On the Transient Behavior of the M/M/1 and M/M/1/M Queues

This paper gives a transient analysis of the classic M/M/1 and M/M/1/K queues. Our results are asymptotic as time and queue length become simultaneously large for the infinite capacity queue, and as the system’s storage capacity K becomes large for the finite capacity queue. We give asymptotic expansions for p_n(t), which is the probability that the system contains n customers at time t. We treat several cases of initial conditions and different traffic intensities. The results are based on (i) asymptotic expansion of an exact integral representation for p_n(t) and (ii) applying the ray method to a scaled form of the forward Kolmogorov equation which describes the time evolution of p_n(t).     

Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3:2:1

We consider a boundary-value problem for the Poisson equation in a thick junction Ω_ε, which is the union of a domain Ω₀ and a large number of ε-periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition ∂_νu_ε + εκ(u_ε)=0 is given on the lateral surfaces of the thin cylinders. The asymptotic analysis of this problem is performed as ε → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove the convergence theorem and show that the nonlinear Robin boundary condition is transformed (as ε → 0) in the blow-up term of the corresponding ordinary differential equation in the region that is filled up by the thin cylinders in the limit passage. The convergence of the energy integral is proved as well. Using the method of matched asymptotic expansions, the approximation for the solution is constructed and the corresponding asymptotic error estimate in the Sobolev space H¹(Ω_ε) is proved. Copyright © 2007 John Wiley & Sons, Ltd.     

Singular perturbations for the exterior three-dimensional slow viscous flow problem

In this paper the rigorous justification of the formal asymptotic expansions constructed by the method of matched inner and outer expansions is established for the three-dimensional steady flow of a viscous, incompressible fluid past an arbitrary obstacle. The justification is based on the series representation of the solution to the Navier-Stokes equations due to Finn, and it involves the reductions of various exterior boundary value problems for the Stokes and Oseen equations to boundary integral equations of the first kind from which existence as well as asymptotic error estimates for the solutions are deduced. In particular, it is shown that the force exerted on the obstacle by the fluid admits the asymptotic representation F = A₀ + A₁Re + O(Re² ln Re⁻¹) as the Reynolds number Re → 0⁺, where the vectors A₀ and A₁ can be obtained from the method of matched inner and outer expansions.     

Systems of sets with multiplicative asymptotic density

The authors define a notion of system of sets with multiplicative asymptotic density in this paper. A criterion and one necessary condition for a given system {A _i } _i=1^∞ to be a system with multiplicative asymptotic density is given. Properties of certain special types of systems of sets with multiplicative asymptotic density are treated. This work is supported by The Ministry of Education, Youth and Sports of the Czech Republic. Project CQR 1M06047.     

Thermodynamic Equilibrium in the System of Chaotic Quantized Vortices in a Weakly Imperfect Bose Gas

In the example of a weakly imperfect Bose gas, we discuss the mechanism of establishing thermodynamic equilibrium for a chaotic set of quantum vortex filaments. We assume that the dynamics of the Bose condensate is described by the Gross–Pitaevsky equation with an additional noise satisfying the fluctuation–dissipation theorem. In considering a vortex filament as the intersection line of surfaces on which the real and imaginary parts of the order parameter (x,t) vanish, we obtain an equation of the Langevin type for elements of the vortex filament with an appropriately transformed random force. The Fokker–Planck equation for the probability density has a solution given by the Gibbs distribution at the temperature of the Bose condensate. In other words, when the Bose condensate is in thermal equilibrium and no other random actions exist, the system of vortices is also in thermal equilibrium.     

Asymptotic analysis and asymptotic domain decomposition for an integral equation of the radiative transfer type

We consider an integral equation of the radiative transfer type stated in the interval [0,τ₀] with the length τ₀1. We construct an asymptotic solution of the problem and we give a method transforming this problem to some similar problems set in the interval with the length dτ₀. Error estimates are proved.     

Localization results for densities associated with stable small-noise diffusions

Summary This paper concerns asymptotic properties of the stationary density associated with small-noise diffusion processes, such as considered in the well-known work of Ventcel and Freidlin [12]. We assume that the origin is a globally attracting asymptotically stable equilibrium point of the underlying deterministic flow. For a bounded domain D, containing the origin, we derive estimates which establish the asymptotic independence, as the size of the noise vanishes, of the equilibrium density in D from the coefficients of the process outside D. These results are applied to generalize a result of Sheu [10] on an asymptotic representation of the equilibrium density.     

The convergence and stability analysis of the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions

In this paper, we apply the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions. Due to existence of integral boundary conditions, after reformulation of this equation in the integral form, the method is proposed for solving the obtained integral equation. Also, the convergence and stability analysis of the proposed method are studied in two main theorems. Furthermore, the optimum degree of convergence in the L₂ norm is obtained for this method. Furthermore, some numerical examples are presented in order to illustrate the performance of the presented method. Finally, an application of the model in control theory is introduced. Copyright © 2015 John Wiley & Sons, Ltd.     

Concentrated matrix Langevin distributions

This paper concerns the matrix Langevin distributions, exponential-type distributions defined on the two manifolds of our interest, the Stiefel manifold V_k,m and the manifold P_k,m−k of m×m orthogonal projection matrices idempotent of rank k which is equivalent to the Grassmann manifold G_k,m−k. Asymptotic theorems are derived when the concentration parameters of the distributions are large. We investigate the asymptotic behavior of distributions of some (matrix) statistics constructed based on the sample mean matrices in connection with testing hypotheses of the orientation parameters, and obtain asymptotic results in the estimation of large concentration parameters and in the classification of the matrix Langevin distributions.     

Linear Differential Equations with Variable Coefficients in Regular and Some Singular Cases

In this paper the asymptotic properties as t → + ∞ for a single linear differential equation of the form x⁽ⁿ⁾ + a₁ (t)x^(n?1)+…. + a_n(t)x = 0, where the coefficients aj (z) are supposed to be of the power order of growth, are considered. The results obtained in the previous publications of the author were related to the so called regular case when a complete set of roots {λ,(t)}, j = 1, 2, …, n of the characteristic polynomial yⁿ + a₁ (t)y^n?1 + … + a_n(t) possesses the property of asymptotic separability. One of the main restrictions of the regular case consists of the demand that the roots of the set {λ,(t)} have not to be equivalent in pairs for t → + ∞. In this paper we consider the some more general case when the set of characteristic roots possesses the property of asymptotic independence which includes the case when the roots may be equivdent in pairs. But some restrictions on the asymptotic behaviour of their differences λ_i(t)→ λ_j(t) are preserved. This case demands more complicated technique of investigation. For this purpose the so called asymptotic spaces were introduced. The theory of asymptotic spaces is used for formal solution of an operator equation of the form x = A(x) and has the analogous meaning as the classical theory of solving this equation in Band spaces. For the considered differential equation, the main asymptotic terms of a fundamental system of solution is given in a simple explicit form and the asymptotic fundamental system is represented in the form of asymptotic Emits for several iterate sequences.     

Steady compressible Navier–Stokes equations with large potential forces via a method of decomposition

We investigate the steady compressible Navier–Stokes equations near the equilibrium state v = 0, ρ = ρ₀ (v the velocity, ρ the density) corresponding to a large potential force. We introduce a method of decomposition for such equations: the velocity field v is split into a non-homogeneous incompressible part u (div (ρ₀u) = (0) and a compressible (irrotational) part ∇ϕ. In such a way, the original complicated mixed elliptic–hyperbolic system is split into several ‘standard’ equations: a Stokes-type system for u, a Poisson-type equation for ϕ and a transport equation for the perturbation of the density σ = ρ − ρ₀. For ρ₀ = const. (zero potential forces), the method coincides with the decomposition of Novotny and Padula [21]. To underline the advantages of the present approach, we give, as an example, a ‘simple’ proof of the existence of isothermal flows in bounded domains with no-slip boundary conditions. The approach is applicable, with some modifications, to more complicated geometries and to more complicated boundary conditions as we will show in forthcoming papers. © 1998 B.G. Teubner Stuttgart–John Wiley & Sons Ltd.     

On Some Second Order Differential Equations with Convergent Solutions

Via a special integral transformation, asymptotic integration results for ordinary differential equations are used to establish accurate asymptotic developments for radial solutions of the elliptic equation Δu + K(|x|)e ^u = 0, |x| > x ₀ > 0, in the bidimensional case.     

Stability of a delayed SIRS epidemic model with a nonlinear incidence rate

In this paper, an SIRS epidemic model with a nonlinear incidence rate and a time delay is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. By comparison arguments, it is proved that if the basic reproductive number R₀<1, the disease-free equilibrium is globally asymptotically stable. If R₀>1, by means of an iteration technique, sufficient conditions are derived for the global asymptotic stability of the endemic equilibrium.     

Ein gemischtes randwertproblem der kreiszylinderschale mit beliebigen ausschnitten

The problem of stress determination in the area of cut-outs in circular cylindrical shells at given loads is of great interest in industrial practice. This work deals with a mixed boundary value problem of a differential equation derived according to the theory of shallow shells. On part C_t1 of the boundary, the displacements are given, whereas the stresses are specified on the remaining part C_t2. Starting from the Betti-Maxwell principle and with the aid of the fundamental solutions for unit loads and unit displacements, integral representations can be derived for the displacement functions as well as the stress functions. The problem is then transformed into an equivalent system of Fredholm integral equations of the first kind with logarithmic kernels as the main part. As the integral equations together with the auxiliary conditions form a strongly elliptical system of pseudo-differential operators, the Galerkin method converges. Assuming that curves C_t1 and C_t1 do not have points of intersection and that the data are sufficiently regular, the required functions are approximated by cubic splines and, for simplicity's sake, the integral equation system is solved by approximation with a collocation method. In view of the complicated terms of the kernel functions, the kernels are split into a regular and a singular part, the regular part being in turn replaced by cubic splines. The remaining integrations are done numerically by means of Gaussian quadrature formulae. The applicability of the method is demonstrated with the example of a cylinder under internal pressure.     

Identification of Linear Error-Models with Projected Dynamical Systems

Linear error models are an integral part of several parameter identification methods for feedforward and feedback control systems and lead in connection with the L ₂-norm to a convex distance measure which has to be minimised for identification purposes. The parameters are hereby often subject to specific restrictions whose intersections span a convex solution set with non-differentiability points on its boundary. For solving these well conditioned problems on-line the paper formulates the solution of the bounded convex minimisation problem as a stable equilibrium set of a proper system of differential equations. The vector field of the corresponding system of differential equations is based on a projection of the negative gradient of the distance measure. A general drawback of this approach is the discontinuous right-hand side of the differential equation caused by the projection transformation. The consequence are difficulties for the verification of the existence, uniqueness and stability of a solution trajectory. Therefore the first subject of this paper is the derivation of an alternative formulation of the projected dynamical system, which exhibits, in contrast to the original formulation, a continuous right-hand side and is thus accessible to conventional analysis methods. For this purpose the multi-dimensional stop operator is used and the existence, uniqueness and stability properties of the solution trajectories are established. The second part of this paper deals with the numerical integration of the projected dynamical system which is used for an implementation of the identification method on a digital signal processor for example. To demonstrate the performance the application of this on-line identification method to the hysteretic filter synthesis with the modified Prandtl-Ishlinskii approach is presented in the last part of this paper.     

Proof of uniform high-frequency asymptotic expansion of Green's function for Helmholtz equation

A method is proposed for construction of the uniform high-frequency asymptotic expansion of the Green's function for the Helmholtz equation describing wave propagation in the ionosphere with electron density N(z, x). The problem reduces to an integral equation of 2nd kind whose Neumann series converges if the asymptotic parameter is k₀ 1 and |N_x| << 1, |N_xx| << 1 for – < x < .Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 125–129, 1993.     

A free boundary problem for three-dimensional harmonic vector fields

We study in this paper a free boundary value problem ( FB ), where a region G_o in R ³ is determined by the condition that there exists a vector field v_o in G_o which satisfies div v_o = e_o, curl v_o = g_o in G_o and v_o = E on the boundary ?G_o with a given scalar function e_o and given vector fields g_o and E. We give two equivalent formulations for this problem. Then we characterize the solutions by a non-linear integral equation. In order to solve the latter by a Newton method we linearize this equation. We investigate the ensuing linear integral equation. In case of axisymmetric configurations this is a singular integral equation whose index can be easily determined from the given data. We obtain a related equation, if we try to construct a field v in a region G which is on the boundary perpendicular to a given field B . Finally we use this method to investigate an astrophysical problem, which arises in the theory of pulsar magnetospheres.     

On the asymptotics with respect to a parameter of solutions of differential systems with coefficients in the class <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>

We consider a system of first-order ordinary linear differential equations with coefficients depending on an arbitrary parameter λ. For large λ, if the coefficients are smooth with respect to x, then there are known classical exponentially asymptotic (with respect to λ) formulas for the solution of the system. We generalize such formulas to the case in which the coefficients belong to the class L _q , q > 1. We use a new method for the reduction of problems to an integral system of special form.     

Coercivity of Combined Boundary Integral Equations in High‐Frequency Scattering

We prove that the standard second‐kind integral equation formulation of the exterior Dirichlet problem for the Helmholtz equation is coercive (i.e., sign‐definite) for all smooth convex domains when the wavenumber k is sufficiently large. (This integral equation involves the so‐called combined potential, or combined field, operator.) This coercivity result yields k‐explicit error estimates when the integral equation is solved using the Galerkin method, regardless of the particular approximation space used (and thus these error estimates apply to several hybrid numerical‐asymptotic methods developed recently). Coercivity also gives k‐explicit bounds on the number of GMRES iterations needed to achieve a prescribed accuracy when the integral equation is solved using the Galerkin method with standard piecewise‐polynomial subspaces. The coercivity result is obtained by using identities for the Helmholtz equation originally introduced by Morawetz in her work on the local energy decay of solutions to the wave equation. © 2015 Wiley Periodicals, Inc.