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On the Asymptotic and Numerical Analysis of Exponentially IllConditioned Singularly Perturbed Boundary Value Problems
, 1995
"... Asymptotic and numerical methods are used to study several classes of singularly perturbed boundary value problems for which the underlying homogeneous operators have exponentially small eigenvalues. Examples considered include the familiar boundary layer resonance problems and some extensions, and ..."
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Cited by 4 (4 self)
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Asymptotic and numerical methods are used to study several classes of singularly perturbed boundary value problems for which the underlying homogeneous operators have exponentially small eigenvalues. Examples considered include the familiar boundary layer resonance problems and some extensions, and certain linearized equations associated with metastable internal layer motion. For the boundary layer resonance problems, a systematic projection method, motivated by the work of De Groen [SIAM J. Math. Anal. 11, (1980), pp. 122], is used to analytically calculate high order asymptotic solutions. This method justifies and extends some previous results obtained from the variational method of Grasman and Matkowsky [SIAM J. Appl. Math. 32, (1977), pp. 588597]. A numerical approach, based on an integral equation formulation, is used to accurately compute boundary layer resonance solutions and their associated exponentially small eigenvalues. For various examples, the numerical results are show...
Exponentially IllConditioned ConvectionDiffusion Equations in MultiDimensional Domains
 in MultiDimensional Domains, submitted, Methods and Appl. of Analy
, 1998
"... The phenomenon of dynamic metastability is analyzed for a class of singularly perturbed linear convectiondiffusion equation in both a onedimensional and a twodimensional domain. The extreme sensitivity,... ..."
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Cited by 2 (2 self)
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The phenomenon of dynamic metastability is analyzed for a class of singularly perturbed linear convectiondiffusion equation in both a onedimensional and a twodimensional domain. The extreme sensitivity,...
Dynamic Metastability and Singular Perturbations
 CRM Proc. Lecture Notes (Ban 95
, 1998
"... . Certain singularly perturbed timedependent partial differential equations exhibit a phenomenon known as dynamic metastability whereby the timedependent solution approaches a steadystate solution only over an an asymptotically exponentially long time interval. This metastable behavior is directl ..."
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Cited by 1 (0 self)
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. Certain singularly perturbed timedependent partial differential equations exhibit a phenomenon known as dynamic metastability whereby the timedependent solution approaches a steadystate solution only over an an asymptotically exponentially long time interval. This metastable behavior is directly related to the occurrence of an asymptotically exponentially small principal eigenvalue for the linearized equation. In this paper, we illustrate metastablebehavior for various classes of perturbedproblems and we show how this behavior can be analyzed asymptotically by supplementing the method of matched asymptotic expansions with certain spectral information associated with the linearized equation. 1. Introduction The method of matched asymptotic expansions is a wellknown and powerful method for systematically calculating asymptotic approximations to solutions of singularly perturbed problems. This method has been used successfully in a wide range of applications (cf. [H], [HO], [KC], [...
Exponential Asymptotics, Boundary Layer Resonance, and Dynamic Metastability
, 1996
"... This paper considers the linear convectiondiffusion equation u t = fflu xx \Gamma xu x , and certain natural generalizations, on fixed bounded spatial domains including the turning point x = 0. For constant boundary values and consistent initial values, asymptotic solutions as ffl ! 0 + converge ..."
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This paper considers the linear convectiondiffusion equation u t = fflu xx \Gamma xu x , and certain natural generalizations, on fixed bounded spatial domains including the turning point x = 0. For constant boundary values and consistent initial values, asymptotic solutions as ffl ! 0 + converge to steady states over asymptotically exponentiallylong time intervals. The occurrence of an asymptotically exponentiallysmall eigenvalue is the reason for such metastability, as well as for the sensitivity of the longtime behavior to small perturbations. The utility of these asymptotic results is illustrated through numerical computations with only moderately small ffl values. 1 Introduction There has been much recent work on the asymptotic solution of exponentially illconditioned boundary value problems for nonlinear singularly perturbed parabolic partial differential equations. Examples of such problems include Burgers equation and the GinzburgLandau equation on bounded spatial doma...