Showing papers on "Method of matched asymptotic expansions published in 2005"
TL;DR: In this paper, the existence, stability, and pulse-splitting behavior of spike patterns in the one-dimensional Gray-Scott model on a finite domain is analyzed in the semi-strong spike-interaction regime.
89 citations
TL;DR: In this paper, asymptotic expansions of the solution of the associated Cauchy problem for parabolic partial differential equation are obtained and the desired error bounds are derived and used to analyze related limit distributions of normalized integral functionals.
87 citations
TL;DR: In this paper, the stability properties of symmetric k-spike equilibrium solutions to the Gray-Scott model on a bounded one-dimensional domain are analyzed with respect to the large eigenvalues >.
Abstract: In a singularly perturbed limit of small diffusivity e of one of the two chemical species, equilibrium spike solutions to the Gray-Scott (GS) model on a bounded one-dimensional domain are constructed asymptotically using the method of matched asymptotic expansions. The equilibria that are constructed are symmetric k-spike patterns where the spikes have equal heights. Two distinguished limits in terms of a dimensionless parameter in the reaction-diffusion system are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime, the solution branches of k-spike equilibria are found to have a saddle-node bifurcation structure. The stability properties of these branches of solutions are analyzed with respect to the large eigenvalues >. in the spectrum of the linearization. These eigenvalues, which have the property that \ = O(1) as e → 0, govern the stability of the solution on an 0(1) time scale. Precise conditions, in terms of the nondimensional parameters, for the stability of symmetric k-spike equilibrium solutions with respect to this class of eigenvalues are obtained. In the low feed-rate regime, it is shown that a large eigenvalue instability leads either to a competition instability, whereby certain spikes in a sequence are annihilated, or to an oscillatory instability (typically synchronous) of the spike amplitudes as a result of a Hopf bifurcation. In the intermediate regime, it is shown that only oscillatory instabilities are possible, and a scaling-law determining the onset of such instabilities is derived. Detailed numerical simulations are performed to confirm the results of the stability theory. It is also shown that there is an equivalence principle between spectral properties of the GS model in the low feed-rate regime and the Gierer-Meinhardt model of morphogenesis. Finally, our results are compared with previous analytical work on the GS model.
66 citations
01 Jan 2005
TL;DR: In this paper, a stability theory for spatially periodic patterns on R was developed for a class of singularly perturbed reaction-diffusion equations that can be represented by the generalized Gierer-Meinhardt equations as "normal form".
Abstract: In this paper we develop a stability theory for spatially periodic patterns on R. Our approach is valid for a class of singularly perturbed reaction-diffusion equations that can be represented by the generalized Gierer-Meinhardt equations as 'normal form'. These equations exhibit a large variety of spatially periodic patterns. We construct an Evans function
43 citations
TL;DR: In this article, a mixed boundary-value problem for the Poisson equation in a plane two-level junction is considered, where the thin rods from each level are e-periodically alternated.
Abstract: We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ωe, which is the union of a domain Ω0 and a large number 2N of thin rods with variable thickness of order . The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are e-periodically alternated. The Robin conditions are given on the lateral boundaries of the thin rods. Using the method of matched asymptotic expansions, we construct the asymptotic approximation for the solution as e → 0 and prove the corresponding estimates in the Sobolev space H1(Ωe).
40 citations
TL;DR: The numerical method constructed for singularly perturbed semilinear differential equations with a discontinuous source term is shown to be uniformly convergent with respect to the singular perturbation parameter.
Abstract: In this paper singularly perturbed semilinear differential equations with a discontinuous source term are examined. A numerical method is constructed for these problems which involves an appropriate piecewise-uniform mesh. The method is shown to be uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented that validate the theoretical results.
38 citations
TL;DR: In this article, a connection between matched asymptotic expansions and geometric singular perturbation theory is established in the context of the simple fold problem, where the blow-up technique is used to derive asymptic expansions of slow manifolds continued beyond the fold point.
Abstract: The method of matched asymptotic expansions and geometric singular perturbation theory are the most common and successful approaches to singular perturbation problems. In this work we establish a connection between the two approaches in the context of the simple fold problem. Using the blow-up technique [5], [12] and the tools of geometric singular perturbation theory we derive asymptotic expansions of slow manifolds continued beyond the fold point. Our analysis explains the structure of the expansion and gives an algorithm for computing its coefficients.
38 citations
01 Jan 2005
TL;DR: In this article, the asymptotic structure of the stabilizing solution of Riccati equation and the optimal feedback gain for a linear quadratic optimization problem associated to a singularly perturbed stochastic linear system were investigated.
Abstract: In this note the asymptotic structure of the stabilizing solution of Riccati equation and the asymptotic structure of the optimal feedback gain for a linear quadratic optimization problem associated to a singularly perturbed stochastic linear system were investigated. To obtain the dominant part of the stabilizing solution of algebraic Riccati equation and the dominant part of the optimal feedback gain, the solutions of two linear quadratic optimization problems of lower dimension and independent of the small param- eter e are involved. Keywords: Linear quadratic optimization problem, Singular perturbations, Linear sto- chastic systems, Algebraic Riccati equations, Asymptotic structure
37 citations
TL;DR: In this paper, the asymptotic behaviour of the frequencies of eigenvibrations of the membrane as the small parameter (which char- acterizes the diameter and density of the concentrated masses) tends to zero is studied.
Abstract: We consider vibrations of a membrane which contains many "light" concentrated masses on the boundary. We study the asymptotic behaviour of the frequencies of eigenvibrations of the membrane as the small parameter (which char- acterizes the diameter and density of the concentrated masses) tends to zero. We construct asymptotic expansions of eigenelements of the corresponding problems and carefully justify these expansions.
36 citations
TL;DR: Four complementary asymptotic expansions are derived which approximate the incomplete gamma functions @C(a,z) and @c(a-z) for large values of their variables a and z with simpler structure than other expansions previously given in the literature.
30 citations
TL;DR: In this paper, the asymptotic behavior of solution for the initial boundary value problems of reaction diffusion equations with boundary perturbation is studied using the theory of differential inequalities under suitable conditions.
Abstract: A nonlinear singularly perturbed problems for reaction diffusion equation with boundary perturbation is considered. Under suitable conditions, the asymptotic behavior of solution for the initial boundary value problems of reaction diffusion equations is studied using the theory of differential inequalities.
TL;DR: In this article, an alternative method to matched asymptotic expansions for the construction of approximate solutions of the Cahn-Hilliard equation suitable for the study of its sharp interface limit is presented.
Abstract: We develop an alternative method to matched asymptotic expansions for the construction of approximate solutions of the Cahn-Hilliard equation suitable for the study of its sharp interface limit. The method is based on the Hilbert expansion used in kinetic theory. Besides its relative simplicity, it leads to calculable higher order corrections to the interface motion.
TL;DR: In this article, four examples of delayed systems which occur in practical models are considered: the delayed recruitment equation, relaxation oscillations in stem cell control, the delayed logistic equation, and density wave oscillation in boilers, the last being a prob- lem of concern in engineering two-phase flows.
Abstract: Asymptotic methods for singularly perturbed delay differential equations are in many ways more chal- lenging to implement than for ordinary differential equations. In this paper, four examples of delayed systems which occur in practical models are considered: the delayed recruitment equation, relaxation oscillations in stem cell control, the delayed logistic equation, and density wave oscillations in boilers, the last of these being a prob- lem of concern in engineering two-phase flows. The ways in which asymptotic methods can be used vary from the straightforward to the perverse, and illustrate the general technical difficulties that delay equations provide for the central technique of the applied mathematician.
TL;DR: In this article, asymptotic properties of higher-order, nonlinear, Emden-Fowler type differential equations were studied. But the authors used the method of change of variables, which allows one to reduce the initial equation of order n to a dynamical system on the (n-1)-dimensional compact sphere.
Abstract: We study asymptotic properties of a higher-order, nonlinear, Emden-Fowler-type differential equation.We investigate asymptotics of all possible solutions of the equation in the cases of regular and singular nonlinearity for n=3 , 4.We use the method of change of variables,which allows one to reduce the initial equation of order n to a dynamical system on the (n-1)-dimensional compact sphere.
TL;DR: In this article, the two-dimensional turbulent wall jet on a flat surface without free stream is analyzed at a large Reynolds number, using the method of matched asymptotic expansions.
Abstract: The two-dimensional turbulent wall jet on a flat surface without free stream is analysed at a large Reynolds number, using the method of matched asymptotic expansions. The open mean equations of the turbulent boundary layer are analysed in the wall and wake layers by the method of matched asymptotic expansions and the results are matched by the Izakson–Millikan–Kolmogorov hypothesis. In the overlap region, the outer wake layer is governed by the velocity defect law (based on U m , the maximum velocity) and the inner layer by the law of the wall. It is shown that the overlap region possesses a non-unique solution, where the power law region simultaneously exists along with the log law region. Analysis of the power law and log law solutions in the overlap region leads to self-consistent relations, where the power law index, α , is of the order of the non-dimensional friction velocity and the power law multiplication constant, C , is of the order of the inverse of the non-dimensional friction velocity. The lowest order wake layer equation has been closed with generalized gradient transport model and a closed form solution is obtained. A comparison of the theory with experimental data is presented.
Journal Article•
TL;DR: A survey of the result concerning the asymptotic behavior of the solutions of △u = f(u) in D which blow up at the boundary is given in this article.
Abstract: In this paper we give a survey of the result concerning the asymptotic behavior of the solutions of △u = f(u) in D which blow up at the boundary. We concentrate ourselves to the case where the blowup occurs on the whole boundary. The main tools to derive sharp estimates are the comparison principle and the method of upper and lower solutions. A list of references is given which are closely related to the specific aspects discussed in this survey. This list is by no means complete. 2000 Mathematics Subject Classification. 35B40, 35J25, 35J65.
TL;DR: In this article, the authors give an asymptotic equivalent at infinity of the unbounded solutions of some boundary layer equations arising in fluid mechanics, where the solution of each boundary layer equation can be found at infinity.
Abstract: We give an asymptotic equivalent at infinity of the unbounded solutions of some boundary layer equations arising in fluid mechanics.
TL;DR: In this paper, the authors illustrate how to obtain hyperasymptotic expansions for solutions of nonlinear ordinary differential equations using transseries expansions and the Riemann sheet structure of the Borel transform of the divergent asymptotic expansion.
Abstract: We illustrate how one can obtain hyperasymptotic expansions for solutions of nonlinear ordinary differential equations. The example is a Riccati equation. The main tools that we need are transseries expansions and the Riemann sheet structure of the Borel transform of the divergent asymptotic expansions. Hyperasymptotic expansions determine the solutions uniquely. A numerical illustration is included.
TL;DR: Asymptotic stability and the complex stability radius of a class of singularly perturbed systems of linear differential-algebraic equations (DAEs) are studied and can be extended to other singular perturbation problems for DAEs of more general form.
TL;DR: In this article, the matched asymptotic expansions were used to construct solutions for the planar steady flow of Oldroyd-B fluids around reentrant corners of angles / (1/2 < 1).
Abstract: The method of matched asymptotic expansions is used to construct solutions for the planar steady flow of Oldroyd-B fluids around re-entrant corners of angles / (1/2<1). Two types of similarity solu...
TL;DR: By means of three examples, it is shown that this method provides uniformly convergent solutions with respect to both the time step and the small perturbation parameter, and these solutions are more accurate and exhibit a higher order of convergence than those obtained with an upwind finite difference scheme in a piecewise uniform mesh that is boundary-layer resolving.
TL;DR: It is shown that this method provides uniformly convergent solutions with respect to the small perturbation parameter, but, if the time step is sufficiently large, then the roots of the characteristic polynomial that defines the exponential solutions of the homogeneous differential equation may become nearly independent of the timestep and the solution may exhibit large errors.
TL;DR: In this article, an asymptotic theory for internal reflection in the plane elastic waveguide, slowly varying along one of the longitudinal directions, is developed by the method of matched ASM expansions.
Journal Article•
TL;DR: In this article, special solutions for linear functional differential equations with small delays are studied. But the importance of special solutions relies on the fact that, under some conditions, they can be used to describe the asymptotic behaviour of all solutions.
Abstract: The present paper deals with special solutions for linear functional differential equations with small delays. The importance of special solutions relies on the fact that, under some conditions, they can be used to describe the asymptotic behaviour of all solutions. Sufficient conditions for the existence of special solutions for nonautonomous equations in $\mathbb R^n$ are given. For autonomous equations in $\mathbb R^n$, results relating special solutions and characteristic values are presented. We also consider linear autonomous functional differential equations in Banach spaces, and use special solutions to study the asymptotic behaviour of their solutions. Several applications are given.
TL;DR: A theory for surface modes at the nematic-isotropic interface in thermotropic nematogen-non-nematogen mixtures is developed, using the dynamical generalization of the Landau-de Gennes model for the orientational (nonconserved) order parameter, coupled with the Cahn-Hilliard equation for concentration (conserved parameter).
Abstract: We develop a theory for surface modes at the nematic-isotropic interface in thermotropic nematogen--non-nematogen mixtures. We employ the dynamical generalization of the Landau--de Gennes model for the orientational (nonconserved) order parameter, coupled with the Cahn-Hilliard equation for concentration (conserved parameter), and include hydrodynamic degrees of freedom. The theory uses a generalized form of the Landau--de Gennes free-energy density to include the coupling between the concentration of the non-nematogen fluid and the orientational order parameter. Two representative phase diagrams are shown. The method of matched asymptotic expansions is used to obtain a generalized dispersion relation. Further analysis is made in particular cases. Orientational order parameter relaxation dominates in the short-wavelength limit, while in the long-wavelength limit viscous damping processes become important. There is an intermediate region (depending on the temperature) in which the interaction between conserved parameter dynamics and hydrodynamics is important.
TL;DR: In this paper, asymptotic expansions for a one-phase soliton-type solution of the Korteweg-de Vries equation with coefficients depending on a small parameter were constructed.
Abstract: We construct asymptotic expansions for a one-phase soliton-type solution of the Korteweg-de Vries equation with coefficients depending on a small parameter.
01 Jan 2005
TL;DR: In this paper, the global almost sure asymptotic stability of the trivial solution of nonlinear stochastic difference equations with in-the-arithmetic-meansense monotone drift part and diffusive part driven by independent (but not necessarily identically distributed) random variables is proven under appropriate conditions in IR1.
Abstract: Global almost sure asymptotic stability of the trivial solution of some nonlinear stochastic difference equations with in-the-arithmetic-meansense monotone drift part and diffusive part driven by independent (but not necessarily identically distributed) random variables is proven under appropriate conditions in IR1. This result can be used to verify asymptotic stability of stochastic-numerical methods such as partially drift-implicit trapezoidal methods for nonlinear stochastic differential equations with variable step sizes.
TL;DR: It is shown that the smooth locally-analytical method is more precise than second-order accurate finite difference discretizations and depends on the kind of nonlinearity and inhomogeneities of singularly perturbed ordinary differential equations, but is always higher than that of exponentially-fitted techniques based on the local solution of advection-diffusion operators.
TL;DR: In this paper, the authors studied the asymptotic expansion for solution of singularly perturbed equation for functional Markovian evolution in road traffic and found the view of regular and singular parts of solution.
Abstract: We study the asymptotic expansion for solution of singularly perturbed equation for functional of Markovian evolution in Rd. The view of regular and singular parts of solution is found.