Showing papers on "Method of matched asymptotic expansions published in 2001"

Motion of a drop in a vertical temperature gradient at small Marangoni number the critical role of inertia

Abstract: When a drop moves in a uniform vertical temperature gradient under the combined action of gravity and thermocapillarity at small values of the thermal Peclet number, it is shown that inclusion of inertia is crucial in the development of an asymptotic solution for the temperature field. If inertia is completely ignored, use of the method of matched asymptotic expansions, employing the Peclet number (known as the Marangoni number) as the small parameter, leads to singular behaviour of the outer temperature field. The origin of this behaviour can be traced to the interaction of the slowly decaying Stokeslet, arising from the gravitational contribution to the motion of the drop, with the temperature gradient field far from the drop. When inertia is included, and the method of matched asymptotic expansions is used, employing the Reynolds number as a small parameter, the singular behaviour of the temperature field is eliminated. A result is obtained for the migration velocity of the drop that is correct to O(Re 2 log Re)

Layer resolving grids and transformations for singular perturbation problems

Abstract: Chapter 1 Introduction to singularly perturbed problems: introduction examples of singularly perturbed problems convection-diffusion problems momentum conservation laws Prandtl equations problem of a thin beam problems of the shock wave structure Burger's equation one dimensional steady reaction-diffusion-convection model Orr-Sommerfeld problem diffusion-drift motion problem idealized problems semilinear problem weakly-coupled systems of ordinary differential equations autonomous equation equation with a power function multiplying the second derivative general idealized problem invariants of equations singular functions definition of the singular functions examples of singular functions layer-type functions notion of layers definition of layers examples of layers partition of layers scale of a layer classification of layers basic approaches to analyze problems with a small parameter method of multivariable asymptotic expansions method of matched asymptotic expansions expansion via differential inequalities numerical methods method of layer-damping transformations comments. Chapter 2 Background for qualitative analysis: introduction differential inequalities scalar problems systems of the second order theorems of inverse monotonicity first order equations second order equations requirements imposed on estimates of the derivatives formulation of an optimal univariate transformation necessary bounds for the first derivative bounds on the higher derivatives uniform bounds on the total variation inequality relations comments. Chapter 3 Estimates of the solution derivatives to semilinear problems: introduction initial problem smooth problem nonsmooth terms second order equations strong ellipticity problem with the condition f(x,u) = xg(x,u) problem of population dynamics theory generalization to mixed boundary conditions and dependence on e equation with a power function affecting the second derivative power singularities exponential singularity generalization to elliptic and parabolic equations estimates of the solution derivatives comments. Chapter 4 Problems for ordinary quasilinear equations: introduction autonomous boundary value problem preliminary results boundary layers interior layers nonautonomous equation estimates of the first derivative graphical chart for localizing the layers example of the problem analysis of the limit solution properties of the limit solution comments. (Part contents).

Geometric Analysis of the Singularly Perturbed Planar Fold

Abstract: The geometric approach to singular perturbation problems is based on powerful methods from dynamical systems theory. These techniques have been very successful in the case of normally hyperbolic critical manifolds. However, at points where normal hyperbolicity fails, e.g. fold points or points of self-intersection of the critical manifold, the well developed geometric theory does not apply. We present a method based on blow-up techniques which leads to a rigorous geometric analysis of these problems. The blow-up method leads to problems which can be analysed by standard methods from the theory of invariant manifolds and global bifurcations. The presentation is in the context of a planar singularly perturbed fold. The blow-up used in the analysis is closely related to the rescalings used in the classical analysis based on matched asymptotic expansions. The relationship between these classical results and our geometric analysis is discussed.

Construction and asymptotic stability of structurally stable internal layer solutions

Abstract: We introduce a geometric/asymptotic method to treat structurally stable internal layer solutions. We consider asymptotic expansions of the internal layer solutions and the critical eigenvalues that determine their stability. Proofs of the existence of exact solutions and eigenvalue-eigenfunctions are outlined. Multi-layered solutions are constructed by a new shooting method through a sequence of pseudo Poincare mappings that do not require the transversality of the flow to cross sections. The critical eigenvalues are determined by a coupling matrix that generates the SLEP matrix. The transversality of the shooting method is related to the nonzeroness of the critical eigenvalues. An equivalent approach is given to mono-layer solutions. They can be determined by the intersection of a fast jump surface and a slow switching curve, which reduces Fenichel’s transversality condition to the slow manifold. The critical eigenvalue is determined by the angle of the intersection. We present three examples. The first treats the critical eigenvalues of the system studied by Angenent, Mallet-Paret & Peletier. The second shows that a key lemma in the SLEP method may not hold. The third is a perturbed activator-inhibitor system that can have any number of mono-layer solutions. Some of the solutions can only be found with the new shooting method.

Robust stability of singularly perturbed state feedback systems using unified approach

Abstract: A method for the decomposition of unified state-feedback singularly perturbed systems is presented. Based on the small-gain theorem, a sufficient condition for robust stability is derived such that the composite state feedback renders the closed-loop singularly perturbed system asymptotically stable. With this method there is no need to consider two different approaches for continuous-time and discrete-time domains. An example illustrates the methodology.

The singularly perturbed problem for combustion reaction diffusion

Abstract: A singularly perturbed combustion reaction diffusion Robin boundary value problem is considered. Using the theory of differential ineaqality, the existence of solution to the problem is proved and the asymptotic estimation of the solution is obtained.

Sub-optimal linear quadratic control for singularly perturbed systems

Abstract: Proposes a methodology to design suboptimal controllers for singularly perturbed systems. The controller is given in terms of the solution of a set of inequalities. An algorithm is given to solve those inequalities through LMI (linear matrix inequality) formulation. An example exhibits that the results are very close to the exact optimal solutions. The main features of this approach are that it takes advantage of LMI to deal with singularly perturbed systems with less conservativeness and can be extended to other robust and multi-objective control problems for singularly perturbed systems. In addition, it is suitable for both standard and nonstandard singularly perturbed systems.

Matched asymptotic expansions and the numerical treatment of viscous-inviscid interaction

Abstract: The paper presents a personal view on the history of viscous-inviscid interaction methods, a history closely related to the evolution of the method of matched asymptotic expansions. The main challenge in solving Prandtl's boundary-layer equations has been to overcome the singularity at a point of steady flow separation. Stewartson's triple-deck theory has inspired a solution to this challenge, and thereby it paved the way for industrial use of viscous-inviscid interaction methods.

Motion of a Curved Vortex Filament: Higher-Order Asymptotics

Abstract: Three-dimensional motion of a slender vortex tube, embedded in an inviscid incompressible fluid, is investigated based on the Euler equations Using the method of matched asymptotic expansions in a small parameter ∈, the ratio of core radius to curvature radius, the velocity of a vortex filament is derived to O(∈ 3), whereby the influence of elliptical deformation of the core due to the self-induced strain is taken into account In the localized induction approximation, this is reducible to a completely integrable evolution equation among the localized induction hierarchy

Complete asymptotic expansions of the Fermi–Dirac integrals Fp(η)=1/Γ(p+1)∫0∞[εp/(1+eε−η)]dε

Abstract: The complete asymptotic expansions, that is to say expansions which include any exponentially small terms lying beyond all orders of the asymptotic power series, are calculated for the Fermi–Dirac integrals. We present two methods to accomplish this, the first in the complex plane utilizing Mellin transforms and Hankel’s representation of the gamma function, and the second on the real line using the known asymptotic expansions of the confluent hypergeometric functions. The complete expansions of Fp(η) are then used to investigate the effect that these traditionally neglected exponentially small terms have on physical systems. It is shown that for a 2 dimensional nonrelativistic ideal Fermi gas, the subdominant exponentially small series becomes dominant.

Uniform Asymptotic Expansions of Symmetric Elliptic Integrals

Abstract: Symmetric standard elliptic integrals are considered when two or more parameters are larger than the others. The distributional approach is used to derive seven expansions of these integrals in inverse powers of the asymptotic parameters. Some of these expansions also involve logarithmic terms in the asymptotic variables. These expansions are uniformly convergent when the asymptotic parameters are greater than the remaining ones. The coefficients of six of these expansions involve hypergeometric functions with less parameters than the original integrals. The coefficients of the seventh expansion again involve elliptic integrals, but with less parameters than the original integrals. The convergence speed of any of these expansions increases for an increasing difference between the asymptotic variables and the remaining ones. All the expansions are accompanied by an error bound at any order of the approximation.

On a Class of Singularly Perturbed Parabolic Equations

Abstract: A method of matched asymptotic expansions has been used to construct an n-term uniformly valid approximate solution for un initiol value problem of a bnear singularly pertarted parabolic equation exhibiting an internal layer behavior. It is shown that each internal layer function caused by a non-smooth initial data can be described by an n-th iteroted integral of the complementary error function.

Turning points for adiabatically perturbed periodic equations

Abstract: We make a study of uniform asymptotic solutions of some general adiabatic differential equations on the intervals containing a single turning point or a pair of turning points. We reduce this to the study of models which can be explicitly solved. We apply these results to the case of an adiabatically perturbed differential equation with periodic coefficients.

Rational chebyshev spectral methods for unbounded solutions on an infinite interval using polynomial-growth special basis functions☆☆☆★

Abstract: In the method of matched asymptotic expansions, one is often forced to compute solutions which grow as a polynomial in y as | y | → ∞. Similarly, the integral or repeated integral of a bounded function f ( y ) is generally unbounded also. The k th integral of a function f ( y ) solves . We describe a two-part algorithm for solving linear differential equations on y ϵ [−∞, ∞] where u ( y ) grows as a polynomial as | y | → ∞. First, perform an explicit, analytic transformation to a new unknown v so that v is bounded. Second, expand v as a rational Chebyshev series and apply a pseudospectral or Galerkin discretization. (For our examples, it is convenient to perform a preliminary step of splitting the problem into uncoupled equations for the parts of u which are symmetric and antisymmetric with respect to y = 0, but although this is very helpful when applicable, it is not necessary.) For the integral and interated integrals and for constant coefficient differential equations in general, the Galerkin matrices are banded with very low bandwidth. We derive an improvement on the “last coefficient error estimate” of the author's book which applies to series with a subgeometric rate of convergence, as is normally true of rational Chebyshev expansions.

A class of nonlocal singularly perturbed problems for nonlinear hyperbolic differential equation

Abstract: The nonlocal singularly perturbed problems for the hyperbolic differential equation are considered. Under suitable conditions, using the fixed point theorem, the asymptotic behavior of solution for the initial boundary value problems is studied.

Asymptotic expansions for bounded solutions to semilinear Fuchsian equations.

Abstract: It is shown that bounded solutions to semilinear elliptic Fuchsian equations obey complete asymptotic expansions in terms of powers and logarithms in the distance to the boundary. For that purpose, Schulze’s notion of asymptotic type for conormal asymptotic expansions near a conical point is refined. This in turn allows to perform explicit computations on asymptotic types — modulo the resolution of the spectral problem for determining the singular exponents in the asymptotic expansions. 2000 Mathematics Subject Classification: Primary: 35J70; Secondary: 35B40, 35J60

On nonoscillatory solutions of differential equations withp-Laplacian

Abstract: The asymptotic behavior of nonoscillatory solutions of the differential equation with p-Lpalacian is investigated. By topological approach there are given conditions for the boundedness and convergence to zero of solutions. The singular solutions are studied too.

Diffusiophoresis of colloidal spheres in nonelectrolyte gradients at small but finite Péclet numbers

Abstract: The diffusiophoretic motion of a spherical particle in a uniform imposed gradient of a nonionic solute is analyzed for small but finite Peclet numbers. The range of the interaction between the solute molecules and the particle surface is assumed to be small relative to the radius of the particle, but the polarization effect of the mobile solute in the thin diffuse layer surrounding the particle caused by the strong adsorption of the solute is incorporated. A normal flux of the solute and a slip velocity of the fluid at the outer edge of the diffuse layer are used as the boundary conditions for the fluid domain outside the diffuse layer. Through the use of a method of matched asymptotic expansions along with these boundary conditions, a set of transport equations governing this problem is solved in the quasisteady situation and an approximate expression for the diffusiophoretic velocity of the particle good to O(Pe 2) is obtained analytically. The analysis shows that the first correction to the particle velocity is O(Pe 2). The normalized particle velocity is found to decrease monotonically with the Peclet number and to increase monotonically with the dimensionless relaxation coefficient. The stronger the polarization effect in the diffuse layer, the weaker the convective effect on the diffusiophoresis.