Showing papers on "Method of matched asymptotic expansions published in 1979"

Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations

Abstract: Recently Aronson [1] extended the concept of asymptotic speed which he and Weinberger [3], [4] had developed for nonlinear diffusion problems in population genetics, combustion and nerve propagation, to an epidemic model proposed by Kendall [11], [12] in 1957 (1965). In this model (which is a spatial version of Kermack's and McKendrick's epidemic model [13]) the aflfected individuals become immediately infectious and are removed at a constant rate. The model does not take into account that with most infectious diseases the affected individuals underlie an incubation period, before they become infective, and that they remain infective for a fixed period only. These features cannot be described by the equation considered by Aronson [1] which contains a derivative with respect to time and an integral with respect to space. It is therefore desirable to extend Aronson's and Weinberger's concept of asymptotic speed to the nonlinear integral equation t J g(u(t-s,x+y))k(s9\\y\\)dsdy

A “sinc-Galerkin” method of solution of boundary value problems

Abstract: : This paper illustrates the application of a Sinc-Galerkin method to the approximate solution of linear and nonlinear second order ordinary differential equations, and to the approximate solution of some linear elliptic and parabolic partial differential equations in the plane. The method is based on approximating functions and their derivatives by use of the Whittaker cardinal function. The DE is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products, the evaluation of which does not require any numerical integration. Using n function evaluations the error in the final approximation to the solution of the DE is 0 exp (-c(n to the 1/2 d power)) where c is independent of n, and d denotes the dimension of the region on which the DE is defined. This rate of convergence is optimal in the class of n-point methods which assume that the solution is analytic in the interior of the interval, and which ignore possible singularities of the solution at the end-points of the interval. (Author)

Valid asymptotic expansions for the likelihood ratio statistic and other perturbed chi-square variables

Abstract: Let {Z n } n≥1 be a sequence of random vectors. Under certain conditions, distributions of statistics which are smooth functions of the mean vector Z n - and whose asymptotic distributions are central Chi-square are shown to possess asymptotic expansions in powers of n -1 As applications, asymptotic expansions of the null distributions of the likelihood ratio statistic, Wald's and Rao's statistics are obtained. The results proved here supplement the recent work of Bhattacharya and Ghosh (1978) and also justify the validity of the formal expansions obtained by Box (1949) and Hayakawa (1977).

Asymptotic Solutions of Model Equations in Nonlinear Acoustics

Abstract: This paper uses the method of matched asymptotic expansions to derive asymptotic solutions to various problems in nonlinear acoustics. Model equations, generalizing the well known Burgers equation to include effects of cylindrical or spherical spreading and of non-equilibrium relaxation, are given and regarded as governing the propagation. Solutions are sought for initial or boundary conditions of N-wave or harmonic wave form. For a thermoviscous medium, the small parameter upon which the asymptotic expansions are predicated is an inverse acoustic Reynolds number; for a relaxing medium it is the product of wave frequency and relaxation time. The complete asymptotic solution for N-waves in a thermoviscous fluid is known, in the case of plane motion, from the Cole—Hopf solution of Burgers’s equation. Here a similarly complete solution is found for spherical N-waves, with the exception of one region of space—time in which an irreducible nonlinear problem remains unsolved. In this region the outer limiting behaviour is, nevertheless, determined, so that the solutions in all other regions are completely fixed. For cylindrical N-waves an irreducible problem again results, but the motion can be followed right through into its ‘old age’ phase aside from an undetermined purely numerical constant. Correct results are obtained here for the ‘correction due to diffusivity9 to the weak-shock theory prediction of shock centre location for plane, cylindrical and spherical N-waves. These results indicate a non-uniformity in weak shock theory at large times, and also, in the case of spherical N-waves, reveal a large time non-uniformity in the Taylor shock solution. Harmonic waves, plane, cylindrical and spherical, in thermoviscous fluids and relaxing fluids are considered, and the asymptotic solutions are found to leading order in most of the many overlapping asymptotic regions of space-time. A single dimensionless function remains undetermined in the important case of spherical harmonic waves. We have also been unable to find scalings and differential equations describing precisely how a discontinuity is formed at the front of a partly dispersed shock in a relaxing gas, though the shock centre is located for both fully and partly dispersed shocks. The harmonic wave solutions unify and extend certain solutions (the Fay, Fubini and old-age solutions) which are well known in the nonlinear acoustics literature, and the amplitude saturation and scaling laws for the old age regime are in accord with experiments on high amplitude spherical waves in water.

On the unsteady squeezing of a viscous fluid from a tube

Abstract: Viscous fluid is squeezed out from a shrinking (or expanding) tube whose radius varies with time as (1 – β t ) ½ . The full Navier–Stokes equations reduce to a non-linear ordinary differential equation governed by a non-dimensional parameter S representing the relative importance of unsteadiness to viscosity. This paper studies the analytic solutions for large | S | through the method of matched asymptotic expansions. A simple numerical scheme for integration is presented. It is found that boundary layers exist near the walls for large | S |. In addition, flow reversals and oscillations of the velocity profile occur for large negative S (fast expansion of the tube).

A class of singularly perturbed, nonlinear, fixed-endpoint control problems

Abstract: Singular perturbation techniques are applied to a class of nonlinear, fixed-endpoint control problems to decompose the full-order problem into three lower-order problems, namely, the reduced problem and the left and right boundary-layer problems. The boundary-layer problems are linear-quadratic and, contrary to previous singular perturbation works, the reduced problem has a simple formulation. The solutions of these lower-order problems are combined to yield an approximate solution to the full nonlinear problem. Based on the properties of the lower-order problems, the full problem is shown to possess an asymptotic series solution.

The Lateral Force on a Small Sphere Sedimenting in a Viscous Fluid Bounded by a Cylindrical Wall

Abstract: A theoretical study is made of the lateral force on a spherical rigid particle sedimenting with a constant velocity in an incompressible viscous fluid bounded by a vertical cylindrical wall. Results are obtained up to the first-order effects of boundaries using the method of matched asymptotic expansions. The sphere experiences a lateral force towards the axis everywhere and its magnitude is proportional to the distance from the axis to the position of the sphere when the distance is not so large and has a maximum value near the wall.

On the asymptotic validity of perturbation methods for hyperbolic differential equations

Abstract: In this work a justification for the following formal perturbation methods is presented: (i) Luke's procedure for equations of the type φtt−φss=ef(φ). (ii) The Chikwendu-Kevorkian perturbation method for equations of the type φtt−φss=ef(φt, φs). (iii) The Krylov-Bogoljubov-Mitropolski-Montgomery-Tidman approach for equations of the type φtt−φss+p2φ=ef(φ, φt, φs).

The cooling of low-heat-resistance cylinders by radiation

Abstract: The method of matched asymptotic expansions is applied to the non-linear radiative cooling of finite or semi-infinite cylinders. It is shown that the method applies when radiation is the limiting factor in the heat-transfer process, i.e. when the heat resistance of the bulk is relatively low. The analysis will be of importance in the fields of crystal growth and the cooling of fins.

Functional differential equations

Abstract: Using a modification of the Tawumikhin method, the authors obtain asymptotic properties of solutions os delay differential equations. In particular using Liapunov functions we obtain sufficient conditions for solutions to approach a constant c as t→∞. Here and f has appropriate smoothness properties to guarantee extendability of solutions. These results are applied to the widely studied class of delay equations , where are continuous γ is ratio of odd integers, e>0, and r>0. When c=0 new results on asymptotic stability are obtained.

Weak Buoyancy in Laminar Vertical Jets

Abstract: The paper is concerned with the influence of weak buoyancy forces on the velocity and temperature distributions of a laminar, vertical and axisymmetric get. It is shown that a regular perturbation method as carried out by Mollendorf and Gebhart ceases to be valid in a certain, outer region of the jet if Pr < 3/2, and that it breaks down entirely if Pr ≦ 1/2. A uniformly valid first approximation is obtained by the method of matched asymptotic expansions for the case 1/2 < Pr < 3/2.

Lifting three-dimensional wings in transonic flow

Abstract: The far field of a lifting three-dimensional wing in transonic flow is analysed. The boundary-value problem governing the flow far from the wing is derived by the method of matched asymptotic expansions. The main result is to show that corrections which are second order in the near field make a first-order contribution to the far field. The present study corrects and simplifies the work of Cheng and Hafez (1975) and Barnwell (1975).

Qualitative dynamics from asymptotic expansions

Abstract: Publisher Summary This chapter focuses on qualitative dynamics from asymptotic expansions. It discusses all that can be said about the qualitative features of the exact solutions given an asymptotic expansion of the general solution of a differential equation, calculated, say, to m terms in a small parameter. It is general practice in applied mathematics to assume that the exact solution shows the same behavior as the approximation, at least when this is not obviously unrealistic. It is possible to pose the problem locally or globally. The chapter presents local results and also presents an approach to the global problem. It chapter also focuses on a nonlinear oscillation problem.

Translational friction coefficient of a permeable cylinder in a sheet of viscous fluid

Abstract: The author calculates the translational friction coefficient and the translational diffusion coefficient of a permeable cylinder moving in a sheet of fluid which is embedded on both sides in a fluid of much lower viscosity. The result, which is an asymptotic expression valid in the limit of small ratios of the viscosities, is derived by the method of matched asymptotic expansions.

On asymptotic expansions in nonlinear, singularly perturbed difference equations ∗

Abstract: The present paper deals with the problem of constructing and proving asymptotic expansions for nonlinear, singularly perturbed difference equations. New methods for the construction of asymptotic expansions are presented and compared with well-known ones. For the proof of their validity, fundamental principles for the treatment of nonlinear singular perturbation problems are applied, based on the concepts of e-stability, formal asymptotic expansions, matching and asymptotic expansions. The results are derived from a general theory of asymptotic expansions of nonlinear operator equations that has been developed recently by the author.

The influence of solid bodies on low mach number vortex sound

Abstract: The mechanisms of sound generation by unsteady, subsonic flows in the presence of solid boundaries are investigated. For that purpose an alternative integral representation for the radiated pressure field is applied which is different from the generally used integral representation introduced by Lighthill and Curle. The main advantage of our method consists in a linear dependence of its integrand on the time derivative of the vorticity fluctuations in the hydrodynamic near field, while the ordinary Green's function has to be substituted by a “vector Green's function.” This vector Green's function can be chosen for the flow fields appropriate in such a way that surface integrals do not appear. In particular the paper is concerned with two‐dimensional flow and sound fields caused by a pair of spinning vortices and superimposed stationary potential flows along a finite plate or around a cylinder. Analytical solutions are determined by applying the method of matched asymptotic expansions.

Singular perturbation for nonlinear boundary-value problems

Abstract: Asymptotic solutions of a class of nonlinear boundary-value problems are studied. The problem is a model arising in nuclear energy distribution. For large values of the parameter, the differential equations are of the singular-perturbation type and approximations are constructed by the method of matched asymptotic expansions.

Asymptotic Expansions for the Discretization Error In Poisson’s Equation on General Domains

Abstract: In this paper we modify the usual five-point-difference discretization near the boundary of a general domain as to guarantee the existence of an asymptotic expansion. We generalize and improve results due to Gerschgorin [9], Collatz [8], Mikeladse [11], Wasow [14] and Pereyra-Proskurowski-Widlund [12].

Explicit perturbative solution of multichannel wave equations

Abstract: It is shown that a general perturbative method can be developed for solving the coupled equations of the multichannel formalism Assuming channel potentials for which Taylor expansions exist and weak channel coupling, we show that two pairs of eigensolutions can be constructed which belong to one and the same set of eigenvalues The solutions can be matched and normalized, and the eigenvalues are calculated explicitly in the form of asymptotic expansions Finally, generalizations and applications are discussed