Showing papers on "Method of matched asymptotic expansions published in 1993"

Phase-field models for anisotropic interfaces.

Abstract: The inclusion of anisotropic surface free energy and anisotropic linear interface kinetics in phase-field models is studied for the solidification of a pure material. The formulation is described for a two-dimensional system with a smooth crystal-melt interface and for a surface free energy that varies smoothly with orientation, in which case a quite general dependence of the surface free energy and kinetic coefficient on orientation can be treated; it is assumed that the anisotropy is mild enough that missing orientations do not occur. The method of matched asymptotic expansions is used to recover the appropriate anisotropic form of the Gibbs-Thomson equation in the sharp-interface limit in which the width of the diffuse interface is thin compared to its local radius of curvature. It is found that the surface free energy and the thickness of the diffuse interface have the same anisotropy, whereas the kinetic coefficient has an anisotropy characterized by the product of the interface thickness with the intrinsic mobility of the phase field.

Two loop two point functions with masses: Asymptotic expansions and Taylor series, in any dimension

Abstract: In all mass cases needed for quark and gluon self-energies, the two-loop master diagram is expanded at large and smallq2, ind dimensions, using identities derived from integration by parts Expansions are given, in terms of hypergeometric series, for all gluon diagrams and for all but one of the quark diagrams; expansions of the latter are obtained from differential equations Pade approximants to truncations of the expansions are shown to be of great utility As an application, we obtain the two-loop photon self-energy, for alld, and achieve highly accelerated convergence of its expansions in powers ofq2/m2 orm2/q2, ford=4

Summing logarithmic expansions for singularly perturbed eigenvalue problems

Abstract: Strong localized perturbations of linear and nonlinear eigenvalue problems in a bounded two-dimensional domain D are considered. The effects on an eigenvalue $\lambda _0 $ of the Lapla-cian, and on the fold point $\lambda _{c0} $ of a nonlinear eigenvalue problem, of removing a small subdomain $D_\epsilon $, of “radius” $\epsilon $, from D and imposing a condition on the boundary of the resulting hole, are determined. Using the method of matched asymptotic expansions, it is shown that the expansions of the eigenvalues and fold points for these perturbed problems start with infinite series in powers of $( - 1/\log [ \epsilon d( \kappa ) ] )$. Here $d( \kappa )$ is a constant that depends on the shape of $D_\epsilon $ and on the precise form of the boundary condition on the hole. In each case, it is shown that the entire infinite series is contained in the solution of a single related problem that does not involve the size or shape of the hole. This related problem is not stiff and can be solved numerically...

Asymptotic and numerical methods for partial differential equations with critical parameters

Abstract: Preface. Part 1: Modeling of Complex Systems with Asymptotic-Enhanced Numerical Methods. Part 2: Asymptotic-Induced Domain Decomposition Methods. Part 3: Multiple-Scale Problems in Scientific Computing. Part 4: Applied and Asymptotic Analysis. Part 5: Symbolic Manipulation Tools for Asymptotic Analysis. Part 6: Numerical Methods, Algorithms, and Architectures. Index.

Scattering of Water Waves by Vertical Cylinders

Abstract: The scattering of small amplitude water waves by an array of vertical cylinders is studied theoretically and experimentally. In the theoretical study, a method of matched asymptotic expansions is first developed to find the reflection and the transmission coefficients without considering real fluid effects. Energy loss due to the flow separation near the cylinders is modeled by adopting a linearized form of the quadratic resistance law. It is shown that this is equivalent to introduce a complex blockage coefficient. The energy loss coefficients for square cylinders and circular cylinders are determined by comparing theoretical results with experimental data.

Euclidean Asymptotic Expansions of Green Functions of Quantum Fields (i) Expansions of Product of Singular Functions

Abstract: The problem of asymptotic expansion of Green functions in perturbative QFT is studied for the class of Euclidean asymptotic regimes. Phenomenological applications are analyzed to obtain a meaningful mathematical formulation of the problem. It is shown that the problem reduces to studying asymptotic expansion of products of a class of singular functions in the sense of the distribution theory. Existence, uniqueness and explicit expressions for such expansions. (As-operation for products of singular functions) in dimensionally regularized form are obtained using the so-called extention principle.

Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions

Abstract: A class of second-order linear differential equations with a large parameter u is considered. It is shown that Liouville-Green type expansions for solutions can be expressed using factorial series in the parameter, and that such expansions converge for Re (u) > 0, uniformly for the independent variable lying in a certain subdomain of the domain of asymptotic validity. The theory is then applied to obtain convergent expansions for modified Bessel functions of large order.

Boundary layer flow of power-law fluids past arbitrary profiles

Abstract: Two-dimensional steady-state boundary layer equations of power-law fluids are derived using a special coordinate system which makes the equations independent of the body shape immersed in the flow. In deriving the boundary layer equations, the method of matched asymptotic expansions is used. The similarity solutions for power-law fluids are much the same as those of Newtonian fluids. Similarity solutions corresponding to the case of parallel flow past a flat plate and stagnation-point flow are presented. Finally, the shear stress is calculated for different geometries

Euclidean Asymptotic Expansions of Green Functions of Quantum Fields (ii) Combinatorics of the as Operation

Abstract: The results of Ref. 1 are used to obtain full asymptotic expansions of Feynman diagrams renormalized within the MS scheme in the regimes when some of the masses and external momenta are large with respect to the others. The large momenta are Euclidean, and the expanded diagrams are regarded as distributions with respect to them. The small masses may be equal to zero. The As operation for integrals is defined and a simple combinatorial technique is developed to study its exponentiation. The As operation is used to obtain the corresponding expansions of arbitrary Green functions. Such expansions generalize and improve upon the well-known short-distance, operator-product expansions, the decoupling theorem etc.: e.g. the low-energy effective Lagrangians are obtained to all orders of the inverse heavy mass. The obtained expansions possess the property of perfect factorization of large and small parameters, which is essential for meaningful applications to phenomenology. As an auxiliary tool, the inversion of the R operation is constructed. The results are valid for arbitrary QFT models.

Scattering of water waves by vertical cylinders with a backwall

Abstract: The scattering of small amplitude water waves by an array of vertical cylinders with a solid vertical backwall is studied theoretically and experimentally. In the theoretical study, a method of matched asymptotic expansions is developed without considering real fluid effects. The energy loss due to flow separation near cylinders is modeled by introducing a complex blockage coefficient. The theories are compared with laboratory data.

On the transient behaviour of the Erlang loss model: heavy usage asymptotics

Abstract: The Erlang loss model, which is the $M/M/m/m$ queue is considered. Asymptotic expansions are constructed for systems with a large number of servers $( m \gg 1 )$ and a large arrival rate that is $O( m )$. Formulas are given for the probability $p_n ( t )$ that n servers are occupied at time t. Several cases are treated of initial conditions and several regions in the $( {n,t} )$ plane. The approximations are constructed by using singular perturbation techniques such as the ray method, boundary layer theory, and the method of matched asymptotic expansions.

Exponentially-improved asymptotic solutions of ordinary differential equations I: the confluent hypergeometric function

Abstract: There has been renewed interest in both formal and rigorous theories of exponentially-small contributions to asymptotic expansions. In particular, a generalized asymptotic expansion was obtained for the confluent hypergeometric function $U(a,a - b + 1,z)$ in which the parameters a and b are complex constants, and z is a large complex variable. This expansion is expressed in terms of generalized exponential integrals and has a larger region of validity and greater accuracy than conventional expansions of Poincare type. The expansion was established by transformations and a re-expansion of an integral representation of $U(a,a - b + 1,z)$. In this paper it is shown how the same result can be achieved by a direct differential-equation approach, thereby laying the foundation for a rigorous theory of generalized asymptotic solutions of linear differential equations.

An extension of the routh array for the asymptotic stability of a system of differential equations with complex coefficients

Abstract: In this paper, we establish two necessary and sufficient conditions for the asymptotic stability of a system of differential equations with complex coefficients. These conditions are expressed in terms of the partial fraction and continued fraction expansions of a certain function related to the characteristic polynomial of the system. We then work out a systematic procedure for testing the asymptotic stability of an nxn complex system, and we show how our results generalize the well-known Routh criterion for the asymptotic stability of an nxn real system.

Asymptotic Solutions of the System of Elasticity Theory for Rod and Frame Structures

Abstract: Full asymptotic solutions are constructed for the system of equations of elasticity theory for a rod (a thin cylinder whose cross-section diameter is a small parameter), a rod structure (a connected aggregate of several rods), and frame domains. The limiting equations and matching conditions are obtained.

Asymptotic behaviour of solutions of linear differential equations with delay

Abstract: This contribution is devoted to the problem of asymptotic behaviour of solutions of scalar linear differential equation with variable bounded delay of the form ẋ(t) = −c(t)x(t− τ (t)) (∗) with positive function c(t). Results concerning the structure of its solutions are obtained with the aid of properties of solutions of auxiliary homogeneous equation ẏ(t) = β(t)[y(t) − y(t − τ (t))] where the function β(t) is positive. A result concerning the behaviour of solutions of Eq. (*) in critical case is given and, moreover, an analogy with behaviour of solutions of the second order ordinary differential equation x ′′(t) + a(t)x(t) = 0 for positive function a(t) in critical case is considered. AMS Subject Classification. 34K15, 34K25.