Singular perturbation approach to stability of a SIRS epidemic system

In this paper, a new method named multiple time scales is introduced to estimate the domain of attraction of the endemic equilibrium point of J. Mena-lorca and H. W. Hethcote’s SIRS model. Simultaneously, the recurrence formulae established by S. Balint, A. Balint and V. Negru are adopted to construct Lyapunov function for the reduced system. Furthermore, the stability for the non-hyperbolic equilibrium point of the reduced system is discussed. In the end, numerical simulations are carried out.     

A general approach to the analysis of controlled perturbation algorithms

Controlled Perturbation (CP, for short) is an approach to obtaining efficient and robust implementations of a large class of geometric algorithms using the computational speed of multiple precision floating point arithmetic (compared to exact arithmetic), while bypassing the precision problems by perturbation. It also allows algorithms to be written without consideration of degenerate cases. CP replaces the input objects by a set of randomly perturbed (moved, scaled, stretched, etc.) objects and protects the evaluation of geometric predicates by guards. The execution is aborted if a guard indicates that the evaluation of a predicate with floating point arithmetic may return an incorrect result. If the execution is aborted, the algorithm is rerun on a new perturbation and maybe with a higher precision of the floating point arithmetic. If the algorithm runs to completion, it returns the correct output for the perturbed input.The analysis of CP algorithms relates various parameters: the perturbation amount, the arithmetic precision, the range of input values, and the number of input objects. We present a general methodology for analyzing CP algorithms. It is powerful enough to analyze all geometric predicates that are formulated as signs of polynomials.     

A singular perturbation approach for a reaction diffusion system arising in nuclear reactor theory

The qualitative behavior of the solutions of a reaction-diffusion system arising in the theory of nuclear reactors is investigated by means of singular perturbation techniques, the small parameter ? representing the inverse of neutron velocity. At the lowest approximation for small ? the solutions exhibit oscillations about the unique strictly positive equilibrium, much as in the case of the associated lumped parameter system. A two-times representation for solutions of small amplitudes is also given.     

Continuous-time linear programming problems revisited: A perturbation approach

The class of continuous-time linear programming problems under the assumption that the constraints are satisfied almost everywhere in the time interval [0,?T]?is taken into account in this article. Under this assumption, its corresponding discretized problems cannot be formulated by equally dividing the time interval [0,?T]?as subintervals of [0,?T]?. In this article, we also introduce the perturbed continuous-time linear programming problems to prove the strong duality theorem when the constraints are assumed to be satisfied a.e. in [0,?T]?.     

Reliability-centred maintenance analysis for a complex military system

The preventive-maintenance (PM) programme is a very importantdocument in the life cycle of a piece of equipment or system.The traditional method used within China to determine PM programmeshas many disadvantages, and some tasks in old programmes haveproved to be unnecessary. Through analysing a complex militarysystem (X system) by means of the reliability-centred maintenance(RCM) analysis, it is found that RCM analysis is more successfulthan the traditional method in deciding the PM programme. Thenew PM programme does not include unnecessary PM tasks includedin the old programme. Thus the performance of the X system isenhanced, because the cost is reduced without compromising availability.It is believed that RCM analysis in military industry will makea valuable contribution to the maintenance of weapon systemsand equipment.     

A simple perturbation approach to Blasius equation

In this paper, we couple the iteration method with the perturbation method to solve the well-known Blasius equation. The obtained approximate analytic solutions are valid for the whole solution domain. Comparison with Howarth’s numerical solution reveals that the proposed method is of high accuracy, the first iteration step leads to 6.8% accuracy, and the second iteration step yields the 0.73% accuracy of initial slop.     

Sliding mode control of uncertain systems: a singular perturbation approach

In this paper we consider a nonlinear control system affectedby deterministic uncertainty and described by a system of ordinarydifferential equations. The uncertainty is modelled by a multivaluedmap whose t-measurable and x-Lipschitz selections representthe possible system dynamics of the uncertain system. We proposea dynamical feedback control design, based on the singular perturbationtheory, which allows all the possible system trajectories correspondingto the system dynamics to have the same prescribed behaviour.Specifically, given a manifold K of the state space, definedas the zeros of a smooth map, the proposed control steers andthen holds, during finite or infinite time intervals, any possiblesystem trajectory to any prescribed neighbourhood of K. A resultensuring the exact attainability of K is also provided. Someexamples illustrating the obtained results are presented.     

Asymptotic solutions of a magnetohydrodynamic system which describe smoothed discontinuities

Asymptotic solutions of a nonlinear magnetohydrodynamic system rapidly varying near moving surfaces are described. It is shown that the motion of jump surfaces is determined from a free boundary problem, while the main part of the asymptotics satisfies a system of equations on the moving surface. In the “nondegenerate” case, this system turns out to be linear, while, under the additional condition that the normal component of the magnetic field vanishes, it becomes nonlinear. In the latter case, the small magnetic field instantaneously increases to a value of order 1.     

A delay-time-based maintenance model of a multi-component system

There is a well established literature on delay-time modelsof regular inspection policies where inspections may or maynotbe perfect, and where the initial point u of a defect arisesas a homogeneous Poisson process. This paper extends the modellingin two ways. The first is to include the observed practice wherethe multi-component system is inspected not only on a plannedbasis, but also when a component fails. The second extensionis to use a nonhomogeneous Poisson process to describe defectarrivals in the system. An inspection–replacement modelbased upon these two extensions is then developed for a multi-componentsystem. The total expected cost per unit time is minimized withrespect to theinspection intervals and the system replacementtime. The likelihood function of the time of failures and thenumber of defects found at inspections is established, in orderto estimate model parameters based upon routinely collectedmaintenance data. As a special case of the general model, aninspection model—based upon a homogeneous Poisson processof defects arising—is also proposed, which has a relativelysimple structure. Both simulated and real-life data of failuresand defects identified at inspections are used to test the modelsand parameter-estimating procedure.     

Analytical analysis of free vibration of non-uniform and non-homogenous beams: Asymptotic perturbation approach

This paper applies the asymptotic perturbation approach (APA) to obtain a simple analytical expression for the free vibration analysis of non-uniform and non-homogenous beams with different boundary conditions. A linear governing equation of non-uniform and non-homogeneous beams is obtained based on the Euler–Bernoulli beam theory. The perturbative theory is employed to derive an asymptotic solution of the natural frequency of the beam. Finally, numerical solutions based on the analytical method are illustrated, where the effect of a variable width ratio on the natural frequency is analyzed. To verify the accuracy of the present method, two examples, piezoelectric laminated trapezoidal beam and axially functionally graded tapered beam, are presented. The results are compared with those results obtained from the finite element method (FEM) simulation and the published literature, respectively, and a good agreement is observed for lower-order beam frequencies.     

Fuzzy perturbation analysis,Part 2: Undirectional perturbation

In this paper, the stability of the solutions of fuzzy relation equations is discussed by the method of undirectional fuzzy perturbation. First, a general definition of ‘fuzzy perturbation’ is given. From this, the concept of fuzzy perturbation equation is introduced, and thus stability of the solutions is defined. Next, the degree of stability of the solutions, a kind of metric of the stability, is advanced. Three ordered quotient sets are induced by use of the degree of stability. These quotient sets have a well chained characteristic.     

Fuzzy perturbation analysis,part I: Directional perturbation

In this paper, a new idea, fuzzy perturbation, is advanced. The stability of the solutions of a fuzzy relation equation and the generalized solution of the unsolvable equation are defined by means of this idea. Thereupon, a part of the theory of fuzzy perturbation analysis, fuzzy directional perturbation, is formed. By use of the theory an approach to solving two open problems in the inverse problem of fuzzy multifactorial decision is made.     

Asymptotic analysis of a perturbation problem

An asymptotic expansion is constructed for the solution of the initial-value problem

when t is restricted to the interval [0,T/ε], where T is any given number. Our analysis is mathematically rigorous; that is, we show that the difference between the true solution u(t,x;ε) and the Nth partial sum of the asymptotic series is bounded by ε^N+1 multiplied by a constant depending on T but not on x and t.     

Stochastic perturbation approach to the wavelet‐based analysis

The wavelet‐based decomposition of random variables and fields is proposed here in the context of application of the stochastic second order perturbation technique. A general methodology is employed for the first two probabilistic moments of a linear algebraic equations system solution, which are obtained instead of a single solution projection in the deterministic case. The perturbation approach application allows determination of the closed formulas for a wavelet decomposition of random fields. Next, these formulas are tested by symbolic projection of some elementary random field. Copyright © 2004 John Wiley & Sons, Ltd.     

A two-level nesting smoothed meshfree method for structural dynamic analysis

This paper presents a novel two-level nesting smoothed meshfree method (NSMM), which significantly improves the computational efficiency of the meshfree Galerkin methods without losing their accuracy, thus facilitates the employment of meshfree methods in applications where background integration cells would be prohibitively expensive. In the NSMM, the system stiffness matrix is calculated using the general smoothing strain technique over the two-level nesting smoothing sub-domains where fewer integration points are used and the costly derivative computation of meshfree shape functions is avoided. The accuracy, efficiency and stability of the present method are assessed by virtue of several numerical examples for problems involving free and forced vibration analysis of the linear elastic continua and dynamic crack response of elastic solid. The results reveal that the NSMM stands out and achieves better performance compared to other existing approaches in the literature.     

Global existence from single-component estimates in a semilinear reaction-diffusion system

For a system of two reaction-diffusion equations coupled by power nonlinearities, we prove that an bound on a single component for suitable is enough to guarantee global existence. Also we provide a strong indication that our condition on is the best possible. Moreover, this continuation result is in contrast with the corresponding necessary and sufficient conditions for local existence obtained earlier by the authors.

     

A unified approach for solving nonlinear regular perturbation problems

This aritcle describes a simple alternative unified method of solving nonlinear regular perturbation problems. The procedure is based upon the manipulation of Taylor's approximation for the expansion of the nonlinear term in the perturbed equation.

An essential feature of this technique is the relative simplicity used and the associated unified computational procedure that is employed. As such it should be of interest to teachers of applied mathematics courses, particularly those courses which include perturbation methods.

One of the merits of this approach is that it leads on naturally to a scheme based on Taylor's expansions and, consequently, allows the regular perturbation method to be introduced into a course much earlier than is currently common.

The method is illustrated by implementing it to four perturbation problems, including two algebraic equations and two initial-value problems. The new approach is compared and contrasted with the traditional perturbation scheme in order to demonstrate its relative merit.     

Quasi-linear production systems: Optimisation of the transportation system

This paper is devoted to the quasi-linear production systems under the following additional hypothesis:

• 1.1. There exists a loading sequence which gives the input order of the parts of the system.
• 2.2. There exists, for each machine, a processing sequence which gives the order for manufacturing the parts.
• 3.3. The transportation system uses carts. A cart is a transportation unit able to carry one part for a machine to another. The parts are loaded on the carts before entering the system and unloaded when the manufacture of the part is finished.

We give an algorithm which leads to the minimal number of carts needed in order to reach the maximal production rate, the loading and processing sequences being known. We also characterize the optimal solution in the case of equal processing times.