Singular perturbations for third-order nonlinear multi-point boundary value problem

This paper is devoted to study the following third-order multi-point singularly perturbed boundary value problem

     

Numerical modeling of perturbation propagation in a supersonic boundary layer

The growth of two-dimensional disturbances generated in a supersonic (M = 6) boundary layer on a flat plate by a periodic perturbation of the injection/suction type is investigated on the basis of a numerical solution of the Navier-Stokes equations. For small initial perturbation amplitudes, the second-mode growth rate obtained from the numerical modeling coincides with the growth rate calculated using linear theory with account for the non-parallelism of the main flow. Calculations performed for large initial perturbation amplitudes reveal the nonlinear dynamics of the perturbation growth downstream, with rapid growth of the higher multiple harmonics.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 33–44. Original Russian Text Copyright © 2004 by Egorov, Sudakov, Fedorov.     

Sequential and parallel synchronous alternating iterative methods

The so-called parallel multisplitting nonstationary iterative Model A was introduced by Bru, Elsner, and Neumann [Linear Algebra and its Applications 103:175-192 (1988)] for solving a nonsingular linear system using a weak nonnegative multisplitting of the first type. In this paper new results are introduced when is a monotone matrix using a weak nonnegative multisplitting of the second type and when is a symmetric positive definite matrix using a -regular multisplitting. Also, nonstationary alternating iterative methods are studied. Finally, combining Model A and alternating iterative methods, two new models of parallel multisplitting nonstationary iterations are introduced. When matrix is monotone and the multisplittings are weak nonnegative of the first or of the second type, both models lead to convergent schemes. Also, when matrix is symmetric positive definite and the multisplittings are -regular, the schemes are also convergent.

     

Perturbations of the Haar wavelet by convolution

In this note we show that the standard convolution regularization of the Haar system generates Riesz bases of smooth functions for , providing in this way an alternative to the approach given by Govil and Zalik [Proc. Amer. Math. Soc. 125 (1997), 3363-3370].

     

Strong approximations for Markovian service networks

Inspired by service systems such as telephone call centers, we develop limit theorems for a large class of stochastic service network models. They are a special family of nonstationary Markov processes where parameters like arrival and service rates, routing topologies for the network, and the number of servers at a given node are all functions of time as well as the current state of the system. Included in our modeling framework are networks of M _t /M _t /n _t queues with abandonment and retrials. The asymptotic limiting regime that we explore for these networks has a natural interpretation of scaling up the number of servers in response to a similar scaling up of the arrival rate for the customers. The individual service rates, however, are not scaled. We employ the theory of strong approximations to obtain functional strong laws of large numbers and functional central limit theorems for these networks. This gives us a tractable set of network fluid and diffusion approximations. A common theme for service network models with features like many servers, priorities, or abandonment is “non-smooth” state dependence that has not been covered systematically by previous work. We prove our central limit theorems in the presence of this non-smoothness by using a new notion of derivative. This revised version was published online in June 2006 with corrections to the Cover Date.     

Strong stability of stationary solutions and Karush-Kuhn-Tucker points in nonlinear optimization

The concepts of strongly stable stationary solutions (in Kojima's sense) and of strongly regular Karush-Kuhn-Tucker points (in Robinson's sense) for optimization problems with twice differentiable data are crucial in theory and applications of nonlinear optimization with data perturbations. In this paper we give interconnections between both concepts and extend some ideas to standard nonlinear programs withC ¹ data (underC ¹ perturbations). The main purpose of this paper is to survey several equivalent characterizations of strong stability in the classical case of programs withC ² data underC ² perturbations. The unified approach proposed here is essentially based on arguments from the analysis of Lipschitzian mappings.Sponsored by the International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria.     

Time scales in linear delayed differential equations

The aim of this paper is to apply and justify the so-called aggregation of variables method for reduction of a complex system of linear delayed differential equations with two time scales: slow and fast. The difference between these time scales makes a parameter ε>0 to appear in the formulation, being a mathematical problem of singular perturbations. The main result of this work consists of demonstrating that, under some hypotheses, the solution to the perturbed problem converges when ε→0 to the solution of an aggregated system whose construction is proposed.     

A survey of numerical techniques for solving singularly perturbed ordinary differential equations

This survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory. Starting from Prandtl's work a large amount of work has been done in the area of singular perturbations. This paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984–2000. The work done in this area during the period 1905–1984 has already been surveyed by the first author of this paper, see [Appl. Math. Comput. 30 (1989) 223] for details. Due to the space constraints we have covered only singularly perturbed one-dimensional problems.     

Correlated random walks

We present a new approach to the calculation of first passage statistics for correlated random walks on one-dimensional discrete systems. The processes may be non-Markovian and also nonstationary. A number of examples are used to demonstrate the theory.