Christopher A. H. Paul

Issues in the numerical solution of evolutionary delay differential equations

Delay differential equations are of sufficient importance in modelling real-life phenomena to merit the attention of numerical analysts. In this paper, we discuss key features of delay differential equations (DDEs) and consider the main issues to be addressed when constructing robust numerical codes for their solution. We provide an introduction to the existing literature and numerical codes, and in particular we indicate the approaches adopted by the authors. We also indicate some of the unresolved issues in the numerical solution of DDEs.

Differential-difference equations in economics: On the numerical solution of vintage capital growth models

In this papel, we examine techniques for the analytical and numerical solution of statedependent differential-difference equations. Such equations occur in the continuous time modelling of vintage capital growth models, which form a particularly important class of models in modern economic growth theory. The theoretical treatment of non-statedependent differential-difference equations in economics has already been discussed by Benhabib and Rustichini (1991). In general, though, the state-dependence of a model prevents its analytical solution in all but the simplest of cases. We review a numerical method for solving state-dependent models, using sorne simple examples to illustrate our discussion. In addition, we analyse the Solow vintage capital growth model. We conclude by mentioning a crucial unresolved issue related to this topic.

Developing a delay differential equation solver

Abstract We discuss briefly various phenomena to be tested when designing a robust code for delay differential equations: choice of interpolant; tracking of discontinuities; vanishing delays; and problems arising from floating point arithmetic.

Mathematical Modelling of the Interleukin-2 T-cell System: A Comparative Study of Approaches Based on Ordinary and Delay Differential Equations

Cell proliferation and differentiation phenomena are key issues in immunology, tumour growth and cell biology. We study the kinetics of cell growth in the immune system using mathematical models formulated in terms of ordinary and delay differential equations. We study how the suitability of the mathematical models depends on the nature of the cell growth data and the types of differential equations by minimizing an objective function to give a best-fit parameterized solution. We show that mathematical models that incorporate a time-lag in the cell division phase are more consistent with certain reported data. They also allow various cell proliferation characteristics to be estimated directly, such as the average cell-doubling time and the rate of commitment of cells to cell division. Specifically, we study the interleukin-2-dependent cell division of phytohemagglutinin stimulated T-cells - the model of which can be considered to be a general model of cell growth. We also review the numerical techniques available for solving delay differential equations and calculating the least-squares best-fit parameterized solution.

Differential algebraic equations with after-effect

In this paper, we are concerned with the solution of delay differential algebraic equations. These are differential algebraic equations with after-effect, or constrained delay differential equations. The general semi-explicit form of the problem consists of a set of delay differential equations combined with a set of constraints that may involve retarded arguments. Even simply stated problems of this type can give rise to difficult analytical and numerical problems. The more tractable examples can be shown to be equivalent to systems of delay or neutral delay differential equations. Our purpose is to highlight some of the complexities and obstacles that can arise when solving these problems, and to indicate problems that require further research.

A Global Convergence Theorem for a Class of Parallel Continuous Explicit Runge--Kutta Methods and Vanishing Lag Delay Differential Equations

Iterated continuous extensions (ICEs) are continuous explicit Runge--Kutta methods developed for the numerical solution of evolutionary problems in ordinary and delay differential equations (DDEs). ICEs have a particular role in the explicit solution of DDEs with vanishing lags. They may be regarded as parallel continuous explicit Runge--Kutta (PCERK) methods, as they allow one to take advantage of parallel architectures. ICEs can also be related to a collocation method. The purpose of this paper is to provide a theorem giving the global order of convergence for variable-step implementations of ICEs applied to state-dependent DDEs with and without vanishing lags. Implications of the theory for the implementation of this class of methods are discussed and demonstrated. The results establish that our approach allows the construction of PCERK methods whose order of convergence is restricted only by the continuity of the solution.

Parallel continuous Runge-Kutta methods and vanishing lag delay differential equations

We present an explicit Runge-Kutta scheme devised for the numerical solution ofdelay differential equations (DDEs) where a delayed argument lies in the current Runge-Kutta interval. This can occur when the lag is small relative to the stepsize, and the more obvious extensions of the explicit Runge-Kutta method produce implicit equations. It transpires that the scheme is suitable forparallel implementation for solving both ODEs and more general DDEs. We associate our method with a Runge-Kutta tableau, from which the order of the method can be determined. Stability will affect the usefulness of the scheme and we derive the stability equations of the scheme when applied to the constant-coefficient test DDEu′(t)=λu(t) +μu(t −τ), where the lagτ and the Runge-Kutta stepsizeHn ≡H are both constant. (The caseμ=0 is treated separately.) In the case thatμ ≠ 0, we consider the two distinct possibilities: (i)τ ≥H and (ii)τ

Designing efficient software for solving delay differential equations

In this paper, the efficient implementation of numerical software for solving delay differential equations is addressed. Several strategies that have been developed over the past 25 years for improving the efficiency of delay differential equation solvers are described. Of particular interest is a new method of automatically constructing the network dependency graph used in tracking derivative discontinuities.