Showing papers on "Asymptotic analysis published in 1975"
TL;DR: In this article, the boundary value problem and Cauchy problem were studied in the exterior of a compact set for hyperbolic operators whose coefficients depend only on x and are constant near infinity.
Abstract: In this paper we study the Cauchy problem and boundary-value problem of general form in the exterior of a compact set for hyperbolic operators L whose coefficients depend only on x and are constant near infinity. Assuming that the wave fronts of the Green's matrix for L go off to infinity as , we determine the asymptotic behaviour of solutions as . For the corresponding stationary problem we obtain the short-wave asymptotic behaviour of solutions for real and complex frequencies.
200 citations
TL;DR: A recursive algorithm for parametric identification of discrete-time systems known as Panuska's method, the approximate maximum likelihood method or the extended matrix method is analyzed and the manner in which the counterexamples are constructed yields insight into the algorithm and provides ideas to improve the convergence properties.
Abstract: A recursive algorithm for parametric identification of discrete-time systems known as Panuska's method, the approximate maximum likelihood method or the extended matrix method, is analyzed. Making use of recently developed theory for asymptotic analysis of recursive stochastic algorithms, dynamic systems, and autoregressive moving average (ARMA) processes are constructed for which this algorithm does not converge. The manner in which the counterexamples are constructed yields insight into the algorithm and provides ideas how to improve the convergence properties.
85 citations
TL;DR: In this paper, the local character of the elastostatic field in plane strain near a point that separates a free from an adjoining fixed segment of a rectilinear boundary component is investigated.
78 citations
77 citations
TL;DR: In this article, the asymptotic character of deterministic and stochastic equations whose solutions have a rapidly varying component is studied. And the class of problems for which the limiting behavior can be described in a contracted and simplified framework is investigated.
Abstract: The asymptotic character of deterministic and stochastic equations whose solutions have a rapidly varying component is studied. Of particular interest is the class of problems for which the limiting behavior can be described in a contracted and simplified framework.
60 citations
TL;DR: In this paper, a new approach was taken in studying the asymptotic behavior of solutions of Eq. (1) by examining the quotient r(t)/q(t) as t + 03, without requiring that the forcing term, Y(Z), be small.
39 citations
TL;DR: In this article, the authors considered the general problem of potential flow past a slender body of revolution, where the flow incident on the body is described by an arbitrary potential function and hence the results presented here extend those obtained by Handelsman & Keller (1967 α).
Abstract: The general problem of potential flow past a slender body of revolution is considered. The flow incident on the body is described by an arbitrary potential function and hence the results presented here extend those obtained by Handels-man & Keller (1967 α). The part of the potential due to the presence of the body is represented as a superposition of potentials due to point singularities (sources, dipoles and higher-order singularities) distributed along a segment of the axis of the body inside the body. The boundary condition on the body leads to a linear integral equation for the density of the singularities. The complete uniform asymptotic expansion of the solution of this equation, as well as the extent of the distribution, is obtained using the method of Handelsman & Keller. The special case of transverse incident flow is considered in detail. Complete expansions for the dipole moment of the distribution and the virtual mass of the body are obtained. Some general comments on the method of Handelsman & Keller are given, and may be useful to others wishing to use their method.
31 citations
01 Jan 1975
TL;DR: A survey of the tools of asymptotic analysis that are presently available and those that are needed for the next stages of development of the theory of special functions can be found in this paper.
Abstract: The first part of this paper surveys the tools of asymptotic analysis that are presently available and those that are needed for the next stages of development of the asymptotic theory of special functions. Methods are grouped according to the number of free variables to which they achieve a uniform reduction. Topics include the approximation of functions defined parametrically by a definite integral or infinite sum, and approximate solutions of linear ordinary differential equations. The main areas in which work appears to be needed are (i) problems of confluence, that is, coalescing saddle points and singularities of integrals, or coalescing turning points and singularities of differential equations; (ii) rigorous error analysis. The second part of the paper discusses the more important special functions in increasing order of the total number of variables and parameters involved. Almost all asymptotic problems concerning special functions of one variable have been solved. For functions of two variables the problems are solved, or can be solved by use of existing asymptotic tools. It is in the case of functions depending on three variables (including parameters) that the most significant work can be expected in the immediate future. Two new asymptotic tools have been developed very recently, one for integrals with three coalescing saddle points, the other for second-order differential equations having two coalescing turning points. Applications of these tools are needed, as well as the development of similar tools for other three-variable problems involving confluence. As the total number of variables increases beyond three, our knowledge of the asymptotic behavior of the special functions becomes more fragmentary. It will undoubtedly be many years before we have complete knowledge of the asymptotic behavior of any of the functions of four variables, including for example, the hypergeometric function F(a,b;c;z).
29 citations
23 citations
16 citations
TL;DR: In this article, a planar steady flame is assumed to propagate through a mixture of fuel, oxidant, diluent and product species, which burns according to a general one-step reversible reaction.
Abstract: A planar steady flame is assumed to propagate through a mixture of fuel, oxidant, diluent and product species, which burns according to a general one-step reversible reaction. On the assumption that the activation energy of the burning reaction is large, the first significant terms in asymptotic series for several quantities of interest are evaluated. As a result it is possible to make some general observations about the effects of mixture strength on a number of features, such as flame propagation speed and reaction-zone behaviour, for example. It is also deduced that chemical-reaction induced amplification of acoustic disturbances is unlikely to occur in pre-mixed flames of this simple type.
TL;DR: In this paper, the authors developed an approximate theory of stationary propagation of the planar front of a two-stage parallel exothermic reaction in a condensed medium and in a gas.
TL;DR: In this paper, the authors considered non-autonomous (1.2) third-order differential equations, where a(t, b(t)y c(t), g, h, p are continuous real-valued functions depending only on the arguments shown.
Abstract: (1.2) x+a(t)f(x, Λ, x)x+b(t)g(x9 x)+c(t)h(x) = p(t, x, i, x) where a(t), b(t)y c(t) are positive continuously differentiate and /, g, h, p are continuous real-valued functions depending only on the arguments shown, and the dots indicate the differentiation with respect to t. The asymptotic property of solutions of third order differential equations has received a considerable amount of attention during the past two decades, particularly when (1.2) is autonomous. Many of these results are summarized in [11]. A few authors have studied non-autonomous third order differential equations. K. E. Swick [13] considered the following equations
Dissertation•
01 Jan 1975
TL;DR: In this article, the authors give results on oscillatory and asymptotic behavior for differential equations with retarded arguments of arbvitrary order, which is a generalization of our results.
Abstract: We give here some results on oscillatory and asymptotic behavior for differential equations with retarded arguments of arbvitrary order. Lemma 1 establishes a comparison priniple from which we derive the oscillatory anbd asymptotic behavior of the solutions by considering simple ordinary differential equations of the form Y (n)+g(t) Y α=0 whose solutions have known behavior. Several known results for diferential equations with retarded arguments and (cf.[1], [6], [8]and[12])are particular cases of ours. Moreover, even in the case of ordinary differential equations, our results appear as generalizations of other ones (cf.[2-4]and[9-11]).
TL;DR: In this paper, the authors discussed the importance of the multiparameter Sturm-Liouville problems in mathematical physics and pointed out that in recent years this field has been relatively neglected in contrast to the single parameter case.
Abstract: The origin and importance of multiparameter Sturm-Liouville problems in mathematical physics has recently been discussed by Atkinson [1, sects. 3 and 4], [2, introduction]. In spite of the importance of such problems, and in spite of the work done in this field by such early investigators as Klein and Hilbert, Atkinson points out that in recent years this field has been relatively neglected in contrast to the single parameter case. As an example, he states that as opposed to the one parameter case, the detailed behaviour of the eigenvalues and eigenfunctions in the multiparameter Sturm-Liouville case is still far from clear.
TL;DR: In this paper, the asymptotic behavior of proper nonoscillator solutions of the nonlinear, third order, ordinary differential equation(*) u′″±tσun=0, where n>1 and σ is an arbitrary real number, is considered.
Abstract: The asymptotic behavior of the proper nonoscillator solutions of the nonlinear, third order, ordinary differential equation(*) u′″±tσun=0, where n>1 and σ is an arbitrary real number, is considered. Cases for σ and n are studied and the possible asymptotic behavior (t→∞) of the solutions of(*) are found and conditions for their existence are demonstrated.
TL;DR: An approximate theory of the stationary distribution of the plane front of a two-stage exothermic consecutive chemical reaction in a condensed medium is developed in this article, where the method of joined asymptotic expansions is used in constructing the solutions.
Abstract: An approximate theory of the stationary distribution of the plane front of a two-stage exothermic consecutive chemical reaction in a condensed medium is developed in the article. The method of joined asymptotic expansions is used in constructing the solutions. The ratio of the sum of the activation energies of the reactions to the final adiabatic combustion temperature is a parameter of the expansion. The characteristic limiting states of the stationary distribution of the wave corresponding to different values of the parameters figuring in the problem are shown. Approximate analytical expressions for the wave velocity and distribution of concentrations are obtained for each of the states.
TL;DR: In this paper, the stability of a supersonic boundary layer over an intensively cooled plate with respect to three-dimensional disturbances is investigated and two neutral stability curves, the existence of which was established in [1], are considered.
Abstract: The stability of a supersonic boundary layer over an intensively cooled plate with respect to three-dimensional disturbances is investigated. Two neutral stability curves, the existence of which was established in [1], are contemplated. It is shown by asymptotic analysis that each of these two neutral stability curves separates into a closed and an ordinary neutral curve in a certain range of disturbance propagation angles. As the surface is cooled, the closed neutral curve contracts to a point. The results of asymptotic analysis were confirmed by numerical integration of the stability equations.
TL;DR: The boundary layer equations are the zeroth−order solutions to the BGK equation when the Chapman−Enskog procedure is coupled with the Prandtl analysis.
Abstract: The boundary layer equations are the zeroth−order solutions to the BGK equation when the Chapman−Enskog procedure is coupled with the Prandtl analysis. The first− and second−order asymptotic solutions are obtained. They are higher−order corrections to the usual boundary layer equations.
TL;DR: In this paper, an asymptotic solution involving a phase function and an amplitude function is constructed for wave resonance of a compressible fluid in a rotating spherical shell is considered.
Abstract: An asymptotic method based upon the ray theory is extended to the study of planetary waves in a fluid of variable depth over the rotating earth. An asymptotic solution involving a phase function and an amplitude function is constructed. Application of the method to the study of wave resonance of a compressible fluid in a rotating spherical shell is considered.
01 Feb 1975
TL;DR: In this paper, it was shown that certain autonomous linear functional differential equations and their perturbations satisfy a generalized type of asymptotic equivalence, and an example is given.
Abstract: It is shown that certain autonomous linear functional differential equations and their perturbations satisfy a generalized type of asymptotic equivalence. An example is given.