Showing papers on "Asymptotology published in 1975"

Asymptotic analysis of deterministic and stochastic equations with rapidly varying components

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.

Asymptotic Expansions of Singularly Perturbed Quasi-Linear Optimal Systems

Abstract: A class of two-point boundary value problems (TPBVP’s) which arise in fixed final time free endpoint optimal control problems is considered. An asymptotic power series solution of the TPBVP is constructed with respect to a parameter whose perturbation changes the differential order of the problem. Based on a stability hypothesis, the proof of asymptotic correctness is accomplished through a successive approximation scheme.

On the asymptotic behavior of solutions of certain non-autonomous differential equations

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

Asymptotic properties of radiation fields in semiinfinite atmospheres

Abstract: The asymptotic results obtained above admit generalizations. First, one can take into account polarization. For an isotropic medium with arbitrary S matrix, allowance for polarization in the framework of the asymptotic theory developed here reduces in fact to simple replacements of the scalar by vector relations. Second, at least some of the results must remain true for a large class of inhomogeneous media. This is because angular relaxation also occurs in inhomogeneous media, and there must be asymptotic separation of the variables.

Asymptotic theory for a class of weakly nonlinear oscillations

Abstract: For a class of weakly non-linear oscillations involving a small parameter e we determine asymptotic solutions as ɛ ↓ 0 which are uniformly valid on some time interval. First, we consider a general initial-value problem in IRn containing a small parameter ɛ. We derive sufficient conditions for asymptotic correctness as ɛ ↓ 0 to be satisfied by formal asymptotic solutions. Next, we consider for the original problem formal asymptotic solutions of a two-variable type. For this type of formal asymptotic solutions the conditions for asymptotic correctness take a form which is very useful in the subsequent development of a construction technique for asymptotic solutions.

Asymptotic methods in the nonlinear theory of viscoelasticity

Abstract: Dynamic and quasistatic problems of the nonlinear theory of viscoelasticity are described by nonlinear integrodifferential and integral equations. Methods of averaging various classes of nonlinear integrodifferential and integral equations are described and asymptotic expansions of the solutions of these equations are given.

Asymptotic Solutions of the Orr-Sommerfeld Equation

Abstract: A rigorous justification is given of work done by Eagles (1969), in which he applied the method of matched asymptotic expansions to the Orr-Sommerfeld equation to obtain formal uniform asymptotic approximations to a certain pair of solutions. (Somewhat more polished formal expansions of the same general kind were subsequently obtained by Reid (1972).) First, a study is made of the asymptotic properties of solutions of a certain differential equation which admits the Orr—Sommerfeld equation as a special case. Previous work on this differential equation by Lin & Rabenstein ( i960, 1969) is extended to develop a theory suited to our main purpose: to prove the validity of Eagles’s approximations. It is then shown how this theory can be used to prove the existence of actual solutions of the Orr—Sommerfeld equation approximated by these formal expansions. In addition, it is verified that these solutions have the properties assumed by Eagles (1969).

Asymptotic expansions for acoustical modes of the homogeneous compressible model

Abstract: The behavior ofp-modes of high degree and high order in the homogeneous compressilbe model is examined. The second-order differential equation of Pekeris is used to construct asymptotic expansions near the centre and near the surface, which are singular points, and near the turning point of that equation. An equation for the frequencies is obtained by requiring the continuity of the asymptotic solutions and of their first derivatives. Numerical applications are considered.

Linearization and asymptotic behavior

Abstract: I f (E) is autonomous, Grobman [7] has shown there is a homeomorphism of X with itself taking solutions of (E) onto solutions of (L) provided that F is bounded, the difference between any two solutions of (E) is unbounded, and A has no eigenvalues with zero real part. This result is extended to nonautonomous systems in Section 2, and in Section 3 results on asymptotic behavior similar to those in [1], [2], [3], and [5] are obtained. Because part of F i s bounded by a function independent of x, these results are, in a sense, extensions of the previous theorems. The final section generalizes a well-known theorem on the asymptotic stability of a periodic orbit for

Some results on functional perturbations to delay-differential systems

Abstract: Sonic sufficient criteria for the convergence of the solutions to a class of riclny-differcntial systems with variable coefficients arc presented. Conditions on / guaranteeing asymptotic stability for x(t)≡0 of (1) are presented which provide on explicit region of asymptotic stability.