Computing stability regions--Runge-Kutta methods for delay differential equations

We discuss the practical determination of stability regionswhen various fixed-stepsize Runge-Kutta (RK) methods, combinedwith continuous extensions, are applied to the linear delaydifferential equation (DDE) y'(t)= y(t)+µ(t–) (t) with fixed delay . It is significant that the delay is not limitedto an integer multiple of the stepsize, and that we considervarious continuous extensions. The stability loci obtained in practice indicate that the standardboundary-locus technique (BLT) can fail to map the RK DDE stabilityregion correctly. The aim of this paper is to present an alternativestability boundary algorithm that overcomes the difficultiesencountered using the BLT. This new algorithm can be used forboth explicit and implicit RK methods.     

A delay-time model for repairable machinery: maximum likelihood estimation of optimum inspection intervals

A delay-time model is presented for a complex repairable systemin which defects arise according to a nonhomogeneous Poissonprocess. The focus of this paper is not on the resulting probabilisticmodel itself, but on the estimation of the model parametersand their errors from records of failure times and number ofdefects found at inspections of a machine which has been operatedfor some time under some (usually suboptimal) inspection regime.The likelihood function of the observed failure and inspectiondata is derived, and a general approach for estimating the optimuminspection interval is discussed. An example relating to themaintenance of medical equipment is used to illustrate the techniquesdescribed in the paper.     

Modelling maintenance practice of production plant using the delay-time concept

In this paper we present a study carried out for a copper productsmanufacturing company, developing and applying the delay-timemodelling technique to model and thus optimize preventive maintenance(PM) of the plant. A key machine in the plant is used to illustratethe modelling process and management reaction. The parameter values of the process by which faults arise andof the delay-time distribution are estimated from maintenancerecord data of failures and faults found at PM, using the methodof maximum likelihood. A test of the model fit to data is carriedout. Based upon the estimated model parameters and the failuredelay time, an inspection model is proposed to describe therelationship between the total downtime and the PM interval.     

Some Applications of the Boundary-Locus Method and the Method of D-Partitions

We summarize aspects of the method of D-partitions and its analogue,the boundary-locus technique. We apply these techniques to adiscussion of the stability of the delay equation where y(t)=(t), t [–, 0]. Some comparisons with resultsvalid for other functional equations are introduced. Of interestare the following features: (i) the results corresponding toour results but applicable to delay-differential equations arenot found in the literature, (ii) some unconventional parameterspaces are introduced, and (iii) the possibility of extendingresults which hold for a basic test equation to more generalequations is established.     

Convergence and Stability Analysis for Modified Runge--Kutta Methods in the Numerical Treatment of Second-Kind Volterra Integral Equations

In this paper modified and conventional Runge—Kutta methodsfor second kind Volterra integral equations are discussed ina uniform way. The modification presented takes into accountthe residual of the previous step with the aim of improvingthe stability behaviour. A general convergence theorem is givenwhich establishes that the modified methods may lose one orderof accuracy. Furthermore, the stability behaviour of the methodsis analysed and explicit stability results are derived. It transpiresthat every A-stable Runge—Kutta method for ordinary differentialequations generates mixed methods which can be made A-stableby a suitable modification.