is investigated by the method of matched asymptotic expansions, and the eigenvalues of an ideal membrane are obtained, independent of thermal effects.
Abstract: The effect of a thermal gradient on the transverse vibration of a prestressed rectangular plate is investigated by the method of matched asymptotic expansions. This class of heated plate is characterised by changing its Young's modulus with temperature. Analytical results for the eigenvalues are presented for fully-clamped and fully-hinged rectangular plates when the bending rigidity is small compared to the in-plane loading. To leading order in ɛ (where ɛ2 denotes the normalized bending rigidity), the eigenvalues of an ideal membrane are obtained, independent of thermal effects.
Journal Article•
TL;DR: Some approximate solutions of the nonlinear equations were considered in this article, where the approximate solution is exact for a particular initial value of each differential equation, i.e., the value of a particular value is constant.