Showing papers on "Method of matched asymptotic expansions published in 1999"
TL;DR: In this article, the asymptotic behavior of solutions for parabolic non-linear evolution equations in R n is studied and the existence of the global attractor in L 2 (R n ) is established.
272 citations
TL;DR: In this article, the authors use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a hyperasymptotic approximation.
Abstract: Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter e which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.
261 citations
Book•
28 Oct 1999
TL;DR: In this paper, a boundary value problem for the Laplacian in a multi-structure is introduced. But the boundary value is not a boundary-value problem for 3D-1D multi-structures.
Abstract: 1. Introduction to compound asymptotic expansions 2. A boundary value problem for the Laplacian in a multi-structure 3. Auxiliary facts from mathematical elasticity 4. Elastic multi-structure 5. Non-degenerate elastic multi-structure 6. Spectral analysis for 3D-1D multi-structures Bibliographical remarks Bibliography Index
161 citations
Book•
01 Jan 1999
TL;DR: In this paper, the authors propose an approximate solution of linear and nonlinear differential equations for boundary layer theory, and asymptotic expansion of Integrals for WKB theory.
Abstract: I Preface. 1 Ordinary Differential Equations. 2 Difference Equations. 3 Approximate Solution of Linear Differential Equations. 4 Approximate Solution of Nonlinear Equations. 5 Approximate Solution of Difference Equations. 6 Asymptotic Expansion of Integrals. 7 Perturbation Series. 8 Summation of Series. 9 Boundary Layer Theory. 10 WKB Theory. 11 Multiple Scales Analysis. Appendix, References, Index
122 citations
TL;DR: The computational singular perturbation method is employed for the solution of stiff PDEs and for the acquisition of the most important physical understanding and the usefulness of the method is demonstrated by analyzing a transient reaction-diffusion problem.
Abstract: The computational singular perturbation (CSP) method is employed for the solution of stiff PDEs and for the acquisition of the most important physical understanding. The usefulness of the method is demonstrated by analyzing a transient reaction-diffusion problem. It is shown that in the regions where the solution exhibits smooth spatial slopes, a simple nonstiff system of equations can be used instead of the full governing equations. From the simplified system, which is numerically provided by CSP and whose structure varies with space and time, important physical information comes to light. The relation of this method to the class of asymptotic expansion methods is explored. It is shown that the CSP results are identical to the ones obtained by the asymptotic methods. The identifications of the nondimensional parameters and the tedious manipulations needed by the asymptotic methods are performed by programmable numerical or analytic computations specified by CSP. Preliminary numerical results are presented validating the theoretical aspects of the proposed algorithm and providing a measure of its usefulness and its accuracy.
113 citations
TL;DR: The authors' attention is focused on the design of a composite linear controller based on the slow and fast problems such that both stability and a prescribed H/sub /spl infin// performance for the full-order system are achieved.
Abstract: This paper deals with the problem of control of singularly perturbed linear continuous-time systems. The authors' attention is focused on the design of a composite linear controller based on the slow and fast problems such that both stability and a prescribed H/sub /spl infin// performance for the full-order system are achieved. The asymptotic behavior of the composite controller is studied, which is independent of the singular perturbation /spl epsiv/ when /spl epsiv/ is sufficiently small. Furthermore, the problem of robust control for the above system with parameter uncertainty is also investigated.
96 citations
TL;DR: A bootstrapping algorithm for the extraction of potentials and arbitrary derivatives of the Cauchy data of regular three-dimensional second order elliptic boundary value problems in connection with corresponding boundary integral equations.
Abstract: In this paper we develop and analyze a bootstrapping algorithm for the extraction of potentials and arbitrary derivatives of the Cauchy data of regular three-dimensional second order elliptic boundary value problems in connection with corresponding boundary integral equations. The method rests on the derivatives of the generalized Green's representation formula, which are expressed in terms of singular boundary integrals as Hadamard's finite parts. Their regularization, together with asymptotic pseudohomogeneous kernel expansions, yields a constructive method for obtaining generalized jump relations. These expansions are obtained via composition of Taylor expansions of the local surface representation, the density functions, differential operators and the fundamental solution of the original problem, together with the use of local polar coordinates in the parameter domain. For boundary integral equations obtained by the direct method, this method allows the recursive numerical extraction of potentials and their derivatives near and up to the boundary surface.
51 citations
TL;DR: In this paper, the authors developed asymptotic expansions for solutions of integro-differential equations arising from transition densities of singularly perturbed switching-diffusion processes.
45 citations
TL;DR: In this article, a system of equations which models the formation of clusters by coagulation, with particles of unit size being injected at a time-dependent rate, was studied, and the criteria under which gelation occurs were the same as for the constant mass and constant monomer cases, which have been studied previously.
Abstract: We study a system of equations which models the formation of clusters by coagulation, with particles of unit size being injected at a time-dependent rate We observe that the criteria under which gelation occurs are the same as for the constant mass and constant monomer cases, which have been studied previously We identify a variety of types of behaviour in the large-time limit, depending on the coagulation kernel and on the rate at which monomer is introduced into the system The results are obtained by means of exact (generating function) techniques, matched asymptotic expansions and numerical simulations
41 citations
TL;DR: This paper shows how to decompose the singularly perturbed algebraic Riccati equation and the corresponding linear-quadratic optimal control problem at steady state in terms of reduced-order pure-slow and pure-fast problems by using the eigenvector approach.
38 citations
TL;DR: In this article, the authors investigated the asymptotic properties of the inhomogeneous nonautonomous evolution equation (d/dt)u(t), where u(t) is a constant.
Abstract: We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation (d/dt)u(t)=Au(t)
TL;DR: In this paper, the front instability sets in when the order of the reaction, m, exceeds some threshold, m c (;τ), that depends on the inverse of the Lewis number, τ.
Abstract: Travelling waves in a model for autocatalytic reactions have, for some parameter regimes, been suggested to have oscillatory instabilities. This instability is confirmed by various methods, including linear stability analysis (exploiting Evan9s function) and direct numerical simulations. The front instability sets in when the order of the reaction, m , exceeds some threshold, m c (;τ), that depends on the inverse of the Lewis number, τ. The stability boundary, m = m c (;τ), is found numerically for m order one. In the limit m ≫ 1 (in which the system becomes similar to combustion systems with Arrhenius kinetics), the method of matched asymptotic expansions is employed to find the asymptotic front speed and show that m c ∼ (;τ – 1) –1 as ;τ → 1. Just beyond the stability boundary, the unstable rocking of the front saturates supercritically. If the order is increased still further, period–doubling bifurcations occur, and, for small tau there is a transition to chaos through intermittency after the disappearance of a period–4 orbit.
TL;DR: Both regular and multiple scales asymptotic techniques are used for the analysis of the Euler forward difference method, and the resulting asymPTotic expansions are verified for several nonlinear, autonomous, ordinary differential equations.
TL;DR: In this paper, the authors considered singularly perturbed systems involving both diffusions and pure jump processes and developed asymptotic expansions for the transition density vectors via a constructive method.
Abstract: Motivated by many problems in optimization and control, this paper is concerned with singularly perturbed systems involving both diffusions and pure jump processes. Two models are treated. In the first model, the jump process changes very rapidly by comparison with the diffusion processes. In the second model, the diffusions change rapidly in comparison with the jump process. Asymptotic expansions are developed for the transition density vectors via a constructive method; justification of the asymptotic expansions and analysis of the remainders are provided.
TL;DR: In this paper, the two dimensional flow of a power-law fluid is studied analytically using a singular perturbation method in order to determine the residual liquid film thickness of powerlaw fluids on the wall of a circular tube or a rectangular channel when displaced by another immiscible fluid.
TL;DR: The sign patterns of the coefficients suggest open problems pertaining to error bounds for the asymptotic expansions of the zeros and stationary values of the Airy functions.
Abstract: The general theories of the derivation of inverses of functions from their power series and asymptotic expansions are discussed and compared. The asymptotic theory is applied to obtain asymptotic expansions of the zeros of the Airy functions and their derivatives, and also of the associated values of the functions or derivatives. A Maple code is constructed to generate exactly the coefficients in these expansions. The only limits on the number of coefficients are those imposed by the capacity of the computer being used and the execution time that is available.
The sign patterns of the coefficients suggest open problems pertaining to error bounds for the asymptotic expansions of the zeros and stationary values of the Airy functions.
TL;DR: A mathematical model of a steady-state diffusion process through a periodic membrane for a wide class of periodic membranes is dealt with and formulas used in physical, chemical, and biological investigations to describe effective membrane properties are analyzed.
Abstract: The paper deals with a mathematical model of a steady-state diffusion process through a periodic membrane. For a wide class of periodic membranes, we define the effective permeability and obtain upper and lower estimates of the effective permeability. For periodic membranes made from two materials with different absorbing properties, we study the asymptotic behavior of the effective permeability when the fraction of one material tends to zero (low concentration asymptotics). When the low fraction material forms homothetically vanishing disperse periodic inclusions in the host material, low concentration approximations are built by the method of matched asymptotic expansions. We also show that our results are consistent with those which can be obtained by a boundary homogenization. Finally, we analyze formulas used in physical, chemical, and biological investigations to describe effective membrane properties.
TL;DR: In this paper, it was shown that the decay rate of the generalized KP equations with a power nonlinearityupux and with a Burgers-type dissipation in the main direction of propagation was asymptotic.
TL;DR: Melenk and Schwab as discussed by the authors considered a singularly perturbed equation of elliptic-elliptic type in two dimensions and provided asymptotic expansions of the solution and new error bounds that are uniform in the perturbation parameter as well as in the expansion order.
Abstract: A singularly perturbed equation of elliptic-elliptic type in two dimensions is considered. We assume analyticity of the input data, i.e., the boundary of the domain is an analytic curve, the boundary data are analytic, and the right-hand side is analytic. We give asymptotic expansions of the solution and new error bounds that are uniform in the perturbation parameter as well as in the expansion order. Additionally, we provide growth estimates for higher derivatives of the solution where the dependence on the perturbation parameter appears explicitly. These error bounds and growth estimates are used in [J. M. Melenk and C. Schwab, SIAM J. Numer. Anal., 35 (1998), pp. 1520--1557] to construct hp versions of the finite element method which feature robust exponential convergence, i.e., the rate of convergence is exponential and independent of the perturbation parameter $\eps$.
Journal Article•
01 Jan 1999
TL;DR: In this paper, the existence and asymptotic behavior of solution for the boundary value problems are studied using the iteration method and the theory of differential inequalities, using the concept of differential inequality.
Abstract: The singularly perturbed nonlinear problem ey" = f(x, y')y + g(x, y'), 0 x 1, 0 e 1, y(0) = A, y(1) = B, where y, f, g, A, B are n-dimensional vectors is considered. Using the iteration method and the theory of differential inequalities, the existence and asymptotic behavior of solution for the boundary value problems are studied.
TL;DR: In this article, matched asymptotic expansions are used to examine the structure of the positive column at low pressure in electronegative gases using the fluid model to describe the positive-ion motion.
Abstract: The method of matched asymptotic expansions is used to examine the structure of the plasma sheath of the positive column at low pressure in electronegative gases using the fluid model to describe the positive-ion motion. It is shown that at low negative-ion concentrations, and at high concentrations, the structure is that of a plasma joined to a thin sheath, but that for the electron/negative-ion temperature ratio Te/Tn ≡ e > 5 + √24, and for a well-defined range of A ≡ nn0/ne0 (the central negative ion to electron density ratio) and for small Debye length, there is a more complex structure with a central negative-ion-dominated plasma surrounded by a quasiplasma in which density oscillations may occur before joining to a sheath. This is in agreement with recent computations using the same model.
TL;DR: A numerical method for singularly perturbed two-point boundary-value problems for second-order ordinary differential equations without a first derivative term arising in the study of chemical catalysis and Michaelis-Menten process in biology is proposed.
11 Feb 1999
TL;DR: In this article, Hale and Krasovskii proved that the asymptotic stability criterion for non-autonomous functional functions is not right even for ordinary differential equations, and they proposed a new criterion for functional functions that are almost periodic in time.
Abstract: Consider a system of functional differential equations dx/dt = f(t, xt) where f is the vector-valued functional. The classical asymptotic stability result for such a system calls for a positive definite functional V (t, φ) and negative definite functional dV /dt. In applications one can construct a positive definite functional V , whose derivative is not negative definite but is less than or equal to zero. Exactly for such cases J. Hale created the effective asymptotic stability criterion if the functional f in functional differential equations is autonomous (f does not depend on t), and N. N. Krasovskii created such criterion for the case where the functional f is periodic in t. For the general case of the non-autonomous functional f V. M. Matrosov proved that this criterion is not right even for ordinary differential equations. The goal of this paper is to prove this criterion for the case when f is almost periodic in t. This case is a particular case of the class of non-autonomous functionals.
TL;DR: The closure problem for combustion waves arises when applying the method of matched asymptotic expansions for large activation energy to many nonsteady combustion problems.
Abstract: The closure problem for combustion waves arises when applying the method of matched asymptotic expansions for large activation energy to many nonsteady combustion problems. The exponential nature of the dependence of the reaction rate on temperature and the large coefficient (activation energy) in the exponent lead to the first order correction for temperature appearing in the equations at leading order. Equations describing the first order correction involve the second order correction, and so on. These terms can be scaled away for steady solutions, but when considering nonsteady propagation, they remain for any constant scaling of temperature. The closure problem refers to the fact that the equations must be solved at all orders before the leading order solution can be determined. One traditional approach to alleviate the problem is to truncate the series. While sacrificing the distinction between scales of temperature variation ahead of and behind the flame, these methods allow the replacement of the d...
TL;DR: In this paper, a degenerate stable solution for a singularly perturbed scalar differential equation of second order and a scalar DDE of first or second order was proposed.
TL;DR: In this paper, a mixed boundary value problem for a system of two second-order nonlinear differential equations where one equation is singularly perturbed is considered, and the authors prove the existence of a solution of the problem and determine its asymptotic behavior with respect to the small parameter.
TL;DR: In this paper, a linear-quadratic optimal control problem with point and distributed delays in state variables and a small control cost in the performance index is considered, and an asymptotic solution of the singularly perturbed set of Riccati-type functional-differential equations, associated with this problem, is constructed and justified.
Abstract: A linear-quadratic optimal control problem with point and distributed delays in state variables and a small control cost in the performance index is considered. An asymptotic solution of the singularly perturbed set of Riccati-type functional-differential equations, associated with this problem, is constructed and justified. Based on this solution, the suboptimal feedback control law is derived. An accuracy of the corresponding value of the performance index is established.
TL;DR: In this article, a class of singularly perturbed initial boundary value problems for the reaction diffusion integral differential, system are considered, using the iteration, method and the differential inequalities, the existence, uniqueness and its asymptotic behavior of solution for the problem are studied.
Abstract: In this paper a class of singularly perturbed initial boundary value problems for the reaction diffusion integral differential, system are considered. Using the iteration, method and the differential inequalities, the existence, uniqueness and its asymptotic behavior of solution for the initial boundary value problems are studied.
TL;DR: The method of matched asymptotic expansions is used to model the voltage--current characteristics for a given flowrate of fuel and predict the position of the flame region that separates an oxygen-rich region from a fuel- rich region.
Abstract: We analyze a model for the combustion of methane in a planar solid oxide fuel cell. The model includes diffusive and advective transport processes, an electrochemical source of oxygen, and the consumption of oxygen and methane through combustion. The effect of the presence of the reaction products and atmospheric nitrogen is also included. Since the combustion takes place in a narrow gap we are able to reduce the problem from three to two dimensions. After assuming that the flow is steady and axisymmetric, we use the method of matched asymptotic expansions to construct solutions in six asymptotic regions. This allows us to model the voltage--current characteristics for a given flowrate of fuel and predict the position of the flame region that separates an oxygen-rich region from a fuel-rich region. Comparisons show that the theory is in reasonable agreement with experiments.