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CONTRIBUTIONS TO NUMERICAL METHODS FOR*DELAY - DIFFERENTIAL EQUATIONS

IP.com Disclosure Number: IPCOM000128265D
Original Publication Date: 1976-Dec-31
Included in the Prior Art Database: 2005-Sep-15
Document File: 6 page(s) / 22K

Publishing Venue

Software Patent Institute

Related People

Jesper Oppelstrup: AUTHOR [+3]

Abstract

The mathematical model of a dynamical system can often be formulated in terms of ordinary differential equations. In such a system, the rate of change depends 'on the present state y(t), [Equation ommitted] In a number of applications, however, such as the heart-lung system in man, and ecological systems with hereditary effects, the changes are influenced by the whole pre-history of the system. Exam~les often encountered are of the kind [Equation ommitted] retarded delay-differential equations, where the delay, or lag, T usually arises from a transport time, as in'the heart-lung.model.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

CONTRIBUTIONS TO NUMERICAL METHODS FOR*DELAY - DIFFERENTIAL EQUATIONS

by

Jesper Oppelstrup

TRITA-NA,-7628

I (summary of the author's doctoral thesis) The thesis consists of this summary and the four papers

[A] One-step Methods with Interpolation for Volterra Functional Differential Equations, Tech.Rep. TRITA-NA-7623 [B] Delay 2 - A Fourth Order Method for Delay-Differential Equations, Tech.Rep. TRITA-NA-7619 [C] The RKFF-!34 Method for Delay-Differential Equations, Tech.Rep. TRITA-DIA-7625, also to appear in Springer Lecture Notes on Mathematics [D] Stability Analysis for a Class of Non-linear Functional Differential Equations and Their Implicit Euler Discretizations, Tech-Rep. TRITA-l"TA-7626, submitted for publication in BIT.,

INTRODUCTION

The mathematical model of a dynamical system can often be formulated in terms of ordinary differential equations. In such a system, the rate of change depends 'on the present state y(t),

(Equation Omitted)

In a number of applications, however, such as the heart-lung system in man, and ecological systems with hereditary effects, the changes are influenced by the whole pre-history of the system. Exam~les often encountered are of the kind

(Equation Omitted)

retarded delay-differential equations, where the delay, or lag, T usually arises from a transport time, as in'the heart-lung.model. Equations like

(Equation Omitted)

(Equation Omitted)

were studied by Volterra as models for interacting predator-prey populations and are known as Volterra integro-differential equations. To include both (2) and (3), we make the DEFINITIO11: A differential system of the kind

(Equation Omitted)

where the operator F depends only on the values of

The Royal Institute of Technology Page 1 Dec 31, 1976

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CONTRIBUTIONS TO NUMERICAL METHODS FOR*DELAY - DIFFERENTIAL EQUATIONS

(Equation Omitted)

is a Volterra functional differential system.

Numerical methods for initial value problems for (4).havebeen given increasing attention in recent years. The paper by dryer in [111 is a survey of the field. Until very recently, however, there was no general purpose software available. This thesis describes a program for initia.1 value problems for (2),. together with some theoretical background for the method used, and some stability results for (4) and the corresponding difference equations. 'The program is intended for use with the SIMNON system 1 21 for computer simulation of general dynamic systems governed,by differential as well as difference equations. An earlier version, briefly described in, ['01 is included in SIMNON, but the present algorithm is a major improvement.

SUMMARY OF THE PAPERS [A] -JD1-

[A) is an analysis of the one-step methodz with interpolation for (4) with application to (2). These methods are pieced together from a onestep method I> for the ordinary differential system (1), and an interpolation procedure Q is used to calculate approxim...