Christopher T. H. Baker

A perspective on the numerical treatment of Volterra equations

Abstract We discuss the properties and numerical treatment of various types of Volterra and Abel–Volterra integral and integro-differential equations.

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.

Retarded differential equations

Retarded differential equations (RDEs) are differential equations having retarded arguments. They arise in many realistic models of problems in science, engineering, and medicine, where there is a time lag or after-effect. Numerical techniques for such problems may be regarded as extensions of dense-output methods for ordinary differential equations (ODEs), but scalar RDEs are inherently infinite dimensional with a richer structure than their ODE counterparts. We give background material, develop a theoretical foundation for the basic numerics, and give some results not previously published.

Stability Regions in the Numerical Treatment of Volterra Integral Equations

“Step-by-step” methods (in which there is the possibility of unstable error propagation) occur in the approximate solution of first and second kind integral equations of Volterra type. We discuss a definition of stability, and its application to determine regions of stability for such methods applied to simple test equations. This gives new insight into the suitability of numerical methods in practice.

The tracking of derivative discontinuities in systems of delay-differential equations

Abstract Evolutionary type delay-differential equations occur widely in dynamical processes in many fields of biology, engineering and the physical sciences. One of their characteristics is that their solutions frequently contain derivative discontinuities. In this paper we propose a model for the tracking of such discontinuities and discuss its influence on the design of numerical software. Our discussion is illustrated by DELSOL, a general purpose delay-differential equation solver developed at the University of Manchester.

Stability properties of a scheme for the approximate solution of a delay-integro-differential equation

Abstract We discuss some stability properties of a numerical scheme applied to a Volterra integro-differential equation with a finite memory (in which the solution is determined by an initial function). The numerical scheme is based on a strongly-stable linear multistep formula and a consistent quadrature rule, applied with a fixed step.

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.

DELSOL: a numerical code for the solution of systems of delay-differential equations

Abstract Delay-differential equations arise widely in many fields of science and engineering. In this paper we present an overview of a general purpose code—DELSOL—written to solve systems of such equations. Special attention is given to the programs organisation and design. A number of numerical examples are also presented alongside a number of technical observations pertinent to the design of delay-differential software.

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.

Perturbation Theory for Discrete Volterra Equations

Fixed point theory is used to investigate nonlinear discrete Volterra equations that are perturbed versions of linear equations. Sufficient conditions are established (i) to ensure that stability (in a sense that is defined) of the solutions of the linear equation implies a corresponding stability of the zero solution of the nonlinear equation and (ii) to ensure the existence of asymptotically periodic solutions.